Contents lists available at SciVerse ScienceDirect

Annals of Pure and Applied Logic

www.elsevier.com/locate/apal

Bare canonicity of representable cylindric and polyadic Cr0SSMark

algebras ^

Jannis Bulian1, Ian Hodkinson *

Department of Computing, Imperial College London, London SW7 2AZ, UK ARTICLE INFO ABSTRACT

Article history:

Received 13 February 2012

Received in revised form 21 March 2013

Accepted 8 April 2013

Available online 15 May 2013

primary 03G15

secondary 03C05, 06E15, 06E25

We show that for finite n ^ 3, every first-order axiomatisation of the varieties of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of non-canonical formulas. We also show that the class of structures for each of these varieties is non-elementary. The proofs employ algebras derived from random graphs.

© 2013 Elsevier B.V. All rights reserved.

Keywords: Canonical variety Canonical axiomatisation Algebras of relations Cylindric algebras Diagonal-free algebras Random graphs

1. Introduction

The notion of the canonical extension of a boolean algebra with operators (or 'BAO') was introduced byJonsson and Tarski in a classical paper [15], generalising a construction of Stone [22]. It is an algebra whose domain is the power set of the set of ultrafilters of the original BAO, and its operations are induced from those of the BAO in a natural way. Canonical extensions are nowadays a key tool in algebraic logic, with a multitude of uses and generalisations.

A class of BAOs is said to be canonical if it is closed under taking canonical extensions. In this paper we are concerned with the classes of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras, for finite n > 3. These four classes are varieties. They are non-finitely axiomatisable, and many further 'negative' results on axiomatisations are known (e.g., [1,21]). However, the classes are canonical. Now [15] already established that positive equations are preserved by canonical extensions, and more generally, Sahlqvist equations are also preserved (see, e.g., [2]). This may suggest that the four classes might be axiomatisable by positive or Sahlqvist equations.

It turned out that the representable cylindric algebras are not Sahlqvist axiomatisable [23, footnote 1]. In this paper, we extend this result to a wider class of axioms and to all four classes. A first-order sentence is said to be canonical if the class of its BAO models is canonical. Although some syntactic classes of canonical sentences (such as Sahlqvist equations)

* Corresponding author.

E-mail addresses: jannis.bulian@cl.cam.ac.uk (J. Bulian), i.hodkinson@imperial.ac.uk (I. Hodkinson). URLs: http://www.cl.cam.ac.uk/~jb782/ (J. Bulian), http://www.doc.ic.ac.uk/~imh/ (I. Hodkinson). 1 Current address: Computer Laboratory, University of Cambridge, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK.

0168-0072/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apal.2013.04.002

are known, canonicity is a semantic property that cannot be easily defined syntactically. For example, there is no algorithm to decide whether an equation is canonical [16, Theorem 9.6.1]. The goal of this paper is to show that there is no canonical axiomatisation of any of the four classes listed above. In fact, we will show that any first-order axiomatisation of any of them contains infinitely many non-canonical sentences. We say that a canonical class of BAOs with this property is barely canonical. Although the class is canonical, its canonicity emerges only 'in the limit' and does not reside in any finite number of axioms for it, however they are phrased.

There are a few related results in the literature. The class of representable relation algebras, proved to be canonical by Monk (reported in [18]), was shown in [13] to be barely canonical. Our proof in the current paper is similar but somewhat simpler: the use of finite combinatorics (finite Ramsey theorem, etc.) in [13] is replaced here by the use of first-order compactness. Bare canonicity of the 'McKinsey-Lemmon' modal logic was shown in [8].

We sketch the rough outline of the proof. Our aim is to convey the idea quickly, and the description will not be completely accurate in detail. Our construction uses polyadic equality-type algebras built from graphs. They are polyadic expansions of cylindric-type algebras constructed from graphs in [12], where it was shown (roughly) that such an algebra is representable if and only if its base graph has infinite chromatic number. (This was used in [12] to prove that the class of structures for the variety of representable n-dimensional cylindric algebras (finite n > 3) is non-elementary, a result generalised to diagonal-free, polyadic, and polyadic equality algebras in Theorem 9.7 below.) Here, we will cast this work in a wider setting by defining an elementary class K of three-sorted structures comprising a polyadic equality-type algebra A, a graph G, and a boolean algebra B of subsets of G, with various relations tying them together quite closely. We will show that representability of A is equivalent to G having infinite chromatic number in the sense of B. Both these properties can be defined by first-order theories, which therefore have the same models modulo the theory defining K. It follows by compactness that if the class of representable algebras had a first-order axiomatisation using only canonical sentences, there would be a function f: a ^ a such that whenever an algebra A has chromatic number at least f (k) (in the sense of some three-sorted structure), its canonical extension has chromatic number at least k. We then borrow from [13] an inverse system of finite (random) graphs of chromatic number m whose inverse limit has chromatic number k, for any chosen 2 < k < m < a. Using some results of Goldblatt [5] connecting canonical extensions with inverse limits, this yields an algebra of chromatic number m whose canonical extension has chromatic number k. Since k, m are arbitrary, no function f as above can exist. A slight extension of the argument, using a little more compactness, shows that any first-order axiomatisation of the representable algebras has infinitely many non-canonical sentences.

Layout of paper. In Section 2 we recall some basic notions of algebras of relations, representability, duality and canonicity. We define polyadic equality-type algebras over graphs in Section 3, and abstract generalisations of them in Section 4, where we also ascertain some of their elementary properties. This is continued in Section 5, where we study their ultrafilters. In Section 6 we introduce approximations to representations by means of systems of ultrafilters called 'ultrafilter networks', and lower-dimensional approximations of them called 'patch systems'. This will allow us to prove in Section 7 that (roughly) an abstract algebra is representable if and only if its associated graph has infinite chromatic number. In Section 8 we introduce some needed material on direct and inverse systems. Assuming an axiomatisation with only finitely many non-canonical formulas, we use direct and inverse systems in Section 9 to build an algebra that satisfies an arbitrary number of axioms, while its canonical extension satisfies only a bounded number, and thus obtain a contradiction. Section 10 lists some open problems.

Notation. We use the following notational conventions. We usually identify (notationally) a structure, algebra, or graph with its domain. For signatures L c L' and an L'-structure M, we write M \ L for the L-reduct of M.

Throughout the paper, the dimension n is a fixed finite positive integer and n is at least 3. It will often be implicit that cylindric algebras, etc. are n-dimensional and that i, j, k, m, etc., denote indices < n. We identify a non-negative integer m with the set {0,1,...,m — 1}. If V is a set, we write [V]m for the set of subsets of size m of V. We write a for the first infinite ordinal number. p(S) denotes the power set of a set S.

For a function f : X ^ Y we write dom f for its domain, im f for its image, and f [X'] for {f (x') | x' e X'} when X' c X. We use similar notation for m-ary functions, for m < a — e.g., in Definition 2.7. For functions f, g, we write f o g for their composition: f o g(x) = f (g(x)). We omit brackets in function applications when we believe it improves readability. By a U, where a is an ordinal, we denote the set of functions from a to U, so an a-ary relation on U is a subset of a U. To keep the syntax similar to the finite case, we write xi for x(i) if x e a U and i < a. Similarly, we write pi for p(i) where p e ]"[i<a Ui and i < a.

2. Algebras of relations

In this paper, we will consider four types of algebra: cylindric-type algebras, diagonal-free cylindric-type algebras, polyadic-type algebras, and polyadic equality-type algebras, all of dimension n. They differ in their signatures and notion of representation. Here, we define them formally and recall some aspects of duality and canonicity for them.

2.1. Signatures and algebras Definition 2.1. We let

1. Lba = {+, —, 0,1} denote the signature of boolean algebras,

2. LcAn = Lba U {ci, djj | i, j < n} denote the signature of n-dimensional cylindric algebras,

3. LDfn = Lba U{cj | i < n}, denote the signature of n-dimensional diagonal-free cylindric algebras,

4. LPAn = Lba U {ci, sa | i, j < n, a : n ^ n} denote the signature of n-dimensional polyadic algebras,

5. LPEAn = Lba U {ci, djj, sa | i, j < n, a : n ^ n} denote the signature of n-dimensional polyadic equality algebras.

Here, the ci ('cylindrifications') and sa ('substitutions') are unary function symbols and the dij ('diagonals') are constants. By a cylindric-type algebra, we mean simply an algebra of signature LCAn. Diagonal-free cylindric-type algebras, polyadic-type algebras, and polyadic equality-type algebras are defined analogously for the other signatures.

Our concern in this paper is with representable algebras of these four kinds, but we briefly note that abstract algebras have been defined as well: namely, cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras. They are algebras of the above types that satisfy in each case a finite set of equations that can be found in [10,11]. In particular, cylindric algebras are defined in [10, Definition 1.1.1]. We will not use the formal definition so we do not recall it here, but the proofs of some later lemmas will be easier for readers familiar with basic computations in cylindric algebras. The material in [10, §1] is easily enough for what we need. Readers not so familiar can easily verify our claims directly in the specific algebras we are working with.

2.2. Representations

Natural examples of each kind of algebra arise from algebras of n-ary relations on a set. Definition 2.2. A polyadic equality set algebra is a polyadic equality-type algebra of the form

(p(V), 0, V, U, V \-,cU, Dj, SU | i, j < n, a :n ^ n), where U is a non-empty set, V = nU, and

1. DU ={x e V | Xi = Xj} for i, j < n,

2. CUX = {x e V |3y e X Vj < n (j = i ^ yj = xj)} for i < n and X c V,

3. SUX = {x e V | x o a e X}, for a : n ^ n and X c V.

A polyadic set algebra (cylindric set algebra) is the reduct of a polyadic equality set algebra to the signature LPAn (respectively, LCAn). Since LDfn has no operations connecting two different dimensions, a diagonal-free cylindric set algebra is defined rather differently, as an LDfn -algebra of the form

(p(V), 0, V, U, \, CV I i < n), where U 0,..., Un-1 =0, V = n<n Ui, and C^X = {x e V |3y e X Vj e n \ {i} (yj = xj)} for i < n and X c V.

The only difference between the polyadic and diagonal-free cylindrifications is what the 'unit' V is. The polyadic operator CU is the same as the diagonal-free operator C( U). In the polyadic case we can simplify the notation to CU, but in the diagonal-free case we have no choice but to write CiV .

Definition 2.3. An LPEAn -algebra is said to be representable if it is isomorphic to a subalgebra of a product of polyadic equality set algebras. The isomorphism is then called a representation. The class of all representable polyadic equality algebras of dimension n is called RPEAn.

Exactly analogous definitions are made for LDfn, LCAn, and LPAn, using the appropriate set algebras in each case. The classes of representable algebras for these are, respectively, RDfn, RCAn, and RPAn.

It is known that RPEAn, RPAn, RCAn, and RDfn are varieties (elementary classes defined by equations): see, e.g., [11, 3.1.108, 5.1.43]. For n > 3, they are not finitely axiomatisable [19,14], and indeed we will see this later in Corollary 9.5.

2.3. Atom structures

We now recall a little duality theory, leading to canonicity, the topic of the paper. For more details, see, e.g., [15,2] and [10, §2.7].

Definition 2.4. Let L 2 LBA be a functional signature (i.e., one with only function symbols and constants). We write L+ for the relational signature consisting of a (k + 1)-ary relation symbol R f for each k-ary function symbol f e L \ LBA. By an (L)-atom structure, we will simply mean an L+-structure. We will sometimes refer to the elements of an atom structure as atoms.

Given an L-atom structure S = (S, Rf | Rf e L+), we write S+ for its complex algebra: S + = (p(S), f | f e L), where each f e Lba is interpreted in the natural way as a boolean operation on p(S), and f (X1,..., Xk) = {s e S | S = R f (x1,...,xk, s) for some x1 e X1,...,xk e Xk}, for each k-ary f e L \ LBA and X1,..., Xk c S. We sometimes identify each s e S with the atom {s} of S+.

For the particular signature LPEAn, we will be defining atom structures in which the RCi are equivalence relations and the R— are functions. So we adopt a slightly different definition of atom structure that is a little easier to specify in practice.

