Electronic Notes in Theoretical Computer Science 12 (1998)

URL: http://www.elsevier.nl/locate/entcs/volumel2.html 51 pages

A Notion of Division for Concrete Monoids

Andrew Solomon

School of Mathematics and Statistics The University of Sydney NSW 2006, Australia

Abstract

A concrete monoid over a category C is a subset of the endomorphisms of an object of C, containing the identity and closed under composition. To contrast, an abstract monoid is just a one object category.

There is a natural notion of division between concrete monoids distinct from the usual division of abstract monoids. This concrete division is identified via two examples, and then defined, giving rise to a bicategory of concrete monoids over C whose arrows are concrete divisions. The Poincare classes of the arrows of this bicategory are found to have a simple and appealing characterization, allowing us to define a category of concrete monoids over C.

These definitions are illustrated with examples from the theories of semigroups, matrices, vines and automata. With the aid of these definitions, we make functo-rial the well known constructions of the action monoid of an automaton, and the endomorphism monoid of an object of a category.

1 Introduction

It is always a delicate matter to discuss mathematics informally, but in questions of motivation it often becomes necessary. Therefore we begin with an informal account of the motivation for this work and ask the reader to bear in mind that when we use expressions such as 'abstract' or 'concrete' or 'interpretation' we do not intend them yet to have a technical meaning.

This paper is based on the premise that there is a natural distinction between the property of being abstract or concrete. To illustrate, one might say that an abstract set is one in which the elements have no particular interpretation, whereas a concrete set is one in which each element is to be interpreted as something. For example, a set with three elements is an abstract set, while a set of three oranges is a concrete set — each of its elements is to be interpreted as a particular orange, real or imaginary.

While it may seem churlish to make this distinction, in the case that the set is the set of elements of a monoid, we give a number of examples which show

©1998 Published by Elsevier Science B. V.

that the natural structural relationships between monoids whose sets of elements are abstract sets are different from the natural structural relationships between monoids whose sets of elements are concrete sets.

The primary examples of structural relationships between abstract monoids are the embedding and the quotient. These are both readily (and often) generalized to division: crudely speaking, the monoid B divides the monoid A if B can be viewed as a submonoid of A when one ignores some of the structure of A. (Formally, B divides A if there is a submonoid C of A with B a quotient of C.) Birkhoff's Variety Theorem for abstract algebras, Eeitermann's Theorem for finite algebras, and the Krohn-Rhodes Theorem all attest to the fundamental importance of the division relationship.

In the following examples, we show that when the structure of the 'things' which make up the set of elements of a concrete monoid are taken into account, the relationship of division is strengthened. This stronger notion of division for concrete monoids which respects the structure of the elements of the monoid will be called a concrete division and this paper is devoted to finding a general definition for this stronger notion. In particular, we use concrete division to construct a very simply defined morphism between concrete monoids which has an associative composition and hence we derive a category of concrete monoids.

The purpose of this paper is threefold,

• To give a general mathematical setting for the ideas discussed above and illustrated by the examples below. This formalization will hopefully account for all particular instances of this type of construction.

• To convince the reader (particularly the semigroup theorist) that a good general question when studying concrete monoids is "What are the concrete divisors of this monoid".

• To put forth the possibility that the category of concrete monoids will be useful in making functorial various constructions of concrete monoids from other mathematical objects.

Example 1.1 [A concrete quotient] Consider the concrete monoid M whose

Then removing the third string gives the concrete quotient Q generated by the single braid a:

with a quotient map defined by a, b a.

In the example above, since M is a submonoid of B3, the braid group on 3 strings, the concrete quotient Q of M also gives rise to an obvious concrete division of B3 by Q. Another example of a concrete division in the case that the monoids are transformation monoids will convince the reader of the diversity of situations in which constructions of this type arise.

Example 1.2 [A concrete division] This example concerns the case when the elements of the monoid are endofunctions of a set, (That is, the monoid is a transformation monoid.)

A monoid A of transformations of a set X will be represented as a table — each row of the table represents an element of the monoid and each column represents an element of the set. The ith entry in the row corresponding to a E A is a(i), the value of a at i. The top row represents the identity element.

2 2 3 2 2 3 2 2 3

Consider the transformation monoids A

3 3 2 3 3 2

he rows of the table

We find B by 'filtering' A — by first deleting some of (not the top row), in such a way that the set of remaining rows is closed under composition, and then deleting some of the columns, ensuring that the remaining entries of the table consist only of elements whose columns have not been deleted, (In the table above, we have deleted column 4 and rows 5-7.) In short, we have found a submonoid N of A and a subset Y of X, such

that for each n e N,y e F, n(y) e Y. To make the remaining part of the table into the transformation monoid B, we collapse identical rows. We now say that BY is a concrete divisor of the monoid Ax.

If this process is repeated with B to give a transformation monoid C, then C may be derived from A by a single step of deleting rows and columns and collapsing identical rows. This defines a composition of concrete divisions.

As an example of where these constructions arise, the concrete quotient construction between permutation groups is employed in algorithms to find the chief series and the svlow subgroups of a permutation group. The problem is solved for several concrete quotients and then the answers are "sewn together" to give an answer for the original permutation group [4], [5].

The first part of this paper is devoted to making this notion of a concrete division precise, and to defining the category Conc(C) whose objects are concrete monoids over C and whose arrows are equivalence classes of concrete divisions (in the example above, there would be an arrow from B to A).

The rather forbidding length of the following formalization is due to the fact that, being of general mathematical interest, the development has been made entirely self contained and assumes no prior knowledge of category theory beyond basic definitions and concepts.

Part A - Theory

This part is a systematic approach to the definition of the category Conc(C) of concrete monoids over C, In Section 2, we recall the definition of a bieategorv and illustrate it in Section 3 with some relevant examples.

The category Reps(C) whose objects are monoid representations over C is defined in Section 4, and several useful theorems are proved. We take a detour in Section 5 to explain how the well established notions of morphism, relational morphism and division of monoid representations over Set fit into the framework constructed so far.

Finally, in Section 6 we study the bieategorv of concrete divisions, whose objects are concrete monoids and whose arrows are the concrete divisions described in the introduction. Through a thorough understanding of this bieategorv, we derive the category Conc(C) which is applied to various diverse situations in Part B,

2 Bicategories

In his seminal work [3] Benabou defines a bieategorv. For completeness, we repeat this definition and an example almost verbatim, A bicategory S is determined by the following data:

B I A set Ob(S) called the set of objects of S,

B II For each pair (A, B) of objects of S, a category S(A, B). An object

S of S(A, B) is called an arrow of S and written A A- B. If S and T are objects of S(A, B), an arrow s of S(A, B) between S and T is called a 2-cell of S and will be written S T or more usually

Composition in the category S (A, B) will thus correspond to the pasting

b ^a/xjxfvxh^r^ a

which is referred to as vertical composition of 2-eells,

B III For each triple (A, B, C) of objects of S, a functor ca,b,cS (A, B) x S (B, C) —S(A, C) which is called the composition functor of S, For

q 7' e

A 4 B and B 4 C, write T o S for ca,b,c(S,T). If S 4 S' is

an arrow of S(A, B) and T =$> T' is an arrow of S(B, C) then write t o s for CA,B,c(s,t). Then the composition functor corresponds to the pasting

V O S'

which is referred to as horizontal composition of 2-eells,

B-IV For each object A of S an object IA of S(A, A) called the identity arrow of A. The identity morphism of IA in S(A, A) is denoted iA and called the identity 2-cell of A.