Definition 2.5. A polyadic equality atom structure is a structure

S = (S, Dij, =i, —a | i, j < n, a :n ^ n),

where Dij c S, =i is an equivalence relation on S, and —a : S ^ S is a function. We regard S as a standard LPEAn-atom structure in the sense of Definition 2.4 by interpreting Rdij as Djj, Rci as =i, and letting Rsa(s, t) iff ta = s.

2.4. Canonicity

One source of atom structures is from boolean algebra with operators (BAOs). These originated in [15], where they were called 'normal BAOs', and they are now familiar: see, e.g., [2]. Let L be a functional signature containing LBA.

Definition 2.6. An L-BAO is an L-structure whose LBA-reduct is a boolean algebra and in which each f e L \ LBA defines a function that is normal (its value is zero whenever any argument is zero) and additive in each argument.

For example, if S is an L-atom structure then S + is an L-BAO (note that constants are vacuously normal and additive). Any algebra in RDfn, RCAn, RPAn, and RPEAn is easily checked to be a BAO for its signature.

Definition 2.7. Let B be an L-BAO. We define the ultrafilter structure B+ to be the L-atom structure which has the set of ultrafilters of (the boolean reduct of) B as domain and, for any k-ary function symbol f e L \ LBA and ¡i0,..., [ik—1, v e B+,

B+= Rf(y,o,...,y,k—i,v) ^^ f[^o,...,^k—i]C v.

The canonical extension of B, denoted by Ba, is the L-BAO (B+)+. A class K of L-BAOs is said to be canonical if B e K implies Ba e K. A first-order L-sentence 6 is said to be canonical if B |= 6 implies Ba = 6 for every L-BAO B.

Canonical extensions were introduced in [15], where it was shown that there is a canonical embedding of B into Ba given by b ^ {v e B+ | b e v}, so justifying the use of 'extension'. Canonical extensions of cylindric algebras are studied in [10, §2.7]. Canonical varieties in general have been intensely studied, for example by Goldblatt [7], and it is not hard to derive the following well-known result. The proof we give follows [7]: [7, Theorem 4.6] proves by a stronger version of the same method that RCAa and ICrsa are canonical varieties for every ordinal a. Canonicity of RCAn is proved in a different way in [10, p. 459].

Proposition 2.8. RDfn, RCAn, RPAn, and RPEAn are canonical varieties.

Proof. Let KDf be the class of all (LDfn)+ -structures of the form (f[ i<nUi, Rci | i < n), where U 0,...,Un—1 =0 and Rci ((u0,..., un—1), (v0,..., vn—1)) iff uj = vj for each j e n \ {i}. Let KPEA be the class of (LPEAn)+-structures of the form (nU, R ci, Rdij, Rsa l i, j < n, a :n ^ n), where U = 0, RCi is defined in the same way as above, Rdi,((u0,..., un—1)) iff ui = uj, and Rsa((u0,..., un—1), (v0,..., vn—1)) iff ui = va(i) for each i < n. Let KPA, KCA be the class of reducts of structures in KPEA to the signatures (LPAn)+ and (LCAn)+, respectively.

We now assume familiarity with the notation of [7]. By Theorem 4.5 of [7], if K is a class of atom structures with PuK c HSUdK, then SCmSUdK is a canonical variety. By Theorem 2.2(2, 5) of [7], PCm = CmUd and SUd = UdS, so SCmSUdK = SPCmSK. Now let K e {KDf, KPEA, KPA, KCA}. Then K is closed under ultraproducts, and under inner substructures (since no structure of the above forms has any proper inner substructures), so PuK c K = SK. Consequently, SPCmK — the closure of {S + | S e K} under subalgebras of products — is a canonical variety. But it follows from the definitions that SPCmKPEA = RPEAn, and similar results hold for the other three classes. □

Notwithstanding this proposition, we will show that any first-order axiomatisation of any of these four varieties requires infinitely many non-canonical sentences.

3. Algebras from graphs

Here we will describe how to obtain polyadic equality-type algebras from graphs. In this paper, graphs are undirected and loop-free. Recall that a set of nodes of a graph is independent if there is no edge between any two nodes in the set.

3.1. Atom structures from graphs

The first step is given by the following definitions (adapted from [12, Definition 3.5]), which construct a polyadic equality atom structure from a graph.

Notation. We let Eq(n) denote the set of equivalence relations on n. If — e Eq(n) and i < n, we will write —i for the restriction of ~ to n \ {i}.

Definition 3.1. Let r = (V, E) be a graph. We let r x n denote the graph

(V x n,{((x, i), (y, j)) e V x n 1 E(x, y) v i = j}) consisting of n copies of r with all possible additional edges between copies.

Each graph r will give rise to an atom structure whose 'atoms' will essentially be the pairs (K, —), where ~ e Eq(n) and K: [n/-]n—1 ^ r x n is a map such that if ~ is equality on n then the image of K is not an independent set in r x n. Of course, K = 0 if [n/-]n—1 =0. For notational simplicity, we will actually represent K by a possibly partial map K : n ^ r x n in the following way. For each j < n let J = {i/~: i e n \ {j}}. If | J| = n — 1 then K(j) = K(J). Otherwise, K(j) is undefined. This leads us to the following formal definition.

Definition 3.2. Fix a graph r. Let S (r) be the set of all pairs (K, —), where K: n ^ r x n is a partial map and ~ an equivalence relation on n that satisfies the following:

1. If |n/~| = n, then dom(K) = n and im(K) is not independent.

2. If |n/~| = n — 1, so that there is a unique —-class {i, j} of size 2 with i < j < n, say, then dom(K) = {i, j} and K(i) =

K (j).

3. Otherwise, i.e. if |n/~| < n — 1, K is nowhere defined.

For (K, —), (K—') e S(r) and i, j < n, we will write K(i) = K'(j) if either K(i) and K'(j) are both undefined, or they are both defined and are equal. According to this, if i — j then K(i) = K(j).

Definition 3.3. Let i < n. A relation ~ e Eq(n) is said to be i-distinguishing if —(j — k) for all distinct j, k e n \ {i}. It is equivalent to saying that —i is equality on n \ {i}. A pair (K, —) e S(r) is said to be i-distinguishing if ~ is i-distinguishing.

Remark. If (K, —) e S(r), then K is defined on i < n if and only if ~ is i-distinguishing.

Definition 3.4. Let r be a graph. The polyadic equality atom structure

At(r) = (S(r), Dij, =i, —a I i, j < n, a : n ^ n) is defined as follows:

1. Dij = {(K, ~) e S(r) | i - j} c S(r), for i, j < n.

2. =i is the equivalence relation on S(r) given by: (K, ~) =i (Kif and only if K(i) = K'(i) and ~i = ~i for i < n.

3. For each a :n ^ n, the map —a : S(r) ^ S(r) is given by: (K, -)a = (Ka, -a), where

• ~a e Eq(n) is defined by i —a j iff a(i) — a(j) (for i, j < n),

• Ka(i) (for i < n) is defined iff —a is i-distinguishing, and in that case, Ka(i) = K(j), where j < n is the unique element satisfying j e a[n \ {i}].

We leave it to the reader to check that (Ka, —a) is well defined and in S(r), and that Ka is determined by K and a even though we cannot in general recover —a from them. Note that if a is one-one then Ka = K o a.

Definition 3.5. We write A(r) for the n-dimensional polyadic equality-type algebra At(r)+. Explicitly,

A(r) = (p(S(r)), U, \, 0, S(r), dij, ci, sa I i, j < n, a :n ^ n), where dij = Dij as above, and for X c S(r),

1. ciX = {(K, ~) e S(r) | 3(K', ) e X((K', -') = (K, -))},

2. saX = {(K, ~) e S(r) | (K, ~)a e X}.

An algebra of the form A(r) will be called an algebra from a graph.

A(r) is the expansion to the signature of polyadic equality algebras of a cylindric-type algebra, also written A(r), that was defined in [12]. So some results proved for it also apply to the A(r) defined above. Here is one (another is in Proposition 4.10 below):

Proposition 3.6. Let r be a graph. Then the cylindric reduct of A(r) is an n-dimensional cylindric algebra.

Proof. This is proved in [17, Claim 3.4 and displayed line (4)]. □

We will need to pick out certain elements of A(r), so that all the elements beneath are i-distinguishing and thus have K(i) defined on them.

Definition 3.7. Let A be a cylindric-type or polyadic equality-type algebra. For i < n, define

Fi = Y[{—djk l j < k < n, j, k = i}.

We generally take Fi to be an element of the algebra under consideration (here, A).

Remark. Clearly, for an algebra from a graph A(r), Fi is just the sum of all the i-distinguishing atoms. For (K, ~) e S(r), K(i) is defined iff (K, ~) e Fi.

4. Algebra-graph systems

Proposition 3.6 establishes a relation between graphs and cylindric algebras. However, we need to study this relationship in a more abstract setting.

4.1. Definitions

Definition 4.1. We denote by LAGS the signature with three sorts (A, G, B) and the following symbols:

1. A-sorted copies of the function symbols 0, 1, +, —, dij, ci, sa of LPEAn for each i, j < n and a : n ^ n (with the obvious arities that make A into a polyadic equality-type algebra);

2. B-sorted copies of the function symbols 0, 1, +, — of LBA;

3. a binary (graph edge) relation symbol E on G;

4. a binary relation symbol H on G;

5. a binary relation symbol e between the elements of G and B;

6. a unary function symbol Ri: A ^ B for each i < n;

7. a unary function symbol Si: B ^ A for each i < n.

We extend the definition of Fi (Definition 3.7) to LAGS-structures in the obvious way. Sometimes we will regard Fi as an Lags-term.

Definition 4.2. For a graph r, let M(F) be the 3-sorted LAGS-structure

(A(r),r x n,p(r x n))

with operations defined as follows:

• The A-sorted and B-sorted symbols are interpreted on A(r), p(F x n) in the natural way.

• E is interpreted as the edge relation on r x n.

• We have H(x, y) if and only if there is I < n such that x, y e r x {£}.

• The relation e denotes membership of elements of r x n in the sets that are elements of p(F x n).

• Finally, we have

Ri (a) = {K(i) | (K, e Fi ■ a} for a e A(r), Si (B) = {(K, e Fi | K (i) e B} for B e p(r x n).

A structure of the form M(r) will be called a structure from a graph.

We now define a theory that helps us talk about the subclass of all the LAGS-structures similar to the ones derived from graphs.

Definition 4.3. A (first-order) LAGS-formula is said to be A-universal if it is of the form (Wxi,...,xm: A)y,

where y is an LAGS-formula with no quantifiers over the A-sort. We define U to be the set of A-universal sentences that are true in all LAGS-structures M(r) for graphs r. An LAGS-structure M that is a model of U is called an algebra-graph system.

This definition ensures that a good number of first-order statements that hold for algebras and structures from graphs, also hold in any algebra-graph system. It will allow us to prove many statements for algebra-graph systems, by just showing they are expressible by A-universal sentences and hold in M(r) for every graph r.

4.2. Basic properties of algebra-graph systems

Lemma 4.4. In any algebra-graph system M = (A, G, B), the cylindric reduct of A is a cylindric algebra, and B is a boolean algebra isomorphic to a subalgebra of p(G).

Proof. We know from Proposition 3.6 that an arbitrary algebra from a graph will satisfy all the axioms for cylindric algebras. These axioms are equations and can be recast in the obvious way as A-universal LAGS-sentences. Since these sentences hold in every M(r), they hold in M. A similar argument shows that B is a boolean algebra. Since the A-universal sentences

VB, B': B(Vp : G(p e B ^ p e B') ^ B = B'), Vp : G(p e 1 A—(p e 0)), VB, B': B Vp : G(p e B + B' ^ p e B v p e B'), VB : B Vp : G(p e-B ^ — (p e B)) are in U and so are true in M, the function B p e G | M = p e B} is a boolean embedding from B into p(G). □

So in any algebra-graph system (A, G, B), Lemma 4.4 allows us to regard B as a boolean algebra of subsets of G, and the Lags-relation symbol 'e' as denoting genuine set membership. Recall that Fi = ]"[ j<k<n ¡ k=i -djk from Definition 3.7.

Lemma 4.5. Let M = (A, G, B) be an algebra-graph system and i, j < n. Then Fi ■ dij < Fj holds in A.

Proof. It is enough to prove the lemma for algebras from graphs, as it is clearly a set of A-universal first-order sentences. But this is easy and was done in [12, Lemma 4.2]. It is also easily seen to hold in cylindric algebras, of which A is one (by Lemma 4.4).

Now we examine the functions Ri, Si.

Lemma 4.6. Let M = (A, G, B) be an algebra-graph system and let i, j < n be distinct. Then:

(i) If a, b e A anda < b then Ri (a) < Ri (b).

(ii) Ifb e A andb < dij then Ri (b) = Rj(b).

(iii) The map f : A ^ B given by f (a) = Ri(a ■ dij) is a boolean homomorphism satisfying f (Fi ■ dij) = 1.

(iv) If Ai is the relativisation of the boolean reduct of A to Fi, then Si: B ^ Ai is a boolean homomorphism.

(v) IfB e B then ciSi (B) = Si(B).

(vi) If a e A anda < Fi, then Si (Ri (a)) > a.

(vii) IfB e B, then Ri (Si (B)) = B. (Hence, Si is infective and Ri surjective.)

Proof. Since the lemma can be easily expressed by A-universal LAGS-sentences, it is again sufficient to show that it is true for any structure M of the form M(r) from Definition 4.2. Recall from that definition that

Ri (a) = {K(i) | (K, e Fi ■ a} for a e A = A(r), S¡(B) = {(K, e Fi |K(i) e B} for B e B = p(r x n). First observe that:

(t) For each p e F x n, there is a unique atom (Kp, &) e Fi ■ dij with Kp(i) = p. (Of course, Kp and & depend also on i, j, which are fixed throughout the lemma.)

For, we may define & e Eq(n) to be the (unique) i-distinguishing relation with i & j and define Kp by

Kp(i) = Kp(j) = p, Kp(k) undefined if k = i, j.

Then (Kp, &) is certainly a valid element of At(F) contained in Fi and dij and with Kp(i) = p, and it is clearly the only such atom.

We now prove the lemma. Parts (i)-(ii) are easy and left to the reader.

(iii) It is clear that f (0) = 0 and f (a + b) = f (a) + f (b) for all a, b e A. Let p e F xn be arbitrary. For any a e A, by (t) we have p e f (a) iff (Kp, &) e a. Hence, p e f (a) iff (Kp, &) e a, iff (Kp, &) e —a, iff p e f (—a). This shows that f (—a) = —f (a). Hence also, f (1) = 1. So f is a boolean homomorphism. If p e F x n then (Kp, &) e Fi ■ dij and so p e f (Fi ■ dij). As p was arbitrary, f (Fi ■ dij) = 1.

(iv) We require that Si(0) = 0 and that Si(B + B') = Si(B) + Si(B') and Si(—B) = Fi — Si(B) for each B, B' e B. Recalling the definition of Si(B), these requirements are easily checked.

(v) Let (K, ~) =i (K', e Si(B), so that (K', e Fi and K'(i) e B. Then = , so (K, ~) e Fi as well, and K(i) = K'(i) e B. Consequently, (K, ~) e Si(B). Hence, ciSi(B) c Si(B), and the converse is trivial.

(vi) Let (K, ~) e a < Fi be arbitrary. Then (K, ~) e Fi, so K(i) is defined, K(i) e Ri(a), and hence

(K, e {(K', e Fi | K '(i) e Ri (a)} = Si (Ri (a)).

This shows that a < Si(Ri(a)).

(vii) Let B e B. First note that

Ri (Si (B)) = {K (i) | (K, e Si (B)} = {K (i) | (K, e Fi, K (i) e B} c B.

For the converse, let p e B be given. By (t) above, (Kp, &) e Si(B), and p = Kp(i) e Ri(Si(B)). This shows that B c Ri(Si(B)).

Next, we examine the substitution operators.

Definition 4.7. For ~ e Eq(n) let = ]"[ i, j<n,i~ jdij ■ FI i, j<n,i* j —dij.

Lemma 4.8. Let M = (A, G, B) be an algebra-graph system, a e A, i, j < n, and a, t :n ^ n.

(i) The map sa : A ^ A is a boolean homomorphism.

(ii) saot = sa ◦ st.

(iii) sadij = da(i)a(j).

(iv) If ~ e Eq(n), then sad^a ^ d

(v) If a [n \ {i}] = n \ {j}, then Rj (sa a) < Ri (a).

(vi) Ifi e im a then cisa a = sa a, and if a is one-one then ca(i)sa a = sa cia.

Proof. Again, it is enough to show that the lemma is true for an arbitrary structure M(r) from a graph r, as all statements are expressible by ^-universal first-order sentences.

(i) By the definitions, sa0 = 0, sa 1 = 1, and for any a, b e A(r), sa(a + b) ={k e S(r) | Ka e a + b} = {K | Ka e a}U[K | e b}= saa + sab, and sa(-a) ={k e S(F) | e -a} = S(F) \ {k | e a} = —saa.

(ii) Let (K, ~) e S(F) be arbitrary. We claim that ((K, ~)a)T = (K, ~)aoT: that is,

((kay, y) = (kOT, OT).

For i, j < n, plainly i (~a)x j iff T(i) t(j) iff a(r(i)) ~ a(r(j)) iff i oT j, so (~a)x = oT. Let i < n. Then (Ka)T(i) is defined iff (~a)T is i-distinguishing, iff oT is i-distinguishing, iff KaoT(i) is defined. In that case, (Ka)T(i) = Ka (j) where j e t[n\{i}]. Then is j-distinguishing and Ka (j) = K(k) where k e o[n\{j}]. But now, k e o o t[n\{i}], so KaoT(i) = K(k) as well. This proves the claim. Consequently, sosTa = {k e S(F) | Ko e sTa} = {k | (k°)t ea} = {K | KooT e a} = sooTa.

(iii) We have sffdij = {(K, -) | (Ko, e dij} = {(K, ~) | o(i) - o(j)}=da(i)a(j).

(iv) By (i) and (iii),

so d-o = so h dij- Y\ -dij) = n so dij- Y[ -so dij

i, j<n, i-° j i, j<n, j i, j<n, i-° j i, j<n,i^° j

= n do(i)o(j) ■ n -do(i)o(j). i, j<n, o(i)—o(j) i, j<n, o(i)^o(j)

The last expression comprises some of the conjuncts (all of them, if a is onto) of d—. So sad—a ^ d— as required (with equality if a is onto).

(v) Let p e Rj(saa), so that p = K(j) for some (K, —) e saa ■ F j. Hence, (Ka, —a) e a, and — is j-distinguishing. As a[n \{i}] = n \{j}, it follows that —a is i-distinguishing. So (Ka, —a) e a ■ Fi, and Ka(i) is defined and is plainly K(j), i.e. p. Hence p e Ri (a) as required.

(vi) Let i e im a and (K, —) =i (K, —') e sa a, so that (K'a, —'a) e a. We show that (Ka, —a) = (K'a, —'a). As — i = —i and i e im a, we have —a = —'a. Take j < n such that —a is j-distinguishing. Then plainly i e a [n \ {j}], so Ka (j) = K (i) = K'(i) = K'a(j). It follows that (Ka, —a) = (K'a, —'a) e a, so (K, —) e saa. This proves that cisaa < saa. The converse is immediate by Lemma 4.4.

Now suppose that a : n ^ n is one-one. Then plainly, for any atoms (K, —), (K, —'), we have (Ka, —a) = (K o a, —a), and (K, —) =a(i) (K', —') iff (K o a, —a) =i (K' o a, —'a).

Let (K, —) be arbitrary. Then (K, —) e ca(i)saa iff there is (K—') with (K, —) =a(i) (K', —') and (K' o a, —'a) e a, iff there is (K', —') with (Koa, —a) =i (K'oa, —'a) and (K'oa, —'a) e a, iff there is (K*, —*) with (Koa, —a) =i (K*, —*) e a, iff (K o a, —a) e qa, iff (K, —) e sacia as required. □

4.3. Simple algebras

Recall that a cylindric algebra A is simple if |A| > 1 and for any algebra A with cylindric signature, any homomorphism y : A ^ A is either trivial or injective. We will see that the cylindric reduct of the algebra part of an algebra-graph system is simple, so that if it is representable, it has a representation that is just an embedding into a single cylindric set algebra.

Definition 4.9. Let C be a class of BAOs of the same signature L. An L-term d(x) satisfying 1 if a > 0,

d(a) = 10 if a = 0. for each a e A e C, is called a discriminator term for C.

Proposition 4.10. The term c\ ...cn_icn_i ...cixisa discriminator term for the class of algebras from graphs {A(r) | r a graph}.

Proof. See line (5) in the proof of [12, Lemma 5.1]. □

We deduce the following in a standard way.

Corollary 4.11. In every algebra-graph system (A, G, B), the cylindric-type reduct A \ LCAn of A is simple, as is each of its subalgebras.

Proof. Let A' be an algebra with cylindric signature, A* ç A \ LCAn, and y : A* ^ A' a homomorphism. Since the statement that d(x) = c1 ...cn_icn_i ...c1x is a discriminator term is A-universal, it follows from Proposition 4.10 that d(x) is a discriminator term for A*. Suppose y is not injective, i.e. there are distinct a, b e A* such that ya = yb. Then (a _ b) + (b _ a) = 0 and therefore

y(1) = y^(a _ b) + (b _ a)) = d((ya _ yb) + (yb _ ya})

= d((ya _ ya) + (ya _ ya)) = y^(a _ a) + (a _ a)) = yd(0) = y(0). So for any a e A*, y(a) = y(a + 0) = y(a) + y(0) = y(a) + y(1) = y(a + 1) = y(1). Thus y is trivial if it is not injective. □

Lemma 4.12. Let A e RCAn be a representable cylindric algebra. If A is simple, then it has a representation that is an embedding into a single cylindric set algebra.

Proof. There is a representation h : A ^ ÜkeK Sk, where K is an index set and for each k e K, Sk is a non-empty base set and

Sk = (p (nSk ), u, 0, nSk, Dj, cSk I i, j < n).

Because h is injective and |A| > 1, the index set K is non-empty. So choose I e K and let n be the projection of ]"[Sk onto S¿. Then n o h is certainly a homomorphism and because

n o h(1) = nS£ = 0 = n o h(0),

it is non-trivial. But because A is simple, n o h : A ^ S¿ is injective and thus a representation that is an embedding into a single cylindric set algebra.

5. Ultrafilters

We now examine ultrafilters in algebra-graph systems.

5.1. Ultrafilter structures from algebra-graph systems

If M = (A, G, B) is an algebra-graph system then A is an LPEAn-BAO, since this statement is expressible by an A-universal sentence true in every structure from a graph. So its ultrafilter structure A+ (see Definition 2.7) is defined; it satisfies

Rdu(v) ^^ dij e v, Rq (/,v) ^^ Cil/A,] = [Cia | ae //}çv, RSa(/,v) ^^ sa [/] = {sa a I ae //}çv. We view A+ as a polyadic equality atom structure (Definition 2.5) by defining

/ =i v ^^ Rq (/,v), va = {a e A I saa e v}.

Lemma 5.1, the comment following it, and Lemma 5.3(v) show that this gives a well-defined polyadic equality atom structure which, when regarded as an LPEAn-atom structure as in Definition 2.5, yields the ultrafilter structure A+ as above.

Lemma 5.1. Let M = (A, G, B) be an algebra-graph system and a, t : n ^ n. Then for any ultrafilter v of A, the set va is also an ultrafilter of A, and va oT = (va)T.