B-V For each quadruple (A, B, C, D) of objects of S, a natural isomorphism a,a,b,c,d between the two functors ca,b,d • (id x cb,c,d) and ca,c,d ■ (ca,b,c X id) from S(A, B) x S(B, c) x S(C, D) to S(A,d). In particular, if (S,T,U) E S(A,B) x S(B,C) x S(C,D) this gives a component isomorphism between U o (T o S) and (U o T) o S.

B-VI For each pair (A, B) of objects of S, natural isomorphisms Ia,b and ta,b, called the left and right identities:

i.A x id

1 x S(A b)

■ s(A, A) x S(A. B)

ca,a,b

s(A, B)

id x iß

S(A B) x 1 -- S(j4, b) x S[B: B)

"1 \ / C-A,B,B

S(A b)

Here we write 1 for the trivial category and 1 S(A, A) for the unique functor with value IA. For each S E S(A,B), the left and right identities give rise to component isomorphisms IAo S = S and SoIB^S.

B-VII The data above are also required to satisfy the following coherence conditions:

(1) If (V,U,T,S) is an object of S(A,B) x S(B,C) x S(C,D) x S(D, E) then the following diagram commutes:

((S O T) O U) O V

a(S,T,U) oid

(S o (T o U j) o V

a(S o T, U, V)

a(S, T o U, V)

(S o T) o (U o Trt

So ((To U) O V)

a(S, T, U o V)

so (to (u o v)) This condition is known as associativity coherence. (2) The following diagram commutes:

0-a,b,b,c(s• iii: t)

(S oIB)oT

>So(IBoT)

rAM >,c(T)

This condition is known as identity coherence.

2.1 From bicategories to categories

A bieategorv is not necessarily a category — composition of arrows is only associative up to isomorphism. However, for any bieategorv S, there are two closely related categories which capture much of the structure of S — the classifying category and the Poincare category of S [3],

The classifying category of S is the easier to define. Its objects are the objects of S and its arrows are isomorphism classes of arrows of S, If S e S(A, B) and T E S(B, C) and if [S] denotes the isomorphism class of S, then the composition of [S] and [T] in the classifying category of S is defined to be [T o 5], The associative law now holds for composition of arrows, since by axiom B-V above, there is a 2-eell isomorphism from (SoT)oU to So (Toll).

To define the Poincare category, recall from [16, page 86] that a category C is said to be connected if for any two objects A,B of C there is a finite

sequence A = Ai, A2,.....1„ = B of objects of C, and for each i < n, an

arrow Ai —Ai+i or an arrow Ajt Ai+i. In [16, page 88, Ex, 7] it is stated that every category is the disjoint union of connected categories, called its connected components. We may define an equivalence relation on the objects of a category by A B if A and B belong to the same connected component. The objects of the Poincare category are the objects of S and its arrows are ^ equivalence classes of arrows of S, Composition is defined as in the classifying category. That this is a category follows by induction from B-III,

3 Examples of Bicategories

Let A and B be objects in a category C, A span from A to B is a triple (/, g) where A 4- S A B is a diagram of C, One of the many examples of a bieategorv cited in [3] is Span(C), the bieategorv of spans in C, Let C be any category with pullbaeks, together with some extra structure - namely, a chosen pullback for every diagram of the form We now define

the bieategorv Span(C) of spans of C with respect to this structure. Since a pullback is only unique up to isomorphism, another choice of pullbaeks would give another bieategorv which is isomorphic to the first,

S-I The objects of Span(C) are the objects of C,

S-II For each pair (A, B) of objects of C, the category Span(. 1. B) has as

objects spans (f,S,g). An arrow of Span(A, B) from A S A- B

lo .1 A S' A II ih an arrow h : S —S' such that f'-h = f and g' • h = g. Composition in Span(. 1. B) is inherited from C,

S—III The composition functor is defined on arrows by taking the chosen pullback as in the following diagram: -4 R C

and then setting (h,T, A;)o(/, S, g) = (f7rs,SxBT,k7rT). Horizontal composition of 2-eells is performed using the universal property of pullbaeks.

The remaining details of this definition are left to the reader.

3.1 Bicategories from a regular category: Rel(C), Rel*(C) and Div(C).

When C is a regular category (a notion we will define presently), it is possible to construct bicategories Rel(C), Rel*(C) and Div(C) whose arrows are, respectively, relations, totally defined relations, and divisions between objects of C.

3.1.1 Regular categories

Fix some category C, Following [13] we define a strong epimorphism of C to be an epimorphism / such that if

is any commuting diagram with i monomorphic, then there is a (necessarily

We repeat here some useful facts about strong epimorphisms which are taken from [13].

Proposition 3.1 Strong epimorphisms are closed under composition and right division. □

Proposition 3.2 An arrow which is both a monomorphism and strong epi-morphism is an isomorphism. □

If / factors as ip where i is a monomorphism and p is a strong epimorphism, then we refer to ip as a canonical factorization of /,

Proposition 3.3 Canonical factorizations are essentially unique. That is to say, if f = ip = jq with i,j monomorphisms and p,q strong epimorphisms, then there is an isomorphism c making the following diagram commute.

Suppose /: A —B has canonical factorization ip with p: A —C and i:C —B, then as a result of Proposition 3,3, we are able to refer to C as the image of /,

Finally, following [7], we are able to define a regular category. Definition 3.4 A regular category is a category in which:

(1) Every arrow has a canonical factorization;

(2) All finite limits exist;

(3) Strong epimorphisms are stable — i.e. in every pullbaek diagram as pictured below, / is a strong epimorphism implies that /' is a strong epimorphism.

Some examples of regular categories:

• Set — the category of sets and functions. The strong epimorphisms (which are all of the epimorphisms) are the surjeetive functions,

• Mon — the category of monoids and monoid homomorphisms. The strong epimorphisms are the surjeetive monoid homomorphisms. These are not, however, all the epimorphisms.

An example (cited in [1]) of an epimorphism in Mon which is not surjeetive is the canonical inclusion (as multiplicative monoids) of the integers into the rationals.

For the remainder of Section 3,1, C will denote a regular category, 3.1.2 The bicategory Rel(C)

We define the bicategory Rel(C), which in the case that C = Set, has sets as objects and an arrow from the set A to the set B will be a subset of Ax B.

Let C be a regular category and let A, B be objects of C, Following [6], a relation R: A ^ B in C is a span from A to B, such that if S: A ^ B is any other span, then there is at most one 2-eell in Span(C) from S to R. It is an easy exercise to see that for any relation R = (f,R,g), the arrow (f,g):R —A x B is monic. Conversely, if i: R ^ A x B is monic, then (•7TAh R, ttbO is a relation.

Definition 3.5 Every span S = (/, S, g), defines a relation: let S A Q A Ax B be the canonical factorization of (/, g): S —A x B. Then Q = (itAi, Q, 7t.b0 is a relation and p is a 2-eell from S to Q. Such Q is unique up to isomorphism, hence we call it the relation defined by S.

The bicategory Rel(C) of relations in C is defined as follows: Eel-I The objects of Rel(C) are the objects of C,

Rel-II The arrows of Rel(C) are the relations in C, The 2-eells are the

2-eells of Span(C), That is, for any objects A, B of C, the category Rel(C)(A, B) is the full subcategory of Span (C) (A, B) whose objects are the relations,

Rel-III The composite of the relations R: A —B and S: B —C is obtained by composing as spans and then taking the relation defined by the composite span.

The remaining details of this definition are left to the reader. As one would hope, if C = Set, then the composite of R and S in Rel(C) is the set {(a, c) | (a, b) E R, (b, c) E S, for some b E B}.

3.1.3 The bicategory Rel*(C)

The bicategory Rel*(C) of 'total relations' or relational morphisms is a sub-bieategorv of Rel (C), The objects are those of C and the category Rel* (C) (A, B) is the full subcategory of Rel(C)(A, B) whose objects are relations (/, R, g) in which the arrow / is a strong epimorphism in C,

One only needs to check that the composite of two such arrows is again an arrow of Rel*(C), Consider the diagram below where < fh', kg' >: RxB S — A x C has the canonical factorization iq. Then the composite relation is (ttAh Q, ^ci)'- A —C, We need to check that 7xAi is a strong epi,

Since h is a strong epi and C is regular, h' is also a strong epi. By construction, fh' = (ttAi)q. But since q is a strong epi and by the fact (Proposition 3,1) that strong epis are closed under right division, 7xAi is a strong epimorphism, as required.

By way of example, the arrows of Rel*(Mon) are relational morphisms as they are usually defined in the literature (see Section 5 for details). The arrows of Rel* (Set) from A to B are relations R C A x B such that for each a E A, the set {b \ (a, b) E R} is nonempty,

3.1.4 The bicategory Div(C)

The bieategorv Div(C) of divisions is a sub-bicategorv of Rel*(C), whose arrows are spans (/, R, g) where / is a strong epimorphism and g is a monomor-phism. Composition is defined as for Rel(C), but it reduces to the composition in Span(C), To see this, note that in the diagram above, since g is monic, so is g' (see [16, p.72, Ex.5]). Thus, kg' is monic. By construction, kg' = TTciq. But since only monos right divide monos, q must also be a monomorphism. By Proposition 3.2 this implies that q is an isomorphism.

In the case that C = Mon, arrows in Div(Mon) become monoid divisions as they are usually defined.

4 Representations of Monoids

In this section, a monoid M will be regarded as a category with one object. If m, is an arrow of M then write m, E M and say that m, is an element of M. A 'monoid homomorphism from M to N is then a functor <f> from M to N.

Fix some category C. A representation of the monoid M over C is a functor T: M —C. We will sometimes embellish the symbol T by writing rM to indicate that T is a representation of M and T^ to indicate that T lands on X (that is to say, T maps the single object of M to X and is therefore a representation of M as endomorphisms of X). If T is any representation, then the symbols rM, and T^ will all refer to the representation T. If there is only one representation of M as endomorphisms of X in sight, we may simply write M \. omitting the name of the representation altogether.

If Tx (resp. Mx) is a representation then for each m E M, T(rn) (resp. Mx(rn)) is an endomorphism of X. The endomorphism T(rn) will be written m if there is no possibility of confusion about which representation of M is used.

The category Reps(C) is defined to have representations over C for objects. For Tx'.M C and AY: N C objects of Reps(C), an arrow js a pajr where f\.M —N is a homomorphism and

r]i : X Y is an arrow of C which is a natural transformation AY • /1,

i.e. such that for every m, E M the following diagram commutes:

A(/i(m))

Let (/2, AY —Az be another arrow. Their composite is defined by

(/2, m) • (A, m) = (/2 ■ f\,m- m);

it is easy to verify that 772 • Wi is a natural transformation Tx Az • /2 • /1

and that these data define a category. If a is an arrow of Reps(C) we will write «Mon for the first element of the pair and ac for the second. The following proposition is easy to verify.

Proposition 4.1 Let (/, r7) he an arrow of Reps(C). Then (/, r7) is an isomorphism if and only if f is an isomorphism of Mon and rj is an isomorphism of C. The inverse of an isomorphism (f,rj) is □

If TM is a monoid representation, we can define a congruence ~r on M by m ~r n if m = n. A concrete monoid over C is a monoid representation T: M —C for which ~r is trivial, or equivalentlv, it is a monoid representation over C which is faithful.

Remark 4.2 As in the motivating example, the most important representing category is Set, for the objects of Reps(Set) are the transformation representations of monoids. Let M be some abstract monoid. Then the subcategory of Reps(Set) whose objects are representations of M and whose arrows all have Mon component 1M is known as the category of M-sets.

Let A be a category. In [1] a faithful functor r : A —C is called a concrete category over C, Let A: II —C be another concrete category, A concrete functor /: T —A between two concrete categories over C is defined to be a functor / : A —B such that A/ = T, When A and B are monoids, a concrete functor is simply an arrow of Reps(C) whose C component is the identity.

Such an arrow of Reps(C) in which the C component is the identity is often called a strict morphism from T to A, as opposed to the general arrow of Reps(C) which is called a lax morphism. Let T and A be objects of Reps(C), Pictured below are (from left to right) a lax and a strict morphism in Reps(C),

In the following lemma we show that the pullbaek of a diagram in Reps(C) can be constructed from the pullbaeks of the corresponding diagrams in Mon and C, From this fact we are able to deduce in Theorem 4,6 that if C is finite complete, then so is Reps(C),

Lemma 4.3 Let

be pullhacks in Mon and C respectively. Then there is a monoid representation '•I'h • P —C with arrows irr and it a which is the pullback of Diagram (1) in Reps(C), with 7rrMon = = itrc = kx, and ttac = irv.

Proof. To define an action of P on W, let p e P and define 7J: W —X by

rypx = I'(771/ (/>))77 v and similarly define 7y = A(ttn(p))tty'. U —Y. Now

acjpx = acT(TTM(p))TTx

= A(nM<m77u(/)))nC77V

so that by the universal property of pullbacks, there is a unique arrow jp: W — W making the following diagram commute,

We now show that setting ^(p) = 7P:W —W defines a representation

A(-/\1oi.'~Y(/>)MC>V

ßCA(7TN(p))7TY

ßcll'

vl'i, : P ->• C.

Firstly, notice that if /> = 1 then jp = since it is the unique arrow making Diagram (2) commute. Let p,q E P. Then we only need to show that jpjq makes Diagram (2) for pq commute, for then, by uniqueness, jpjq = jpq as required. First notice that

lxlq = T(irM{p))/Kxlq

= r(TTM(p))T'(TTM(q))TTx = T('KM(pq))'Kx = 7f

Similarly, j^j11 = 7^?. But 7^ = 7txlv and 7y = ttyJp so that

Tfj^'yP'y^

■JfytyP<y1

and by uniqueness, 7P79 = jpq as required.

Having defined a representation ^ we proceed to show that it is a pullbaek of Diagram (1) in Reps(C), By definition of 7Tr := (ttm,ttx) and tta '■= (•7Tjv,7ry) are both arrows of Reps(C) and it is immediate that ontr = a. It only remains to show that ^ has the universal property. Suppose that the diagram

commutes. Then there are corresponding commuting diagrams in Mon and in C, Thus, there are arrows £M„n: Q —P and V —W by the universal properties of P and W. We certainly have that if (£Mon,£e) is an arrow of Reps(C), then it is the unique arrow of Reps(C) which makes the diagram commute, so it only remains to show that, for each q E Q,

Aq : = ^(CMo„(g))Cc: * ^ W

Bq := icO(<l): V W

are equal. It is sufficient, by the universal property of W in C, to show that TTxAq = Tlx Bq ] V —X and KyAq = TTy Bq] V —1

for then both Aq and Bq

are the unique arrow making the following diagram commute,

7TXA„ = 77 V Ii.

TTYAV = TTYB„

Tlx Aq = TTx^ (ÎMon (q) ) £e

= r(77Ai(CMon(g)))7rxCc

= r(5Mon(ç)Me = 5ctt(q) = 77\t,c{-l('l)

= 7TXBq

and similarly, TïyAq = ^yBq as required.