Proof. By Lemma 4.8, sa : A ^ A is a boolean homomorphism. It is well known and easily seen that for boolean algebras B1, B2, the preimage of an ultrafilter of B2 under a boolean homomorphism f : B1 ^ B2 is an ultrafilter of B1. So va is an ultrafilter of A. By Lemma 4.8(ii),

va oT = {a e A I sa oT a e v }= {a e A \ sa(sT a) e v} = {a e A \ sT ae va} = (va)r. □ It follows that Rsa(/, v) iff / ç va, iff va = / since both are ultrafilters.

5.2. Projections of ultrafilters

Definition 5.2. Let M = (A, G, B) be an algebra-graph system, let / be an ultrafilter of A, and let i < n. We write /(i) for the set Rj [/] = {Rj (a) | a e /}çB — the 'ith projection of /'. We say that / is i-distinguishing if it contains Fj.

Clearly, / is i-distinguishing iff it does not contain any of the djk for distinct j, k e n \ {i}. In this case, /(i) turns out to be an ultrafilter of B. The following lemma establishes this and other facts about projections of ultrafilters.

Lemma 5.3. Let M = (A, G, B) be an algebra-graph system, let i < n, and let/, v be ultrafilters of A.

(i) The projection /(i) is an ultrafilter of B if / is i-distinguishing, and B (that is, the improper filter on B), otherwise.

(ii) If j < n and dij e /, then /(i) = /( j).

(iii) If i = j < n and Ii is an ultrafilter of B, then a = {a e A I Ri (a ■ dij) e /)} is the unique ultrafilter of A with Fi, dij e a and

a(i) = I.

(iv) / =i v iff (a) djk e / iff djk e v for all j, k e n \{i}, and (b) /(i) = v(i).

(v) =i is an equivalence relation on A+.

(vi) If a : n ^ n and a [n \ {i}] = n \ {j}, then /a (i) = /(j).

Proof. (i) If Fi e /, then /(i) = {B e B | Si(B) e /}. For, if Si(B) e / then by Lemma 4.6(vii), B = Ri(Si (B)) e /(i). Conversely, if B e /(i) then B = Ri(a) for some a e / with a < Fi (since Fi e /). By Lemma 4.6(vi), Si(B) = Si(Ri(a)) > a so Si(B) e /. Let Ai be the relativisation of the boolean reduct of A to Fi. By Lemma 4.6(iv), Si : B ^ Ai is a boolean homomorphism. Now Fi e /, so / n Ai is an ultrafilter of Ai. So its preimage under Si, namely /(i), is an ultrafilter of B.

If -Fi e /, then for any B e B we have -Fi + Si(B) e /. The statement that Ri(-Fi + a) = Ri(a) for all a is A-universal and true in every structure from a graph, so it holds for M. By Lemma 4.6(vii), Ri (—Fi + Si ( B)) = Ri (Si (B )) = B .So B e /(i). Since B was arbitrary, /(i) = B.

(ii) This is obvious if i = j, so suppose i = j. Assume dij e /. Let Ri(a) be an element of ¡x(i) for some a e /. Define b = a ■ dij e /. It follows from Lemma 4.6(i), (ii) that Ri(a) > Ri(b) = Rj(b) e /(j). By (i), /(j) is always a filter, so Ri(a) e /(j). Thus /(i) c ¡(j). The converse inclusion holds by symmetry, so /(i) = /(j).

(iii) By Lemma 4.6(iii), the map a ^ Ri (a ■ dij) is a boolean homomorphism from A to B. As a is the preimage of fi under this map, it is an ultrafilter of A. The lemma also shows that Ri(Fi ■ dij) = 1, so Fi ■ dij e a. Plainly, a(i) c fi, so as fi is an ultrafilter of B, by (i) we have a(i) = fi.

Let a' be any ultrafilter of A with Fi, dij e a' and a'(i) = fi .If a e a', then a ■ dij e a', so Ri (a ■ dij) e fi. Hence, a e{a e A | Ri(a ■ dij) e fi} = a. So a' c a, and since both sides are ultrafilters of A, they are equal.

(iv) Assume / =i v. For each j, k = i, we have djk e / ^ djk = cidjk e v, and — djk e / ^ — djk = ci —djk e v (these equations are easily established by rewriting them as A-universal sentences true in every structure from a graph, or using basic properties of cylindric algebras: see, e.g., [10, 1.3.3, 1.2.12]). As / and v are ultrafilters, this proves (a). Hence also, Fi e / iff Fi e v.

We prove (b). If — Fi e /, part (i) gives /(i) = B = v(i), proving (b). Assume then that Fi e /. Then ¡(i) and v(i) are ultrafilters by part (i), so it is enough to show /(i) c v(i). Let B e /(i) be arbitrary. Take a e / such that B = Ri(a). By assumption, cia e v. Note that the following holds for all structures from graphs:

Va: A( Ri (a) = Ri (ad)).

So B = Ri (a) = Ri(cia) e v(i) as required.

For the converse, assume the hypotheses and let

D = n djk ■ n —j

j,ken\{i}, djke/ j,ken\{i}, djk// so D e / n v by (a). Now the following statement holds in structures from graphs:

Va, b : A(0 < a < D A Ri(b) < Ri(a) ^ b ■ D < cia).

For let (K, —) e b ■ D. If K(i) is defined, then K(i) e Ri(b) < Ri(a), so we may pick (K', —') e a with K'(i) = K(i). If K(i) is undefined, let (K', —') e a be arbitrary (we use a > 0 here). Since (K, —), (K', —') e D, we have — i = —i, hence in the second case K'(i) is also undefined and K'(i) = K(i). So (K, —) =i (K', —'), yielding (K, —) e cia.

Since the statement is A-universal, it holds for M. So if a e /, then a ■ D e / and Ri(a ■ D) e /(i) = v(i), so Ri(a ■ D) = Ri (b) for some b e v .By the above, b ■ D < ci (a ■ D) < cia, and as b ■ D e v, we have cia e v as well. So / =i v by definition.

(v) Immediate from (iv).

(vi) Let B e ¡a(i). Then B = Ri(a) for some a e ¡a, so saa e / and Rj(saa) e ¡(j). By Lemma 4.8(v), which applies since a[n\{i}] = n\{j}, we have Rj(saa) < Ri(a) = B. As /(j) is a filter, B e /(j) as well. As B was arbitrary, ¡a(i) c ¡(j).

So by part (i), it only remains to show that if /a (i) is an ultrafilter of B then so is /(j). But as a[n \ {i}] = n \ {j}, the definition of Fi and Lemma 4.8(i), (iii) yield

11 —dkl) = EI da(k)a(l) = Yl —dkl = Fj.

k,len\{i}, k=l ' k,len\{i}, k=l k,len\{j}, k=l

So Fi e ¡a iff sa Fi e /, iff Fj e /. By part (i), ¡a (i) is an ultrafilter of B iff /(j) is. □

6. Networks and patch systems

In this section we introduce approximations to representations, called ultrafilter networks. They will be part of the game to construct representations. We will approximate the networks themselves by lower-dimensional objects that we call patch systems.

6.1. Ultrafilter networks

Definition 6.1. Let X be a set, i < n, and v e nX.

1. For w e nX, we say v =i w if vj = Wj for all j < n, j = i.

2. If v j = Vk for all distinct j, k e n \ {i}, then v is called i-distinguishing.

Definition 6.2. Let M = (A, G, B) be an algebra-graph system. A cylindric ultrafilter network over A is a pair N = (N1, N2), where N1 is a set and N2 : nN1 ^ A+ is a map that satisfies the following for any v, w e nN1:

1. For i, j < n, we have dij e N2(v) if and only if vi = vj.

2. If i < n and v =i w, then N2(v) =i N2(w).

sa Fi — sa

N is said to be a polyadic ultrafilter network if in addition:

3. For each a :n ^ n we have N2(v o a) = N2(v)a.

If N = (Ni, N2) and M = (M1, M2) are ultrafilter networks, we write N c M to denote N1 c Mi and M2 f nN1 = N2. For a chain N0 c N1 c ■■■ of ultrafilter networks Nk = (Nk, N2), we write Uk<mNk for the ultrafilter network (Uk<m Nf, Uk<m n2) (here we view the maps N2 formally as sets of ordered pairs). We will often write N for both N1 and N2.

6.2. Patch systems

Patch systems provide a way to assign ultrafilters on a graph to (n — 1)-sized subsets, or 'patches', of a set of nodes.

Definition 6.3. Let M = (A, G, B) be an algebra-graph system. A patch system for B is a pair P = (P1, P2), where P1 is a set and P2: [P 1]n—1 ^ B+ assigns an ultrafilter of B to each subset of P1 of size n — 1. (If | P11 < n — 1, then P2 = 0.) A set V = {v0,..., vn—1} e [P 1]n is said to be P-coherent if the following is satisfied: For any Bj e P2(V \ {vi}) (i < n), there are pi e G with pi e Bi for each i < n, such that {p0,..., pn—1} is not an independent subset of G. The patch system P is said to be coherent if every set V c P1 of size n is P-coherent.

Lemma 6.4. Let M = (A, G, B) be an algebra-graph system and P = (P1, P 2) a patch system for B. LetV = {v 0,..., vn—1} e [P 1]n and for each i < n, let Vi = V \{vi}. Then V is P -coherent if and only if there exists an ultrafilter i of A that is i-distinguishing and with ¡(i) = P2(Vi) for each i < n.

Proof.

) Assume V is P-coherent. Define

/ = {Si(B) j i < n, B e P2(Vi)} c A.

We claim that /0 has the finite intersection property. By Lemma 4.6(iv) and because P2(Vi) is an ultrafilter of B, for each i < n the set {Si(B) | B e P2(Vi)} is closed under finite intersection. So it is sufficient to consider arbitrary Bi e P2(Vi) and prove that S0(B0) • Si(Bi) ••• Sn-i(Bn-i) = 0. By the P-coherence of V, we can find pi e Bi for each i < n such that {p0,...,pn-i} is not an independent set. Now the following A-universal sentence holds in structures M(r), because there is an atom (K, ~) that is i-distinguishing and such that K(i) = pi, for each i < n:

VB0,...,Bn-i: b{ 3p0,..., pn-i: s(/\ pi e Bi a \/ E(pu pA ^ Si (Bi) > 0).

i< ]<n

We showed that the left-hand side of the implication is satisfied, so the right-hand side gives us that /x0 has the finite intersection property, as claimed.

By the boolean prime ideal theorem, fi0 extends to an ultrafilter /x of A. Since plainly Fi = Si (1B) e //, we have that // is i-distinguishing for all i < n. Moreover, if B e P2(Vi), then Si(B) e /0 c /, so by Lemma 4.6(vii), B = Ri(Si(B)) e /x(i). Therefore P2(Vi) = /x(i) by Lemma 5.3(i), since both sides are ultrafilters of B.

(^) Assume / is an ultrafilter of A that is i-distinguishing for all i < n and with /x(i) = P2(Vi) for each i < n. Choose arbitrary Bi e P2(Vi) for each i < n. For each i < n, we can choose bi e / such that Ri(bi) = Bi. Let b = ]"[i<n(bi • Fi) e /. Now the following A-universal sentence holds by definition in structures from graphs, because we can take (K, ~) e x, and then imK is not independent and K(i) e Ri(x) for each i:

Vx: a( 0 < x Fi ^3p0,..., pn-i: g(/\ pí e Ri (x) a \J E (pi, p¡ ^Y

i<n i<n i<j<n

So we can choose p0,..., pn-1 with pi e Ri(b) c Ri(bi) = Bi and such that {p0,..., pn-1} is not independent. We conclude that V is P-coherent.

6.3. Patch systems from cylindric networks

Here we show how to construct a coherent patch system from a cylindric ultrafilter network. We will need the following lemma to show that it is well defined. We adopt the standard notation that if i, j < n then [i/j]: n ^ n denotes the function given by [i/j](i) = j and [i/j](k) = k for k = i.

Lemma 6.5. Let M = (A, G, B) be an algebra-graph system and N = (N1, N2) a cylindric ultrafilter network over A. Let i, j < n and v, w e nN1. Then:

(i) N2(v) is i-distinguishing if and only ifv is i-distinguishing.

(ii) v is i-distinguishing iff v o [i/j] is j-distinguishing.