If C has an initial object, denote it by 0 and if it has a terminal object, denote it by 1, Note that we denote the trivial monoid by the bold 1 to avoid confusion. Then the following are immediate.

Proposition 4.4 If C has an initial object, then the functor 1 —C landing on the initial object of G is the initial object of Reps(C). □

Proposition 4.5 If C has a terminal object, then the functor 1 —C landing on the terminal object of G is the terminal object of Reps(C). □

Proposition 4,5 and Lemma 4,3 yield

Theorem 4.6 If C is finite complete, then so is Reps(C). □

The following trivialities are often useful.

Lemma 4.7 Let Mx be an object of Reps(C).

(i) If f : N —ï M is an arrow of Mon then (/,!): N0 Mx is an arrow of Reps(C), where !: 0 —X is the unique arrow in C and N0 is the unique functor so denoted.

(ii) If 7 : Z —ï X is an arrow of C then (¿,7):lz Mx is an arrow of Reps(C) where i\ 1 —M is the unique arrow in Mon, and 1 z is the unique functor so denoted.

Next we identify the monomorphisms and strong epimorphisms of Reps(C) when we plaee certain conditions on C,

Proposition 4.8 Let C be a category with an initial object. An arrow (/, a) of Reps(C) is monic if and only if f and a are monic in Mon and C respectively.

Proof. If / is monic in Mon and a is monic in C then (/, a) is certainly monic in Reps(C),

Suppose (/, a) : Mx —NY is monic, and g,h: L —M are arrows of Mon, Let (g,!), (h,!): L0 —Mx be the arrows of Reps(C) given by Lemma 4,7 (i). If f ■ h = f ■ g then

(/,«)• (g, !) = (/• g,l) = (f-h,l) = (/,«)• (h,!) so that g = h as required.

In a similar way, suppose 7,/3: Z ^ X are arrows of C, Let (t, 7), (t, /3) : 1 z —Mx be the arrows given by Lemma 4,7 (ii). As above, if a • 7 = a • ¡3 then

(/, «) ' 0,7) = 0, a ' 7) = {t,a- ¡3)

= (/,<*)• W),

giving the result that 7 = (3. □

Proposition 4.9 Let C be a regular category with initial object. An arrow (f,a) of Reps(C) is a strong epimorphism if and only if f and a are strong epis in Mon and C respectively.

Remark 4.10 The statement of the proposition may be strengthened as it is enough to assume that C has canonical factorizations and initial and terminal objects.

Proof. Let (j) = (/, a) be an arrow . 1 \ —^ By in Reps(C) with / and a both strong epimorphisms. Suppose that the following diagram commutes in Reps(C) where 1 is monic,

Then since C has an initial object, Proposition 4,8 gives us that ¿Mon and lq are monic, so that there are commuting diagrams in Mon and C as follows.

¿Moil

It is routine to verify that (g, (3) is an arrow of Reps(C) from BY to Cz- Thus 4> is a strong epimorphism.

Conversely, let (j> = (/, a) be a strong epimorphism. We will show that both / and a are strong epimorphisms.

Suppose the following diagram commutes in Mon with i a monomorphism.

In order to find an arrow B C making the diagram commute, we construct a corresponding diagram in Reps(C), In the following diagram Ci and are the unique representations of C and D as endomorphisms of the terminal object of C, 7 = (g,!), S = (h,!) and i =

Since (j> is a strong epimorphism, there is a unique arrow from BY to Ci whose Mon component is the required arrow.

We have shown that the Mon component of a strong epi in Reps(C) is a strong epi. For the C component, suppose the following diagram commutes in C with i a monomorphism.

Since C is regular, we may write the canonical factorization of a as x 4 v 4

Let b E B. Since / has been shown to be surjeetive, there is an a E A with /(a) = b. Define Sh to be the fill-in making the following diagram commute,

I A-10.

It is easy to see that putting Bv(b) = 4 defines a representation of B as endomorphisms of V. In Reps(C) we therefore have arrows (/, rf) \ Ax —By and (1 b, ft)'- By —By with <f> = (1B, //)(/, rf).

To show that a is a strong epi, it is sufficient to show that ¡jl is an isomorphism, Since (/, a) and (f,rj) are both strong epis and by Proposition 3,1, (1 b, ¡j) is a strong epi. Thus (1 b, ¡j) is a strong epi and a mono, and therefore

an iso. By Proposition 4,1, this implies that ¡jl is an isomorphism, as required,

We will see in the next section that the following theorem is a powerful tool for constructing bicategories of relations and divisions from Reps(C),

Theorem 4.11 If C is a regular category with initial object, then so is Reps(C).

Proof. Let C be a regular category with initial object. That Reps(C) has an initial object is simply Proposition 4,4, We proceed by showing that Reps(C) satisfies the three axioms of a regular category,

(1) Let (f,a):Ax By be an arrow of Reps(C), let / have canonical factorization A 4 C 4 B and let a have canonical factorization X 4 Z 4 Y. We construct an object Cz of Reps(C) such that (p, rf) is an arrow from Ax to Cz and (i, t) is an arrow from Cz to By. Then by Propositions 4,8 and 4,9, (i, t) is a mono and (p, rf) is a strong epi so that we have a canonical factorization of (/, a). Let c E C. Since p is a strong epi (surjection in Mon) there is some a E A with p(a) = c. Let jc be the fill-in making the following diagram commute,

BrMc))

Setting Cz(c) = 7C defines the required object of Reps(C), and verifying that (p, rf) and (i, t) are arrows is routine,

(2) That Reps(C) is finite complete is the statement of Theorem 4,6,

(3) Let (j> be a strong epimorphism. By Proposition 4,9, its Mon and C

components are both strong epis. If • —• A ■ is any diagram in Reps(C), then Lemma 4,3 implies that there is a pullbaek diagram in Reps(C) as in the following diagram where </>' is a strong epimorphism in both the Mon and C components, and hence a strong epimorphism in Reps(C), using the regularity of C and Mon,

If there is another pullbaek diagram in Reps(C) in which (j> pulls back to <j>", then there is an isomorphism e such that <f>" = o'r. Since strong epimorphisms are closed under composition with isomorphisms, <f>" is a strong epimorphism, proving that strong epimorphisms of Reps(C) are stable, □

5 Old Definitions From New

The standard definitions of relation, relational morphism and division of sets, abstract monoids and transformation representations of monoids may be exhibited as special cases of the constructions of the earlier sections,

A relation of sets f : X —Y is a function from X to the power set 2Y of Y. The composite of two relations /: X —Y and g: Y —Z is the relation g o / where g o f(x) = {z \ z E g(y) for some y E f(x)}. The following definitions are taken directly from [21]. A relation /: X —t Y is:

(a) fully defined if f(x) ^ 0 for all x E X;

(b) injective if x ^ x' implies f(x) fl f(x') = 0; and

(c) surjective if Uf(X) = Y.

The graph of / is denoted jj/ = {(x, y) \ y E f(x)}.

For this section only, regard a monoid not as a one object category, but as a set with an associative binary operation, satisfying the usual axioms. Let M and N be monoids, A relation <f>: M ^ N of monoids is a relation of sets whose graph jj<f> is a submonoid of M x N. A relational morphism of monoids is a fully defined relation of monoids. This includes the usual morphism of monoids, A division of monoids is an injeetive relational morphism of monoids. The reader may easily verify that:

(a) a relation of sets is precisely an arrow of Rel(Set);

(b) a relation of monoids is precisely an arrow of Rel(Mon);

(c) a relational morphism of monoids is precisely an arrow of Rel*(Mon); and

(d) a division of monoids is precisely an arrow of Div(Mon),

It is also easy to see that composition of the relations in (a), (b), (c) and (d) correspond to horizontal composition in the bicategories Rel(Set), Rel(Mon), Rel*(Mon) and Div(Mon) respectively.

There are corresponding notions of morphism, relational morphism and division when discussing transformation representations of monoids. These standard definitions are taken from Nehaniv's recent paper [17], Let Mx and Ny be transformation representations of monoids (i.e. objects of Reps(Set)), A morphism Mx —Ny is defined in [17] as a pair (/, rf) where /: M —N is a monoid homomorphism, and // : .V —Y is a function, such that for each x E X and each m E M, f(m)(rj(x)) = rj(m(x)). It is immediate that this is precisely an arrow of Reps (Set),

A relational morphism from Mx to Ny is a pair (/, rf) where / is a totally defined monoid relation, r] is a totally defined relation of sets and for all x EX and all m, E M,

y E f](x),p E f(rn) =>- p(y) E 7](rn(x)).

As noted in Theorem 4,11, we may speak of the bieategorv Rel*(Reps(Set)) because Set, and therefore Reps(Set), is regular with initial object. It is now routine, though tedious, to see that a relational morphism as defined in [17] is precisely an arrow of Rel*(Reps(Set)),

A division of monoid representations is a relational morphism in which both the set relation and the monoid relation are injeetive. Again, a division of monoid relations is simply an arrow of Div(Reps(Set)),

6 The Bicategory of Concrete Divisions

In this section we construct the bicategory CDiv(C) whose arrows are concrete divisions — an example of this construction was given in the introduction. This bicategory is essentially the sub-bieategorv of Div(Reps(C)) whose objects are the concrete monoids over C and whose arrows are divisions whose strong epic leg are strict morphisms of Reps(C) (see Remark 4,2),

The bicategory CDiv(C) has much to recommend it: the arrows are more

comprehensible than an arbitrary division in Div(Reps(C)) (see the examples in the introduction); there is a simple description of the hom categories (see Theorem 7,4) and the Poineare category Conc(C) of CDiv(C) has arrows which may be regarded as arrows of C making computation extremely easy (see Definition 7,5), In fact, the bieategorv CDiv(C) is so much simpler than Div(Reps(C)) that there is no need for C to be regular.

Fix a category C with initial and terminal objects. Define the bicategory CDiv(C) of concrete divisions as follows:

CDiv-I The objects of CDiv(C) are the concrete monoids over C, i.e. those objects of Reps(C) which are faithful,

CDiv-II In a similar way to Div(Reps(C)), an arrow of CDiv(C) from Ax to By is an object Sx of Reps(C) together with a mono i: Sx ^ BY and a strong epi f:Sx —Ax, such that fc = 1 v (i.e. the strong epi is strict). We shall often abuse notation by referring to the arrow (f,Sx,i) simply as S. Just as in Div(Reps(C)), a 2-cell from (f,Sx,i) to (g,Tx,j) is an arrow a: Sx —Tx such that got = / and ja = i. The following useful facts about the category CDiv(C)(AX,BY) are immediate.

Proposition 6.1 In the category CDiv(C)(AX,BY):

(i) the C component of every arrow is the identity;

(ii) the Mon component of every arrow is monic; and

(iii) every diagram commutes — that is to say, CDiv(C)(AX,BY) is a preorder.

Remark 6.2 As usual, we will sometimes identify this preorder with the corresponding poset, writing S < T if there is a 2-cell S^T.

CDiv-III The composition functor is defined as in the Span construction, and in the same manner we must make a choice of pullbaeks as part of the definition. For every diagram X —Y A Y in C, the diagram

is a pullbaek. Let D be a diagram of the form • —• ■ in Reps(C)

whose C component is of the form X A Y Y. Then by Lemma 4,3 and the completeness of Mon, we may choose a pullbaek for D

whose C component is of the form shown in Diagram 3,

Let Ax S ->• By and By T ->• Cz be arrows of CDiv(C). By the above choice of pullbaeks, their composite is given by Ax ^ BY ^

(SxB T)x

where (ttt, ic) is mono (since monos pull back to monos) and (tts, lx) is a strong epi (since Mon is regular).

Write the composite of S and T as ToS. To show that composition is a functor we need to show that if a: S =>■ S', a': S' =>■ S", /3: T =>■ T and /?': T ^ T" are 2-cells, then (a'a) o ((3'(3) and (a' o ¡3') (a o ¡3) are equal 2-cells from T o S to T" o S". But this is immediate since CDiv(C)(Ax, Cz) is a preorder and therefore any two parallel arrows are equal,

CDiv-IV The identity arrow of CDiv(C)(Ax, ax) is simply (1 ax,ax, 1 ax) and the identity 2-cell is 1ax-

CDiv-V To define the associativity isomorphism we need to find, for every object (S, T, U) of CDiv(C j {A, B) x CDiv(C) (B, C) x CDiv(C) (C, D), a 2-cell isomorphism from (UoT)oS to Uo (ToS), such that the resulting family of arrows is natural in (S, T, U). Since CDiv(C)(A, D) is a preorder, naturalitv is trivial, so it is enough to find an isomorphism from (U o T) o S to U o (T o S). Again, since CDiv(C)(A, D) is a preorder, it is enough to find any 2-cells from (IJ o T) o S to Uo (ToS) and from U o (T o S) to (U o T) o S.

(U oT)oS

U o(ToS)

To find the 2-cell from (U o T) o S to U o (T o S) observe that the diagram

(U oT)oS

commutes, so by the universal property of (T o S) there is a unique arrow S:(U oT)o5->To5of Reps(C), Since diagrams F and G both commute, the diagram

(U oT)oS

commutes. By the universal property of Uo (ToS) there is an arrow 7 : (U o T) o S ^ U o (T o S) in Reps(C), To see that 7 is a 2-cell simply notice that

(fs^s^Tos)l = fs^sS

= fsns

l(j7T(jy — IjjTTijTTiJoT

as required.

Since this argument in no way depended upon any fact about the particular spans involved, a symmetric argument shows that there is another 2-cell from U o (T o S) to (IJ o T) o S, and we are done,

CDiv-VI In the same manner as the definition above of the associativity isomorphism, it is enough to show that for any objects A and B of CDiv(C), and any arrow S:A B, there is a 2-cell isomorphism from 14 o S to S.

Consider the following diagram,

(-4 xA S)x

The arrow I is clearly a 2-cell from 1A o S to S. Since isos in Mon are monic strong epis, the arrow I is an iso in the Mon component (and the identity in the C component). Proposition 4,1 gives us that I is an isomorphism. The right identity is defined in a similar way,

CDiv-VII The coherence conditions hold trivially by the fact that each of the categories CDiv(C)(A, B) is a preorder.

Remark 6.3 It has been pointed out by Ross Street (in personal communication) that our purposes would be served equally well if, in the definition of CDiv(C), instead of stipulating that the arrows (f,Cx,i) had fc = 1 v we could simply insist that / be a cartesian arrow of the fibration of Reps(C) over Mon given by the domain functor d, which is equivalent to making fc an isomorphism. This is a somewhat prettier defintion as it makes no reference to the components of the arrows of Reps (C) and it might lead to a generalization of the CDiv construction to arbitrary fibrations over Mon, Unfortunately, it would somewhat complicate the proof of Theorem 7,4 and it would extend the text unjustifiably to give an full exposition of these matters therefore we refer the interested reader to [19] for the background to this remark.