(iii) N2(v)(i) = N2(v o[i/j])(j).

(iv) ¡f {vk | i = k < n} = {wk | j = k < n} then N2(v)(i) = N2(w)(j).

Proof. (i) We have that N2(v) 3 Fi if and only if it does not contain djk for j < k < n and j, k = i. But this is true if and only if v is i-distinguishing by the definition of cylindric ultrafilter networks.

(ii) Observe that {[i/j](k) | k e n \ {j}} = n \ {i}. So v is i-distinguishing iff |{vk | k e n \ {i}}| = n — 1, iff |{v[i/j](k) | k e n \ {j}}| = n — 1, iff |{(v o [i/j])k | k e n \ {j}}| = n — 1, iff v o [i/j] is j-distinguishing.

(iii) Write w = v o [i/j]. Then w =i v and wi = vj = wj. By the definition of ultrafilter network we have N2(v) =i N2(w) and dij e N2(w). So by Lemma 5.3(iv) we have N2(v)(i) = N2(w)(i), and by (ii) of the same lemma, N2(w)(i) = N2(w)(j).

(iv) Assume the hypothesis. Now v is i-distinguishing iff |{vk | i = k < n}| = n — 1, and similarly for w. So if v is not i-distinguishing then neither is w j-distinguishing, and by part (i) and Lemma 5.3(i), N2(v)(i) = B = N2(w)(j) as required. So assume that v is i-distinguishing, and hence that w is j-distinguishing. We may suppose without loss of generality that i = j = 0 (by (ii), (iii), we can just replace v by v o [i/0] and w by w o [j/0]).

The proof is by induction on the highest number v, w disagree on: d(v, w) = max{k < n | vk = wk}. If they agree on everything or d(v, w) = 0, then v =0 w, so N2(v) =0 N2(w) and Lemma 5.3(iv) gives us N2(v)(0) = N2(w)(0).

Assume now that d(v, w) = k > 0 and the claim holds if d(v, w) < k. Since {vi | 0 = I < n} = {wi | 0 = I < n}, wk = vj for some 0 < j < n. We have j = k by definition of k. If j > k, then wj = vj = wk, contradicting that w is 0-distinguishing. So 0 < j < k. Now 'swap' the k and j entries of v — that is, define

v' = v o [0/k] o [k/j] o [j/0].

By (iii), N2(v)(0) = N2(v')(0). By (ii), v' is also 0-distinguishing, and clearly {vi | 0 = i< n} = {wi | 0 = i< n}. Also v'k = vj = wk, and vi = vi = wi for all I > k, so d(v', w) < k. So, using the induction hypothesis, we get N2(v)(0) = N2(v')(0) = N2( w )(0). □

The last part in the above lemma says that the ith projection is independent from the ith coordinate and the order of the elements in the vector. This allows us to define the following:

Definition 6.6. Let M = (A, G, B) be an algebra-graph system and N = (N1, N2) a cylindric ultrafilter network over A. We define dN to be the patch system (N1, P2), where

P2: [Nt]n—1 ^ B+,

{v0,..., vi —1, vi+1,..., v„ — 1} N2(v)(i), for each i < n and i-distinguishing v e nN1.

Proposition 6.7. Let M = (A, G, B) be an algebra-graph system and N = (N1, N2) a cylindric ultrafilter network over A. Then dN is a well-defined and coherent patch system for B.

Proof. Let dN = (N1, P2) as above. By Lemma 6.5(iv), P2({v0,..., vi—1, vi+1,..., vn— 1}) = N2(v)(i) is independent of the choice of v, i. By (i) of the lemma, N2(v) is i-distinguishing, so by Lemma 5.3(i), N2(v)(i) e B+. So dN is well defined. Let V = {v0,..., vn—1} e [N1]n, Vi = V \ {vi} e [N1]n—1 for i < n, and / = N2(v0,..., vn—1). By Lemma 6.5(i), / is i-distinguish-ing, and by definition of dN, P2(Vi) = /(i), for every i < n. By Lemma 6.4, V is dN-coherent. As V was arbitrary, dN is coherent.

6.4. Polyadic networks from patch systems

A patch system contains a lot of the information in an ultrafilter network. Here we show that given a coherent patch system P = (P1, P2), we can always find ultrafilters to assign to n-tuples of P1 respecting P2, and under fairly minimal conditions, they form a polyadic ultrafilter network.

Lemma 6.8. Let M = (A, G, B) bean algebra-graph system and P = (P1, P 2) a coherent patch system for B. Letv e nP 1. Then there is an ultrafilter / of A such that

1. For i, j < n, we have dij e / if and only if vi = vj.

2. /(i) = P2 ({v j | j e n \ {i}}) for each i < n such that v is i-distinguishing.

Proof. There are three cases.

(a) If lim(v)| = n, then by Lemma 6.4 there is an ultrafilter / of A that is i-distinguishing and with /(i) = P2({vj | i = j < n}), for all i < n.

(b) If lim(v)| =n - 1, there are unique i < j < n such that vi = Vj, and v is k-distinguishing iff k e {i, j}. By Lemma 5.3(iii),

/ = {a e A | Ri (a • dj e P 2 (im(v))}

is an ultrafilter of A with Fi, dij e / (and hence Fj e / by Lemma 4.5), so for each k, l < n we have dk¡ e / iff Vk = v¡. Also, /(i) = P2(imv). By Lemma 5.3(ii), /(j) = P2(imv) as well.

(c) If |im(v)| < n - 1, define D = f[i< j<n, vi=vj dij • Üi<j<n, vi=vj -dij. In an algebra from a graph, D would just be {(0, ~)} where i ~ j if and only if vi = vj. So D is an atom of A, since this statement is A-universal. We define / to be the principal ultrafilter of A generated by D. Condition 2 holds vacuously as v is never i-distinguishing.

Lemma 6.9. Let M = (A, G, B) be an algebra-graph system and P = (N1, P) a coherent patch system for B. Suppose N2: nN1 ^ A+ is a function satisfying the following, for any v e nN1:

1. For i, j < n, we have dij e N2(v) if and only if vi = v j.

2. N2(v )(i) = P ({vj | j e n \ {i}}) for each i < n such that v is i-distinguishing.

3. If a : n ^ n and v o a : n ^ N1 is one-one, then N2(v o a) = N2(v )a.

Then (N1, N2) is apolyadic ultrafilter network.

Proof. We check the conditions from Definition 6.2 defining ultrafilter networks. The first condition, that dij e N2(v) if and only if vi = vj (for v e nN1 and i, j < n), is given to us. It follows that N2(v) is i-distinguishing iff v is i-distinguishing.

For the second condition, take i < n and v, w e nN1 with v =i w. We require N2(v) =i N2(w). By assumption (2) of the lemma, if v, w are i-distinguishing we have

N2(v)(i) = P2 ({vj | i = j < n}) = P2 ({wj | i = j < n}) = N2(w)(i),

and if they are not, then by Lemma 5.3(i) we have N2(v)(i) = B = N2(w)(i). So by Lemma 5.3(iv), N2(v) =i N2(w).

Lastly we check the third condition for ultrafilter networks. Let a : n ^ n, let w = v o a, and let ~ e Eq(n) be given by i ~ j iff vi = vj. Observe that i ~a j iff wi = wj. We check that

N2( w) = N2(v)a.

There are three cases. If n/^111 =n, then v o a is one-one and the result is given.

Suppose that |n/~a| =n - 1. Let {i, j} be the unique ~a-class of size 2. By condition 1 of the lemma, Fi, dij e N2(w). Also, if k, l < n then dkl e N2(w) iff k ~a l, iff va(k) = va(¡), iff sadn = da(k)a(¡) e N2(v) by Lemma 4.8(iii), iff dn e N2(v)a. Therefore, Fi, dij e N2(v)a as well. So by the uniqueness part of Lemma 5.3(iii), it remains only to show that N2(w)(i) = (N2(v )a)(i).

Now if k, l e n \ {i} and a(k) = a(l), then certainly k ~a l, so k = l by assumption on ~a. Hence, a is one-one on n \ {i}, so a[n \ {i}] = n \{l} for some l < n. We now obtain

N2(w)(i) = P2({wk | k e n \ {i}}) by condition 2, since w is i-distinguishing

= P2({va(k) | k e n \ {i}}) by definition of w

= P2 ({vk | k e n \{}}) since a [n \ {i}] = n \{l}

= N2 (v)(l) by condition 2, since v is l-distinguishing

= (N2 (v)a)(i) by Lemma 5.3(vi).

Finally suppose that |n/~a | < n - 1. Then d^a is an atom of A — this is true in algebras from graphs, because we have d^a = {(0, ~a)}, so it holds for M since the statement is A-universal. So N2(w) is the principal ultrafilter generated by d^a.

Let a e N2(v)a be arbitrary, so that saa e N2(v). By the first part, d^ e N2(v), so saa • d~ > 0. By Lemma 4.8(i) and (iv), sa(a • dr^a) = saa • sad^a ^ saa • d~ > 0, so a • d^a > 0 as well. As d^a is an atom, we obtain a > d^a and a e N2(w). This shows that N2(v)a c N2(w), and equality follows since by Lemma 5.1 both sides are ultrafilters of A. □

7. Chromatic number and representability

Here we show that the chromatic number of a graph r and the representability of A(r) and its reducts are tied together.

Recall that the chromatic number x(r) of a graph r is the size of the smallest partition of the set of nodes of r into finitely many independent sets, or <x if no such partition exists. Although the chromatic number is in general not first-order definable, we can define an analogue for algebra-graph systems with the following formula.

Definition 7.1. For each k < w, we define the following LAGS-sentence:

0k = VB o,..., Bk-i: Bi = 1 ^3p, q : d E (p, q) A \/ (p e Bi A q e Bi )

and 0 = {Ok | k < a}.

Then M = (A, G, B) = Ok iff the chromatic number of G is larger than k 'as far as B can tell'. The true chromatic number of G may in principle be smaller, but B contains no independent sets witnessing this. However, B's estimate is correct when B = p(G), as in structures of the form M(r), and in a number of other circumstances too.

Definition 7.2. If M = (A, G, B) is an algebra-graph system, we will say an element B e B is an independent set, if there are no p, q e B such that E(p, q).

71. Representable implies infinite chromatic number

This direction can be proved without further help, apart from some of the machinery from the preceding section and Ramsey's theorem. It generalises [12, Proposition 5.4].

Theorem 7.3. Let M = (A, G, B) be an algebra-graph system in which B is infinite. If the diagonal-free reduct of A is representable, then M = 0.

Proof. Suppose for a contradiction that the reduct of A to the signature LDfn of diagonal-free cylindric algebras is representable but M = Ok for some k < a.

Recall (e.g., from [10, §1.6]) that for a e A, Aa = {i < n | cia = a}. Define D = {a e A | Aa = n}, and let A' be the closure of D under the boolean operations. We first claim that A' is a subalgebra of A. By Lemma 4.4, the cylindric reduct of A is a cylindric algebra. By basic cylindric algebra, or by A-universal sentences, A0 = A1 = Adii = 0 for i < n; also, Adij = {i, j} for distinct i, j < n, so since n > 3, Adij = n; finally, if a e A' and i < n then i e Acia. So all these elements are in D and hence in A'. Obviously, A' is closed under + and —. By Lemma 4.8(vi), D is closed under each sa, so by Lemma 4.8(i), so is A'. This proves the claim.

Now let N = (A!, G, B). We claim next that N is a substructure of M. Inspecting the function symbols of LAGS, it suffices to show that Si(B) e A' for every B e B and i < n. But by Lemma 4.6(v), i e ASi(B), so Si(B) e D c A'. This proves the claim.