7 The Category of Concrete Monoids

The Poineare category of the bieategorv CDiv(C) is the category which we will refer to as the category of concrete monoids over C, denoted Conc(C),

For the rest of this section we work towards a characterization of the connected components of CDiv(C)(Ax, BY) in order to understand the arrows of Conc(C), We shall use lower case Greek letters to denote arrows of CDiv(C) in order to avoid confusion with objects of Mon,

Definition 7.1 ll(f)= (f,Cx,i) is an arrow . 1 \ —^ By in CDiv(C) then write x((/>) for the monomorphism i&.X —Y.

Lemma 7.2 If <f> and i/j are objects of the same connected component of CDiv(C){Ax,By) then = xW-

Proof. Let (j> = (f,Cx,i) and ip = (g,Dx,j) and suppose there is an arrow (2-cell) a: </>—>• V- Now \ (o) = ¡c- But

ic = otcjc

since ac is forced to be the identity as noted in Proposition 6,l(i), The result now follows by a simple induction, □

The following proposition is now easy to verify:

Proposition 7.3 The construction of Definition 7.1 defines a functor x- Conc(C) — C. □

Note that x likewise defines a functor from the classifying category of CDiv(C) to C.

For any category C and any object c of C, define Subc(c) to be the full subcategory of the comma category C/c whose objects are the monic arrows into c.

Theorem 7.4 Let 4> and ip be objects of CD\v(C)(Ax, BY). The following are equivalent:

(1) 4> and if) are in the same connected component of the category CDiv(C)(Ax, By);

(2) 4> and i/j have an upper bound in the poset CDiv(C)(AX, BY);

(3) 4> and ip have a least upper bound in the poset CDiv(C)(AX, BY);

(4) XW = XW-

Proof. It is immediate that (3) =>■ (2) =>■ (1), and by Lemma 7,2 (1) =>■ (4), It only remains to prove that (4) =>■ (3),

Let (j> = (/, C\. i) and ip = (g, D\.j) be such that // = \ (/>) = xW'- X Y. We proceed in three steps:

(1) We construct an object (Ex,q) of SubR,eps(c)(-Br)-

(2) Define pMon- E A in Mon making (pMon, lx) : Ex Ax an arrow of Reps(C) and 7 = ((pMonAx), Ex,q): Ax BY an arrow of CDiv(C). Set p = (pMon, lx) in Reps(C),

(3) Show that (p, Ex,q) is a least upper bound of (j> and ip. Step 1. We have the diagram

in Mon, Let (^,5Mon) be the coproduct of (C, ¿Mon) and (D,jMon) in SubMon(-B) — that is, the submonoid of B generated by the images of C and D — with coprojections a and /3. The situation in Mon is shown in Figure 1,

Define Ex(e) as follows. If e = a(c), for some c E C then define Ex(e) = Cx(c). If e = /3(d), for some d E D then define Ex(e) = Dx(d). Otherwise, e can be expressed as a word in elements of a(C) and /3(D). Define Ex on such words by homomorphic extension. Suppose that

a(ui)(3(vi)... a(uk)(3(vk) = a(a,i)/3(bi)... a(am)/3(bm)

in E. To show that Ex is a well-defined functor, it is enough to show that Ex(a(ui)[3(vi)... a(uk)[3(vk)) = Ex(a(al)i5(bl)... a(arn)(3(bm)).

r]Ex(a(ui)(3(vi). ..a(uk)[3(vk))=rjCx(ui)Dx(vi)... Cx(uk)Dx(vk)

= BY{iman{ui))r]Dx(vi).. .Cx{uk)Dx(vk)

= BY(iMon(ui)jMon(vi) . . . iMon(uk)jMon(vk))r]

r]EX(a(ai)(3(bi) . . . a(arn)(3(bm)) = By (¿Mon(Ol)jMon(&l) ■ ■ ■ iMon(Om).7Mon(&m))»7-Since

a(ui)(3(vi)... a{uk)(5{vk) = a(oi)/3(6i)... a{am)(5{bm) applying qMon on both sides gives

iMon(«l)jMon(fl) ■ ■ ■ ?Mon(Wfe)iMon(^fe) = ^Mon (®1) jMon (^1) ■ ■ ■ ¿Mon (oTO)j]yion (Wn) ■

Therefore

r]Ex(a(ui)(3(vi)... a(uk)[3(vk)) = r}Ex(a(al)i5(bl)... a(arn)(3(bm))

and by the fact that r] is monic

Ex(a{ui)(5{vi)... a{uk)(5{vk)) = Ex(a{ai)(5{bi)... a{am)(5{bm)).

We have therefore constructed a well-defined functor Ex.

To see that (gMon, v) '1S an arrow Ex —BY of Reps(C), let e = a(oi),,, f3(bm) E E and simply notice that

r]Ex(e)=r]Cx(ai). ,.Dx(bm)

= ¿MiMon(ai)) ... BY(jMon(bm))t] = BY(qMon(a(ai)... f3(bm)))rj = BY(qmoil(e))r]

( , "Mon^ ^ < 'Mon

Fig. 1. The join of C and D in Mon constructed as the coproduct of (C, ¿Mon) and (A ¿Mon) in SubM on (B).

as required. Henceforth, q will refer to (qMon,v)-

Step 2. In this step we will omit the subscript Mon from the names of some arrows as the context makes clear the fact that we are speaking of an arrow of Mon, Define p(a(c)) = f(c) and p(/3(d)) = g(d) and extend homomorphieally. We need to show that this is well-defined. Set w\ = a(ui)fi(vi),,, a(uk)/3(vk) and w2 = a(oi)/3(6i),,, a(am)/3(bm). Suppose w\ = w2 in E, then we need to show that p(wi) = p(w2). By faithfulness of Ax it is sufficient to show that Ax(p(wi)) = Ax(p(w2)). By definition

Ax(p(wl)) = Ax(f(ui)g(vi)... g(vk)) = Cx{ui)... Dx(vk) and similarly, Ax(p(w2)) = Cx(ai)... Dx(brn). Then

r]Ax(p(wi)) = rjCxM ... Dx(vk)

= BY(i(ui)j(vi).. .j(vk))ri and likewise r]Ax(p(w2)) = BY(i(ai)j(bi),, .j{bm))rj. Now

wi = w2 => a(v,i)[3(vi)... a{uk)(5{vk) = a(oi)/3(6i)... a{am)(5{bm)

q(a(ui)(3(vi)... a(uk)(3(vk)) = q(a(al)i5(bl)... a(arn)(3(bm)) =>i(ui)j(vi).. .j(vk) = i(ai)j(bi).. .j(bm)

BY{i(ui)j(vi). ..j{vk))rj = BY(i(a,i)i(bi). ..j{bm))rj =^r]Ax(p(wi)) = r]Ax(p(w2)) (as shown above) =>■ Ax(p(wi)) = Ax(p(w2)) (by the fact that r] is monic)

as required. Thus p is a well-defined homomorphism of E onto A (it is onto since both / and g are onto).

To see that (j?Mon, lx) is an arrow Ex —Ax of Reps(C),

lxEx(a(ai)... (3(bk)) = Cx(ai)... Dx(bk)

= 1 xAx{f{al)...g{bk)) = Ax(p(a{ai)... (5{bk))lx

as required. Henceforth p will refer to (j?Mon, lx)-

Step 3. It is clear that in Reps(C), pa = / and p/3 = g, and by definition of a, ¡3 that qa = i and q/3 = j. Thus 7 = (p,Ex,q) is certainly an upper bound of (j> and ip.