As M = U and all sentences in U are A-universal, it follows that N |= U. So N is also an algebra-graph system in which B is infinite. By Lemma 4.4 and Corollary 4.11, the cylindric reduct A' \ LCAn is a simple cylindric algebra. It is generated by D, and its diagonal-free reduct is representable (since the diagonal-free reduct of A is). It follows from a theorem of Johnson [14, Theorem 1.8(i)] that A' \ LCAn is representable as a cylindric algebra. So by Lemma 4.12, there is a cylindric

representation h that embeds A' \ LCAn into a single cylindric set algebra S = (p(nS), U, \, 0, nS, DSj, CS)i,j<n with base

set S. n ij i

Let N be the ultrafilter network with nodes S and N(s) ={a e A' | s e h(a)} e A+, for s e nS. This is easily seen to be a well-defined cylindric ultrafilter network over A'. Furthermore, by Proposition 6.7 we can make it into a well-defined and coherent patch system dN.

Now M = Ok means that the following is true in M and therefore N:

So G is the union of k independent sets from B: say, B0,..., Bk—1.

Since B is infinite, by Lemma 4.6(vii) A is also infinite. As h is injective, S is infinite and therefore S as well. So we can choose infinitely many pairwise distinct elements s0, s1,... from S. Now define a map f: [w]n— 1 ^ k by letting f ({i1,...,in—1}) be the least j < k such that Bj e dN ({si1,..., sin—1}). By Ramsey's theorem [20], we can choose the elements so that f has constant value c, say. Now consider {s0,..., sn—1}. Since f is constant, Bc e dN({sj | i = j < n}) for all i < n. Because dN is coherent, we can choose p0,..., pn—1 e Bc so that {p0,..., pn— 1} is not an independent set. But this is impossible since Bc is independent.

7.2. Infinite chromatic number implies representable

3 Bo,..., Bk-i: B Bi = 1 A Vp, q : (p e Bi A q e Bi ^-E (p, q))

i<k i<k

For the other direction, we define a game that allows us to build a polyadic representation for A if M = (A, G, B) |= 0 (i.e., G has infinite chromatic number in the sense of B).

Definition 7.4. Let M = (A, G, B) be an algebra-graph system. A game G(A) is an infinite sequence of polyadic ultrafilter networks

M) cN1 c^^

built by the following rules. There are two players, named V and 3. The game begins with the (unique) one-point network M0. There are a rounds. In round t < a, the current network (at the start of the round) is Mt and player V chooses an n-tuple v e nMt, a number i < n and an element a e A such that cia e Mt(v). The other player 3 then has to respond with a polyadic ultrafilter network Mt+1 2 Mt such that there is w e nNt+1 with w =iv and a e Mt+1 (w). She wins the game if she can play a network that satisfies these constraints in each round.

Lemma 7.5. Let M = (A, G, B) be an algebra-graph system. If 3 has a winning strategy in the game G (A), then A is a representable polyadic equality algebra.

Proof. By the downward Lowenheim-Skolem-Tarski theorem (see e.g. [3]), there is a countable elementary subalgebra A0 of A. Let M0 c M1 c ••• be a play of the game G (A) in which V plays every possible move in A0 and 3 uses her winning strategy in G (A) to respond. Define M = Ut<m Mt. This is certainly a polyadic ultrafilter network over A, as all the Mt are polyadic ultrafilter networks. Now define:

h: A0 ^ (p(„M), U, \, 0, nM, dN, Cf, Sf | i, j < n, a : n ^ n) a ^ {v e „M | a e M(v)}.

It can be checked that h is a homomorphism. Recall from Corollary 4.11 that A0 is simple. So, since h(1) = nM = 0 = h(0), the map h is injective. This shows that A0 is representable, and because (by Proposition 2.8) RPEAn is a variety, A is representable as well.

Remark. The converse of the lemma also holds, but is not needed here.

Lemma 7.6. In any algebra-graph system M = (A, G, B), H defines an equivalence relation on G with n classes, each of which is in B.

Proof. The statement that H defines an equivalence relation with n classes is an A-universal statement true in every structure M(r), and hence it is true in M. Each equivalence class is in B since the following A-universal sentence is true in every M(r), and hence in M:

Vx : G 3B : B Vy : G(y e B ^ H(x, y)). □

Lemma 7.7. Let M = (A, G, B) be an algebra-graph system such that M = 0. Let X be an equivalence class of H. Then there is an ultrafilter v of B that contains X but contains no independent sets.

Proof. Let v0 = {B e B | X - B is independent}. Then v0 contains X (clearly), and has the finite intersection property: Suppose for a contradiction that for B0,..., Bk-1 e v0 we have B0 • B1 ••• Bk-1 = 0. Then

X = X - (B0 • B1 ••• Bk-1) = (X - B0) + (X - B1) + ••• + (X - Bk-1).

So X is the union of k independent sets in B. Now in any structure M(r), if an H-class is the union of k independent sets in B, then copies of these sets for every H-class lie in B, so that r is the union of nk independent sets in B — that is, M(r) = — 0nk. This implication is A-universal, so it holds in M. Hence, M = 6nk, a contradiction to M = 0. Thus v0 has the finite intersection property and, by the boolean prime ideal theorem, it can be extended to an ultrafilter v, which contains X but no independent set (because it contains the complement).

Remark. The converse of Lemma 7.7 also holds, but is not needed here.

The following 'converse' of Theorem 7.3 generalises [12, Proposition 5.2].

Theorem 7.8. Let M = (A, G, B) be an algebra-graph system. IfM = 0, then A is representable as a polyadic equality algebra.

Proof. By Lemma 7.5 it is sufficient to show that player 3 has a winning strategy in the game G(A). Suppose we are in round t and the current polyadic ultrafilter network is Mt. According to the rules, player V chooses a e A, i < n and v e nMt with cia e Mt(v). The other player 3 now has to respond with a network Mt+1 2 Mt that contains some tuple w e nMt+-[ such that v =i w and a e Mt+1(w). If there is already such a w in nMt then she can just respond with the unchanged network Mt. So we assume in the following that there is no such w.

Step 1. Let Nt+i = Nt U {z}, where z e Nt is a new node. Let the tuple w be defined by w =i v and wi = z. We will first try to find an ultrafilter of A for w. To help 3 win the game, the ultrafilter should contain a. We achieve this by showing that the following set has the finite intersection property:

1^0 = {a} U {—dij| i = j < n}U {cib | b e Nt(v Let D = ]"[j=i —dij. We claim that ci(a ■ D) e Nt(v). Assume for contradiction that ci(a ■ D) e Nt(v). Clearly, D + Y2j=i dij = 1. Therefore, cia = ci(a ■ D) + ^= ci (a ■ dij) e Nt(v). So there is j = i such that ci(a ■ dij) e Nt(v). Let v' = v o[i/ j]. Then v =i v', so by definition of ultrafilter networks, Nt(v) =i Nt(v'). So ci (a ■ dij) = cici (a ■ dij) e Nt(v') as well. But vj = vj, and therefore by definition of ultrafilter networks, dij e Nt(v'). Thus dij ■ ci(a ■ dij) e Nt(v'). In algebras from graphs (and in cylindric algebras generally) we certainly have

Va: A (dij ■ ci (a ■ dij) < a).

This is A-universal, and hence a e Nt(v'). But this contradicts our assumption that no suitable tuple w exists in nNt. So we must have ci (a ■ D) e Nt (v) as claimed.

Now, if ¡0 failed the finite intersection property, there would be b0,..., bm—1 e Nt(v) such that a ■ D ■ cib0 ■ ■ ■ cibm—1 = 0. Then by cylindric algebra, 0 = ci (a ■ D ■ cib0 ■■ ■cibm—1) = ci (a ■ D) ■ cib0 ■■ ■cibm—1 e Nt (v), a contradiction. Thus ¡0 has the finite intersection property.

By the boolean prime ideal theorem, player 3 can choose an ultrafilter ¡ of A that contains ¡0. By construction, Nt (v) =i Moreover,

djk e ¡ ^^ wj = wk (*)

for all j, k < n, because for j = i we have wi = wj and — dij e and for j, k = i,

wj = wk ^ vj = vk ^ djk e Nt(v) ^ djk = cidjk e wj = wk ^ vj = vk ^ —djk e Nt(v) ^ —djk = ci —djk e

Step 2. 3 also needs to define ultrafilters for all the remaining new tuples containing z. She can do this with the help of the patch system P = (Nt+i, P2), defined as follows.

• For each set of'old' nodes V e [Nt]n—1, we define P2(V) = dNt(V).

• For each j < n, define Wj = {wk | j = k < n}. For each Wj of size n — 1, she has to define P2(Wj).

For the case j = i, if |Wi| =n — 1 then because Wi c Nt, she already defined P2(Wi) = Nt(v)(i) = ¡(i) (by Lemma 5.3(iv)).

Now consider the j = i with | Wj | = n — 1. Then z e Wj Nt. We showed in (*) that ¡ is j-distinguishing if w is, so ¡(j) is an ultrafilter of B in that case. So we define P2(Wj) = ¡(j). Note that this is well defined, because if there is k = i, j such that Wk = Wj, then wj = wk, and thus by (*) djk e ¡ and by Lemma 5.3(ii), ¡(j) = ¡(k).

• For the remaining W e [Nt+1]n—1 that contain z, but that are not contained in im(w), we use a single ultrafilter constructed as follows. Recall from Lemma 7.6 that H is an equivalence relation on G with exactly n equivalence classes, say G:,..., Gn, which are contained in B. In structures from graphs we have:

Vx, y: G(-H(x, y) ^ E(x, y)). (t)

As (t) is A-universal, it is true for H on G. Now each of the ¡(j) for j = i, if an ultrafilter of B, contains exactly one of the Gk. There are at most n — 1 such j, so there must be at least one G¿ that is not contained in any ¡(j) that is an ultrafilter. We are given that M = 0, so by Lemma 7.7 there is an ultrafilter v of B containing G¿ and no independent sets. We define P2(W) = v for all the remaining W e [Nt+i]n—

We check that P is a coherent patch system. Let U = {u0,..., un—i} e [Nt+i]n and write Uj for U \ {uj} for each j < n. We need to check that U is P-coherent:

• If z e U, then U c Nt and U is P-coherent because Nt is a polyadic, hence cylindric network, so by Proposition 6.7, dNt is coherent.

• If U = im(w), then U is P-coherent by Lemma 6.4.

• In the case where z e U and |U nim(w)| =n — 1, we can find j, k < n such that z e Uj = U nim(w) and z e Uk im(w). Then, by the above, Gi e v = P2(Uk). Moreover, by the choice of £, there is m = £ such that Gm e P2(Uj).

Take any Xr e P2(Ur) for each r < n. Choose pr e Xr, for each r < n, with p j e Xj ■ Gm and pk e Xk ■ G£. Since l = m and therefore H(pj, pk) does not hold, we have E(pj, pk) by (t). Thus {p0,..., pn—1} is not independent.

• In the remaining cases, z e U and |U nim(w)| < n — 1. Then there are distinct j, k < n such that z e Uj, Uk im(w). So by the above, we have P2(Uj) = P2(Uk) = v.

Take any Xr e P2(Ur) for each r < n. Then Xj, Xk e v, and thus Xj ■ Xk e v and is therefore not independent. So there are pj, pk e Xj ■ Xk such that E(pj, pk). For the other s = j, k just choose any ps e Xs. Then {p0,..., pn— 1} is not independent.

This shows that P is coherent.

We are nearly ready to define Nt+1. First, define an equivalence relation ~ on the set of one-one tuples in nNt+1 \ nNt, by: u ~ u' iff there is a permutation a of n such that u o a = u'. Choose a representative ue of each ~-class e, ensuring that if w is one-one then it is chosen as a representative. We now define an ultrafilter Nt+1(u) of A for each u e nNt+1 as follows.

U1. If u e nNt we set Nt+1 (u) = Nt (u). U2. Define Nt+1(w) = ¡¡.

U3. If u e nNt+1 \ (nNt U {w}) is the representative of its ~-class or is not one-one, we use Lemma 6.8 to choose any

ultrafilter Nt+1(u) of A satisfying the properties of that lemma. (These properties are exactly L1-L2 below.) U4. Each remaining tuple u is one-one but is not the representative ue of its ~-class e. There is a unique a : n ^ n such that u = ue o a, and we set Nt+1 (u) = Nt+1(ue)a.