To see that 7 is a least upper bound of 4> and ip, suppose S = (h, II k) is another arrow Ax BY and a': ip S and /3': 4> =>■ S are 2-cells, By Lemma 7,2 kc = V- By the universal property of (E,qMon) in SubMon(-B) there is a

unique arrOW /•.' ^ II in Mon SUCh that C^Mon = «Mon ail<i C/^Mon = /?Mon-

The pair (£, 1^) is certainly an arrow of Reps(C) since

lxEx(a(ai)... (3{bm)) = lxCx(oi) ■ ■ ■ Dx(brn)

= Hx(a'(ai))...Hx(/3'(bm))lx = Hx(t(a(ai)...[3(bm))lx.

To see that 7 is a least upper bound, it only remains to show that (£, lx) is

a 2-eell 7 S. As is an arrow from (H, ^Mon) to (E,qMon) in SubMon(-B), ZcMonC = '/Mon- Thus lx) = q, so we only need to show that /i(£, lx) = p. This is eertainlv true in the C component, so we prove it for the Mon component. Leaving off subscripts for brevity, we wish to prove that = p. In Mon, h£a = ha' = / = pa and similarly, hi, ) = pi. Thus for any element a(oi),,, f3(bm) E E, we have h£(a(ai)... f3(bm)) = p(a(ai)... f3(bm)) as required, □

As a result of Theorem 7,4 we are able to give an equivalent definition of the category of concrete monoids over C,

Definition 7.5 The category Conc(C) of concrete monoids over C has concrete monoids as objects, and an arrow from Ax to BY is a monomorphism i:X —Y in C such that there is an arrow (j> : Ax —BY of CDiv(C) with x{4>) = i- To be more explicit, it is a monomorphism i: X —Y in C such that there is an object Sx and arrows (/, lx) : Sx —Ax and (j, i) : Sx —BY in Reps(C) with / surjeetive and j injeetive. Composition is the same as in C,

As an immediate consequence of this observation we have

Corollary 7.6 All arrows of Conc(C) are monic. □

The next theorem relates functors between representing categories and functors between the categories Reps and Cone of representations over them. This theorem is illustrated in Subsection 13,2,

Theorem 7.7 (Change of Base) Let C and D be categories, and F : C ^

D a functor. Then F induces a functor F^eps: Reps(C) —Reps(D), which is faithful if F is faithful. If F is faithful and preserves monos, then it also induces a faithful functor FConc : Conc(C) —Conc(D).

Proof. Let M C be an object of Reps(C), and (/, rf) \ $x an

arrow of Reps(C), Define i7Reps($x) = F$x- M —D, Then F$x maps the object of M to F(X) in D, Write F$x = Define FReps(f, rj) = (/, Frj). To see that this is a valid arrow $ fx —^ fy in Reps(D) we need to show that for each m, E M, ^FY(f(fn))Fr] = Frj$Fx(m)- Since (/, 77) is an arrow of Reps(C) we already have that ^Y(f(m))r] = rj<&x(m), whence

Fr^Fx{rn) = Fri(F$x)(rn) = Fr]F($x(m)) = F(rj$x(rn)) = F(^Y(f(rn))rl) = F(yY{f(m)))Fr1 = fFYFrj

proving that FReps(f,r]) = (f,Frj) is an arrow of Reps(D), The funetorialitv of this definition is immediate. Clearly FReps is faithful if F is.

Suppose now that F is faithful and mono preserving. Then if $x: M —C is an object of Conc(C) then F$x is certainly a faithful functor M —D so de-

fine Fconc($x) = F®x- Let a: —be an arrow of Conc(C) with representative ((/, lx), TX, (¿Mod, ¿c)) ^ CDiv(C). Then ((/, /-'1 y).FTx, (¿Mon, Fic)) is certainly an arrow f$x —f^y in CDiv(D). Define FConc(o:) to be the Poincare class of this arrow, Equivalentlv, by Theorem 7,4 and Definition 7,5, Fconc(ct) = F(ic) and funetorialitv and faithfulness are immediate, □

Finally, we characterize isomorphisms between finite concrete monoids in Conc(C)".

Theorem 7.8 Let C be any category, M and N finite monoids, and f: Mx — Ny an arrow of Conc(C). Then f is an isomorphism of Conc(C) precisely when M and N are isomorphic monoids and f is an isomorphism of C.

Proof. Suppose f:Mx —NY is an isomorphism of Conc(C), Then / is clearly an isomorphism of C, Furthermore M and N (abstractly) divide each other, and since they are finite, they are isomorphic.

Conversely, suppose / is an isomorphism of C and M and N are isomorphic, By assumption, there is an arrow ((</>, lx), Px, /)) of CDiv(.\/y. NY). Since M and N are finite, P is isomorphic to both M and N and i is an isomorphism, Therefore by inverting (t, /) we have the arrow ((In, If), Ny, ((f), lx) • (rl,f_1)) of CDiv(.\y. ,\/v) whose Poincare class is characterized by in C, making the inverse of / in Conc(C), □

Part В - Examples

8 The Endomorphism Monoid Functor

Definition 8.1 Let С be a category. Say that С has the endomorphism extension property if for all monomorphisms // : .V —Y in С and all en-domorphisms ex: X —X, there is an endomorphism ey:Y —Y such that r]ex = еуг].

In any category С there is a concrete monoid associated with every object X, namely the monoid End(X)x of all endomorphisms of that object. In this section we suppose that С has the endomorphism extension property and show that the construction which gives, for every object X of C, the concrete monoid End(X)x is a functor from Cm to Conc(C) where Cm is the subcategory of С whose arrows are all the monies of C,

Let r] : X —Y be an arrow of Cm. Let M be the set of elements of End(Y) consisting of those elements d E End(Y) such that End(Y)Y(d)rj = r]End(X)x (cd) for some cd E End(X).

Proposition 8.2 For each d E M there is a unique cd in End(X) such that

r]Eiid(X)x(cd) = End{Y)Y{d)r).

Proof. Suppose there were cd and c'd satisfying the criterion. Then rjEnd(X)x(cd) = r]End(X)x (c'd), But since r] is monic and End(X)x is a faithful representation, cd = c'd. □

Note also that since С has the endomorphism extension property, for every с E End(X), there is some d in M such that с = cd. To see that M is a monoid, suppose d, d! E M. Then cdd> exists and equals cdcd> because

End(Y)v(dd')r] = End{Y)Y{d)End{Y)Y{d')r] = r]End(X)x(cd)End(X)x(cd/) = r]End(X)x(cdcdi) which proves dd' E M. Clearly 1Y E M.

Define a function f:M End(X) by f(d) = cd. The argument above shows that / is an epimorphism of monoids. Define a representation of M by endomorphisms of X by Mx(d) = End(X)x(f(d)).