We check that Nt+1 is a polyadic ultrafilter network. It is sufficient to check that each u e nNt+1 satisfies the conditions of Lemma 6.9, namely:

L1. For j, k < n, we have djk e Nt+1(u) if and only if uj = uk.

L2. Nt+1(u)(j) = P2({uk I k e n \ {j}}) for each j < n such that u is j-distinguishing.

L3. If a : n ^ n and u o a : n ^ Nt+1 is one-one, then Nt+1 (u o a) = Nt+1 (u)a.

If u e nNt this is immediate because Nt is a polyadic ultrafilter network and by definition of P .If u = w, L1 holds by choice of ¡, L2 by definition of P and because i =j Nt(v), and L3 by U4 above, since w is the representative of its ~-class. If u is not one-one then L1 and L2 hold by choice of Nt+1(u) in U3, and L3 holds vacuously. All that remains is the case where u e nNt U {w} is one-one. Let e be the ~-class of u, and let u = ue o t for some (unique) t : n ^ n. Trivially if u = ue, and by U4 otherwise, Nt+1(u) = Nt+1(ue)T. Below, j, k range over n.

• For L1, djk e Nt+1 (u) = Nt+1(u e)T iff Stdjk = dT(j)T(k) e Nt+1(ue ) by (iii), iff (ue)T(j) = (ue)r(k) by choice of Nt+1(ue), iff uj = uk as required.

• We check L2. Suppose that u is j-distinguishing. Plainly, t is one-one, so t[n \{ j}] = n \{t( j)}. Consequently,

Nt+1 (u)( j) = Nt+1 (u e)T( j) by definition of Nt+1 (u) in U4

= Nt+\(ue)(t(j)) by Lemma 5.3(vi)

= P 2({(ue)k I ke n \{t(j)}}) by choice of Nt+1 (ue)

= P 2 ({(ue)T(k) I k e n\{j}}} as n \ {r(j)} = t [n \ {j}]

= P2({uk 1 k e n \{j}}) as u = ue o r.

• For L3, suppose that a : n ^ n and u oa is one-one. We check that Nt+1(u oa) = Nt+1(u)a. Plainly, u oa = ue o r oa e e and t o a is one-one. Using the definitions and Lemma 5.1,

Nt+1(u o a) = Nt+1(ue)roa = (Nt+1(ue)r}a = Nt+1(u)a, as required.

So by Lemma 6.9, Nt+1 is a polyadic ultrafilter network. We also have Nt+1 2 Nt, w =j v, and a e i = Nt+;i(w). The network Nt+1 is 3's response to V's move in round t. So she is able to respond to any move made by V — she has a winning strategy.

8. Direct and inverse systems and duality

Here we examine morphisms between graphs, atoms structures, and algebras. Our algebras are constructed from atom structures based on graphs, so we will need to transform graph p-morphisms into p-morphisms of atom structures, and then, using duality, to embeddings of algebras. We will also consider direct and inverse systems, and their limits.

Definition 8.1. Let r, A be graphs. A map f : r ^ A is said to be a graph p-morphism if for each x e r, f maps the set of neighbours of x in r surjectively onto the set of neighbours of f (x) in A.

Definition 8.2. Let L 2 LBA be a functional signature and let S = (S, Rf | f e L \ LBA) and S' = (S', Rf | f e L \ LBA) be L-atom structures. Let g : S ^ S ' be a function. We say that g : S ^ S ' is a p-morphism of atom structures if for each k-ary f e L \ LBA, we have:

Forth: g is an L+-homomorphism: for every x1,...,xk, y e S, if Rf (x1,...,xk, y) then Rf (g(x1),..., g(xk), g(y)). Back: if y e S, x'1,...,x'k e S', and Rf (x'1,...,xk, g(y)), then there are x1,...,xk e S such that R f (x1,...,xk, y) and g(xi) = xi for i = 1,... k.

Our first lemma is straightforward.

Lemma 8.3. Let r, A be graphs and f : r ^ A a surjective graph p-morphism. Let fx : r x n ^ A x n be given by fx (p, i) = (f (p), i) for (p, i) e r x n. Define

J: At(r) ^ At(A), (K, ^ (f x o K, Then f is a surjective p-morphism of atom structures.

Proof. Plainly, fx : r x n ^ A x n is a surjective graph p-morphism. We need to check the following:

(i) if (K, ~) e At(r), then f(K, ~) e At(A);

(ii) surjectivity of f;

(iii) the forth property of the cylindrification relations, i.e. if we have i < n and (K1, ~1) =i (K2, ~2) then f (K1, ~1) =i f (K2, ~2); f

(iv) the back property of the cylindrification relations, i.e. if we have i < n and (J2, ~2) =i f (K1, ~1), then there is

(K2, ~2) e At(r) such that f (K2, ~2) = (J2, ~2) and (K2, ~2) =i (K1, -1);

(v) diagonals are preserved, i.e. (K, ~) e Dij ^^ f (K, ~) e Dij;

(vi) substitutions are preserved: f ((K, ~)°) = (f (K, ~))CT.

For (i), suppose (K, ~) e At(r) and |n/~| = n. Clearly the domain of K is preserved by f. Moreover, since imK is not independent and fx is a graph p-morphism, imK' is not independent either. The other cases follow directly from the definition of f.

To show (ii) let (K', ~) e At(A). If K' is not defined anywhere, we let K be undefined everywhere as well. If there are i < j < n such that i ~ j and K'(i) = K'(j) is defined, then as fx is surjective, there is p e r x n such that f x(p) = K'(i). Define K(i) = K(j) = p and let K be undefined for the remaining values in that case. Finally, if K' is defined on all values i < n, then im(K') is not independent, so there are i < j < n such that there is an edge from K'(i) to K'(j). Since fx is surjective, there is pi e r x n such that f x(pi) = K'(i). As fx is a graph p-morphism, there is pj e r x n such that there is an edge between pj and pi and f x(pj) = K'(j). For the remaining s = i, j, using surjectivity we take any vertices ps e r x n such that f x(ps) = K'(s). Now define K(s) = ps for each s < n. By construction, (K, ~) e At(r) in all three cases, and J(K, ~) = (K', ~).

For (iii) we have for (K1, ~1), (K2, ~2) e At(r) and i < n that

(K=i (K2, ~2)

K 1(i) = K 2(i) and =~2 fx( K1 (i)) = f x(K 2(i)) and = J(K-1) =i J(K2, ~2).

For (iv), suppose that (K1, -1) e At(r), (J2, ~2) e At(A), i < n, and f(K1, -1) =i (J2, ~2). Then

f x (K 1(i)) = J2(i) and =~2.

Now take (K2, ~2) such that K2(i) = K 1(i) (which may be undefined), and if j = i, we choose K2(j) from the f x-pre-image of J2(j) if J2 is defined for j, and otherwise we leave K2(j) undefined. It is not hard to do this in such a way that if j ~2 k then K2(j) = K2(k), and if K2 is total then imK2 is not independent (here we use that im J2 is not independent and fx is a graph p-morphism). Then (K2, ~2) e At(r), f(K2, ~2) = (J2, ~2), and (K 1,~1) =i (K2, ~2).

To see that diagonals are preserved (v), note that (K, ~) e Dij ^ i ~ j -o- f (K, ~) e Dij. For (vi), we have

j ((K, ) = (f xo K),

(f (K, ~))a = ((fx o k)a, ).

Recall that in general, Ka(i) is defined iff is i-distinguishing and is then K(j), where j e o[n \{i}]. So fx o Ka(i) is defined iff (fx o K)o(i) is defined, and in that case,

f x o Ko(i) = f x (K(j)) = (f x o K)(j) = (f x o K)o(i). So indeed, f((K, ~)°) = (f(K, ~))°. □

Next, we move from p-morphisms of atom structures to algebra embeddings. Lemma 8.4. Let g: At(r) i At(A) be a surjective p-morphism. Then the map

g+ : A(A) i A(r), Y i {x e At(r)Ig(x) e Y} is an algebra embedding. If f : A(A) i A(r) is an embedding, then the map

f+ : A(r)+i A(A)+, i i {a e A(A) | f (a) e ¡} is a surjective p-morphism.

Proof. This is standard duality: see, e.g., [2, Theorem 5.47]. □

We now scale up these results to direct and inverse systems of embeddings and p-morphisms, and their limits. We will need to consider only very special cases, so our definitions are highly restricted. More general definitions can be found in, e.g., [5, 11.5, 11.1]. We assume familiarity with direct products of arbitrary model-theoretic structures: see, for example, [3, Exercise 4.1.12].

Definition 8.5. Let L 2 LBA be a functional signature.

1. A direct system of L-algebras and embeddings is a family A = (Ak, hlk | k < l < a), where for each k < l < m < a we have: Ak is an L-algebra, hlk : Ak ^ Ai is an algebra embedding, hk is the identity map on Ak, and hm o hlk = hm.

Its direct limit lim A is the L-algebra defined as follows. Let D be the disjoint union of the domains of the algebras Ak (k < a), and define a relation ~ on D by a ~ b iff for some k, l, m < a with k, l < m we have a e Ak, b e Al, and hm(a) = hm(b). This is an equivalence relation. The domain of lim A is defined to be the set Dof ~-equivalence classes. Its algebra structure is defined as follows. Let f e L have arity r, and let ai/~,...,ar/~ e D/~. Suppose ai e Akt for i = 1,..., r. Let m = max{k1,..., kr}, and let bi = hm (ai ) e Am for each i = 1,...,r. Then we define f (a1/~,..., ar/~) = (fAm(b1,...,br))/~. This can be checked to be well defined.

2. An inverse system of finite graphs and surjective graph p-morphisms is a family of the form G = r, flk | k < l < a), where for each k < l < m < a we have: rk is a graph, flk : r ^ rk is a surjective graph p-morphism, fjk is the identity map on rk, and fk o fm = fm . Its inverse limit ljm G is the subgraph of ]"[ k<a rk with domain {p e f[ k<a rk: flk (pl ) = pk for each k < l < a}.

3. An inverse system of L-atom structures and surjective p-morphisms is a family of the form S = (Sk, glk | k < l < a), where for each k < l < m < a we have: Sk is an L-atom structure, gk : Si ^ Sk is a surjective p-morphism, g£ is the identity map on Sk, and glk o gm = gm. Its inverse limit ljm S is the sub-atom structure of ]"[<a Sk with domain {s e ]"[k<a Sk | gk (si ) = sk for each k < l < a}.

Our earlier work allows us to transform some kinds of system into others.

Definition 8.6. Let G = (rk, vlk | k < l < m) be an inverse system of graphs and surjective p-morphisms. In the notation of Lemmas 8.3 and 8.4, define

At(G) = (At(rk), v'k I k < l < m},

A(G) = (A(rk),vk + | k < l <m},

A(G)+ = (A(rk)+, (V[+}+ | k < l <m}.

It is almost immediate from these lemmas that At(G) is an inverse system of atom structures and surjective p-morph-isms, A(G) is a direct system of BAOs and embeddings, and A(G)+ is again an inverse system of atom structures and surjective p-morphisms.

Proposition 8.7. Let G = (Fk, vlk | k < l < m) be an inverse system of finite graphs and surjective p-morphisms. Then:

(i) (lirpA(G))+= ljm(A(G)+);

(ii) A(G)+ = At(G), where isomorphism of inverse systems is defined in the obvious way;

(iii) ljmAt(G) = AtOjm G);

(iv) (ljmA(G))+ = At(ljm G).

Proof. Part (i) is a consequence of important results of Goldblatt [5, Theorems 10.7, 11.2, 11.6]. Goldblatt proved these results for modal algebras, but they generalise easily to BAOs.

For part (ii), as each AtD is finite, AD+ = At(rk) (see, e.g., [5, Theorems 9.2, 10.7]), and this can be easily extended to show that A(G)+ = At(G).

For part (iii), write r = ljm G. We define maps

f : At(r) ^ lpAt(G), g: ljmAt(G) ^ At(r)

as follows. Let (K, ~) e At(r) be arbitrary. Thus, K: n ^ r x n is a partial map satisfying the conditions of Definition 3.2. For k <w define a partial map Kk : n ^ x n with the same domain as K, by Kk(i) = (Pk, j), where i e domK and K(i) = (P, j) e (]"[k<M^k) x n. It can easily be checked that Kk also meets the conditions of Definition 3.2, so that (Kk, ~) e AtD. Define f ((K, ~)) = ((Kk, ~) | k < w). Clearly, this value is in ljmAt(&).

We now define g. Let a e ljmAt(G) be arbitrary. So a has the form ((Kk, ~) | k < w) where (Kk, ~) e Atrk and Kk =

(v'k)x o Ki for each k < l < w. The relation ~ and the set D = dom Kk do not depend on k. We define a map K: D ^ r x n as follows. For each i e D, there are p e r and s < n such that Kk(i) = (Pk, s) for each k < w. Define K(i) = (p, s) e r x n. It can be verified that (K, ~) e At r. We will check the requirement that if D = n then imK is not an independent set in r x n. For each k < w, since (Kk, ~) e At rk, imKk is not independent and there are ik < jk < n such that (Kk (ik), Kk (jk)) is an edge of rk x n. Choose i < j < n such that (i, j) = (ik, jk) for infinitely many k < w. Since for each k < l < w, the map (vk)x : r x n ^ rk x n preserves graph edges, it follows that (Kk(i), Kk(j)) is an edge of rk x n for every k < w, and hence that (K(i), K(j)) is an edge in r x n, as required. The other requirements are easy to check. So indeed, (K, ~) e Atr. We now define g(a) = (K, ~).

We leave it to the reader to check that f, g are mutual inverses and preserve Rdij, =i, and —a. For part (iv), by parts (i), (ii) and (iii) we have

(1—m A(&)) + = ljm (A(©}+) = ljm At(&) = At(ljm G). □ 9. Applications

Here we prove our two main theorems (Theorems 9.4 and 9.7 below). 91. Canonical axiomatisations

Here, we use direct and inverse systems to build a certain algebra, and apply the results from the previous sections to show that it can be made to satisfy an arbitrary number of representability axioms, while its canonical extension only satisfies a bounded number. It will follow that any first-order axiomatisation of the representable cylindric algebras (and various other classes) has infinitely many non-canonical axioms.

Our argument is based on the following result. It is from [13, Lemma 4.1], but it can be proved in a rather simpler way by modifying the argument of [9, Theorem 4]. Both proofs use similar random graphs. Recall that x(r) denotes the chromatic number of a graph r.

Theorem 9.1. Suppose that 2 < i < k <w. Then there exists an inverse system of finite graphs

fo ^ _ f12 lo -11 ----,

where the fj are surjective graph p-morphisms, such that x (rs) = kfor every s < w, and x (lj— rs) = I.

Definition 9.2. Let us define some LAGS-theories.

1. Fix a universal axiomatisation n of RPEAn — such an axiomatisation exists because RPEAn is a variety (Proposition 2.8).

2. Also fix any first-order axiomatisation A of RDfn.

We regard n and A as A-sorted LAcS-theories in the obvious way.

3. Let & be the following LAGS-theory, expressing that B is infinite:

^ = {0m I m <w} where = 3Bo,..., Bm—1: M /\ Bi = Bj J.

i< j<m

Also recall from Definition 7.1 that 0 = {0k\k < w} expresses that G has infinite chromatic number in the B-sense. The theory U defining algebra-graph systems was laid down in Definition 4.3.

Definition 9.3. For LDfn c L c LPEAn, we write RL for the class of L-algebras having a representation respecting all the L-operations.

We can now prove the main result of the paper.

Theorem 9.4. Let L be a signature satisfying LDfn c L c LPEAn. Then any first-order axiomatisation of RL contains infinitely many non-canonical axioms.

Proof. Suppose for a contradiction that T = TC U TNC is a first-order axiomatisation of RL, where every sentence in TC is canonical and TNC is finite. We regard T equally as an A-sorted LAGS-theory in the natural way. Referring to Definition 9.2, we plainly have n \= T = A. Also, by Theorem 7.3 we have U U & U A = 0, and by Theorem 7.8 we have U U 0 = n. Using this and first-order compactness, and bearing in mind that Ok = O¡ whenever l < k < w, we see that:

1. there is i<w such that U U {O¿} \= TNC,

2. there is a finite T0 c TC such that U U & U T0 U TNC = Oi+1,

3. there is a finite n0 c n such that n0 = T0,

4. there is k <w such that k > Í and U U {Ok} = n0.

Using Theorem 9.1, take finite graphs r0,r1,... such that x(rs) = k + 1 for all s < w,

r, f0 r^ f1

Jo^-J 1 ----,

where the fj are surjective graph p-morphisms, and, writing r = ljmrs, we have x(r) = Í + 1. Using Lemmas 8.3 and 8.4, we obtain embeddings:

AD^ Ar)<-+....

Define A = limA(rs). Then, because x(rs) = k + 1, we have M(rs) = U U {Ok}, so A(rs) = n0 for each s < w. As the sentences in n are universal, they are preserved by direct limits, and we therefore have A = n0 and hence A = T0. As all sentences in T0 are canonical, Aa = T0 as well. Moreover, from Proposition 8.7(iv) we get

A+ = (1-m A(rs))+ = At (ljm rs) = At(r),

and thus Aa = A(r) and M(r) = (Aa,r,p(r)). We chose the graphs so that x(r) = Í + 1. So M(r) = U U {O£} and hence Aa = TNC. As r is plainly infinite, p(F) is also infinite, and so M(F) = U U & U T0 U TNC and hence M(F) = Oi+1. So x(r) > Í + 1, a contradiction. □

Corollary 9.5. Any first-order axiomatisation (for example, any equational axiomatisation) of any of the following classes has infinitely many non-canonical sentences:

1. the class RDfn of representable n-dimensional diagonal-free cylindric algebras,

2. the class RCAn of representable n-dimensional cylindric algebras,

3. the class RPAn of representable n-dimensional polyadic algebras,

4. the class RPEAn of representable n-dimensional polyadic equality algebras.

Hence, none of the classes is finitely axiomatisable, nor does it have an axiomatisation where only finitely many axioms are not Sahlqvist equations.

Proof. Immediate from Theorem 9.4 and because Sahlqvist equations are canonical. □ 9.2. Strongly representable atom structures

The main result of [12, Theorem 6.1] showed that for each finite n > 3, the class StrRCAn of strongly representable n-dimensional cylindric algebra atom structures is non-elementary. We finish with a generalisation of this to other signatures. It has already been proved by Sayed Ahmed (draft of untitled monograph, 2010) using the same algebras.

Definition 9.6. Let LDfn c L c LPEAn. An L-atom structure S is said to be strongly representable if S + e RL (see Definition 9.3 for RL). n n

Theorem 9.7. For any LDfn C L c LPEAn, the class of strongly representable L-atom structures is non-elementary. In another common notation, the class StrRL of structures for RL is non-elementary.

Proof. A celebrated result of Erdos [4] shows that for all k <o there is a finite graph Gk with chromatic number and girth (length of the shortest cycle) both at least k. Let rk be the disjoint union of the G¿ for k < I < r: this time, no edges are added between copies. Plainly, rk has infinite chromatic number, and its girth is at least k. By Theorem 7.8 applied to M(rk), A(rk) \ L e RL, so that (Atrk) \ L+ is strongly representable.

Now let r be a non-principal ultraproduct of the rk. Then r is infinite, and by tos's theorem it has girth at least k for all finite k, since this property is first-order definable. Hence, r has no cycles, so its chromatic number is at most two. By Theorem 7.3, the diagonal-free reduct of A(r) is not representable, and hence neither is its L-reduct. So (At r) \ L+ is not strongly representable.

But it is easily seen that the operation At(-) commutes with ultraproducts, and it follows that (At r) \ L+ is isomorphic to an ultraproduct of the (At rk) \ L+. This shows that the class of strongly representable L-atom structures is not closed under ultraproducts and so cannot be elementary.

10. Conclusion

We have proved that every variety of representable algebras of relations whose signature lies between that of RDfn and RPEAn (for finite n > 3) is barely canonical, in that (although canonical) it cannot be axiomatised by first-order sentences only finitely many of which are not themselves canonical. As far as we know, it is an open question whether various other varieties of algebras of relations are also barely canonical, including infinite-dimensional diagonal-free, cylindric, polyadic (equality) and quasi-polyadic (equality) algebras, classes of relativised set algebras such as Crsn, Dn, Gn (n > 3), and various classes of neat reducts, such as SNrnCAm for 3 < n < m < o, and SRaCAn for 5 < n < o. Some of these (such as Gr) are not even known to be varieties. A wider question is to find a more general method for proving bare canonicity.

Acknowledgements

We thank the referee for a very perceptive and helpful report, and the editor for handling the paper.

References

[1] H. Andréka, Complexity of equations valid in algebras of relations, Part I: Strong non-finitizability, Ann. Pure Appl. Logic 89 (1997) 149-209.

[2] P. Blackburn, M. de Rijke, Y. Venema, Modal Logic, Cambridge University Press, Cambridge, 2001.

[3] C.C. Chang, H.J. Keisler, Model Theory, 3rd edition, Elsevier Science, Amsterdam, 1990.

[4] P. Erdos, Graph theory and probability, Canad. J. Math. 11 (1959) 34-38.

[5] R. Goldblatt, Metamathematics of modal logic, Rep. Math. Logic 6 (1976) 41-77, reprinted in [6].

[6] R. Goldblatt, Mathematics of Modality, CSLI Lecture Notes, vol. 43, CSLI Publications, Stanford, CA, 1993.

[7] R. Goldblatt, Elementary generation and canonicity for varieties of boolean algebras with operators, Algebra Universalis 34 (1995) 551-607.

[8] R. Goldblatt, I. Hodkinson, The McKinsey-Lemmon logic is barely canonical, Australas. J. Logic 5 (2007) 1-19.

[9] P. Hell, J. Nesetril, The core of a graph, Discrete Math. 109 (1992) 117-126.

[10] L. Henkin, J.D. Monk, A. Tarski, Cylindric Algebras, Part I, North-Holland, Amsterdam, London, 1971.

[11] L. Henkin, J.D. Monk, A. Tarski, Cylindric Algebras, Part II, North-Holland, Amsterdam, London, 1985.

[12] R. Hirsch, I. Hodkinson, Strongly representable atom structures of cylindric algebras, J. Symbolic Logic 74 (2009) 811-828.

[13] I. Hodkinson, Y. Venema, Canonical varieties with no canonical axiomatisation, Trans. Amer. Math. Soc. 357 (2005) 4579-4605.

[14] J.S. Johnson, Nonfinitizability of classes of representable polyadic algebras, J. Symbolic Logic 34 (1969) 344-352.

[15] B. Jónsson, A. Tarski, Boolean algebras with operators I, Amer. J. Math. 73 (1951) 891-939.

[16] M. Kracht, Tools and Techniques in Modal Logic, Stud. Logic Found. Math., vol. 142, Elsevier, Amsterdam, 1999.

[17] A. Kurucz, On the complexity of modal axiomatisations over many-dimensional structures, in: L. Beklemishev, V. Goranko, V. Shehtman (Eds.), Advances in Modal Logic, vol. 8, College Publications, 2010, pp. 241-254.

[18] R. McKenzie, The representation of relation algebras, Ph.D. thesis, University of Colorado at Boulder, 1966.

[19] J.D. Monk, Nonfinitizability of classes of representable cylindric algebras, J. Symbolic Logic 34 (1969) 331-343.

[20] F.P. Ramsey, On a problem of formal logic, Proc. Lond. Math. Soc. s2-30 (1930) 264-286.

[21] T. Sayed Ahmed, On the complexity of axiomatizations of the class of representable quasi-polyadic equality algebras, Math. Log. Q. 57 (2011) 384-394.

[22] M. Stone, The theory of representations for boolean algebras, Trans. Amer. Math. Soc. 40 (1936) 37-111.

[23] Y. Venema, Atom structures and Sahlqvist equations, Algebra Universalis 38 (1997) 185-199.