Let i : M —End(Y) be the natural inclusion. It is then easy to show that the pair (г, rf) is an arrow from Mx to End(Y)Y in Reps(C) and that the pair (/, lx) is an arrow from Mx to End(X)x in Reps(C), Thus ((/, lx), Mx, (г, rf)) is a concrete division defined by г/: X Y and setting End(rj) = [(/, lx), Mx, (i, rj)] makes End into a functor. In summary, we have proved the following

Theorem 8.3 Let С be a category with the endomorphism extension property. For each monic arrow // : .V ^ Y of C, defining End(77) = rj: End(X)x — End(F)y makes End into a functor from Crn to Conc(C). □

It is immediate that there is functor G: Conc(C) —Cm which maps any monoid M v to X, and any concrete division represented by //: .V —Y to the arrow r] in Cm. Note that for any concrete monoid M\. the arrow 1 v of С is an arrow from Mx to End(X)x in Conc(C), It is now elementary to show that the family of arrows whose component at Mx is

1 Mx —End(X)x = EndG(Mx)

defines a natural transformation lconc(c) ^ EndG and that G is left adjoint to End so that Crn is a reflective subcategory of Conc(C) (see [16]),

9 The Monoid of Order Preserving Transformations of a Chain

In [16] A denotes the category of finite ordinals and order preserving maps between them. To be explicit, the objects of A are the natural numbers 0,1,,,, and an arrow a: n —m, is a function [n] —[m] such that i < j implies a(i) < a(j), where [n] denotes the set {1,,,, , n}, It is clear that A has the endomorphism extension property so the functor End defined above from Am to Conc(A) gives, for each object n of A the monoid On of all order preserving transformations of the set [n]. Notice that the monies of A are precisely the injeetive maps.

The problem of characterizing the abstract divisors of On has occupied a number of mathematicians since the question was posed by J, -E, Pin in 1987, and the results obtained so far make it seem like a genuinely difficult problem (see Higgins' survev [11], and the more recent work of Repnitskii and Volkov [18]).

Here we pose and answer a far more tractable problem which is to characterize the concrete divisors of On, both in the category Conc(A) and in the category Conc(Set),

Theorem 9.1 The concrete divisors of On in Conc(A) are precisely the monoids of order preserving maps on sets [m] where m < n.

Proof. Suppose Mm is a concrete monoid in Cone (A) which divides On. Then there is a monic order preserving function rn —n, which implies that m < n.

Conversely, suppose Mm is a concrete monoid in Conc(A) with m < n. As noted at the end of the last section, the unit of the adjunction between End and G is a map from M,„ to Orn in Conc(A) and the image of the natural inclusion i\m n under End is an arrow from Orn to On. Composing these, there is an arrow from M,„ to On as required, □

Theorem 9.2 The concrete divisors of On in Conc(Set) are transformation monoids which are isomorphic in Conc(Set) to monoids of order preserving maps on sets [m] with m < n.

Proof. If Pm is isomorphic in Conc(Set) to a monoid Mm of order preserving transformations where m, < n, then Pm is certainly a concrete divisor of On since, by the theorem above, there is an arrow in Conc(A) (and therefore Conc(Set)) from Mm to On.

Conversely, suppose that Pm is a concrete divisor of On. Then there is an injection / : [m] —[n] in Set such that ((</>, l[m]), Qm, (i, /)) is an arrow of CDiv(Set) from Pm to On. Write [m] = {¿i,,,,, im} such that /(¿1) < /(¿2) < • • • < f(im) and define a permutation tt of [m] by ij j.

Define P —Set to be another representation of P as transformations of [m] by ^m(p) = ^Pm(pfor each p E P. Then is certainly functorial and (lp,7r) is an isomorphism in Reps(Set) from Pm to ^m, By Theorem 7,8 and the fact that ((lp, l[m])> Pm, (lp, 7r)) is an arrow from Pm to in CDiv(Set), we have that Pm and are isomorphic in Conc(Set),

We complete the proof by showing that ^m is a (faithful) representation of P by order preserving maps. Suppose j < k E [m\. Then for each p E P

*m(p)(J)=nPm(p)K~1(J) = nPm(p)(ij) = nQm{q){ij)

for some fixed q E 4>^l{p). Similarly ^rn(p)(k) = ttQm(<l)(ik) for the same q E (t>'l{p). Now fQm(q)(i j) — On(i(q)) f (ij) and fQm

But it is easy to verify that tt = gf where g:[n] [m] is some order preserving map. Therefore, ^m(p)(j) = gOn^q))/^) while ^m(p)(k) = gOn(i(q))f(ik). By assumption, f(ij) < f(ik) and On(i(q)) and g are order preserving. Thus ^m(p)(j) < ^m(p)(k) as required. □

10 Concrete Matrix Monoids

Let if be a field. Then Mat,/, is the category whose objects are the natural numbers 0,1, 2,... and arrows m, n are the n x m, matrices over K.

In this section we exhibit an arrow in the category Conc(Mat^:). Informally, a concrete monoid Arn over Mat,/, can be viewed as a set of m x m matrices (including the identity) which is closed under matrix multiplication. Suppose there is a submonoid Q. form

m of Am whose elements are matrices of the

for some n + p = rn. Multiplication of two matrices of this form is given by

ßß' *

This observation yields concrete monoids B„ and C„ where B„ is the set of

top left n x n corners of matrices in Qrn and Cp is the set of bottom right p x p corners of matrices in Qrn. There are arrows Bn —Arn and Cp —Arn in CDiv(Mat^) as follows.

Formally, let A be an abstract monoid such that Am is a faithful representation of A in Mat if. Let Q be the abstract submonoid of A whose elements are precisely those whose representation under Arn are in the set Qrn of matrices described above. Let Qn be the (not necessarily faithful) representation of Q by mapping each q E Q to the top left n x n corner of Arn(q). Let ¿Mon^ Q —A be the natural inclusion, and

Let /moii: Q —B simply map each q E Q to the element of B which is represented by the matrix Qn(q). The reader may easily verify that putting /MatA- = Inxn makes (f,Qn,i) into an arrow Bn —Arn of CDiv(Mat/f), and hence ¿MatA- is an arrow Bn —Arn in Conc(Matif), Similarly

0 nxp Ipxp

is an arrow from Cp to Arn in Conc(Matif),

(Note: There is a well known faithful functor from Mat,/, to the category Vect^ of vector spaces over K. Theorem 7,7 therefore allows us to regard the above arrows as arrows of Conc(Vectif),)

11 Transformation Monoids

Questions about monoid representations over Set motivated the development of this work where the category Conc(Set) was discovered as the codomain of the functor which corresponds to the Catalan construction of [20], Though it is the motivating example, the Catalan construction is not discussed here. Rather, we focus on two issues of more central importance to semigroup theory: the Cavlev representation and the subgroup structure of a transformation monoid.

11.1 The Cayley construction for Set

The well known Cayley construction can be viewed as a functor Cay. Mon — Reps(Set), The Cayley construction gives, for any monoid M a representation Cay(M): M Set where the object of M is mapped to the set \M\ of elements of M. If / : M —N is a monoid homomorphism, then denote by |/| the function from the set \M\ to the set |iV| which is induced by /, Then

it is easy to verify that (/, |/|) is an arrow Cay(M) —Cay(N) in Reps(Set), and that setting Cay(f) = (/, |/|) makes Cay into a functor,

11.2 Subgroups of transformation monoids

When studying semigroups, Green's relations are an important source of structural information (see [12]), In particular, Green's % relation identifies the maximal subgroups to be among the basic building blocks of a semigroup. Here we show that in a transformation monoid, the subgroups are permutation groups which can be found as concrete divisors of the original transformation monoid.

Let M be a monoid, and let S be a subsemigroup of M. not necessarily containing the identity. We say that S is a subgroup of M if there is an idempotent e E S, such that for each s E S:

(i) es = se = s] and

(ii) there is a unique t E S with st = ts = e.

The following result formalizes in our setting the fact (folklorie in semigroup theory) that if S is a subgroup of a transformation semigroup M\. then M v restricts to a faithful representation of S as permutations of a subset Y ofX.

Theorem 11.1 Let Mx be a transformation monoid and S an (abstract) subgroup of M. Then there is an arrow

Sy —^ Mx

in Conc(Set), where Sy is some faithful permutation group representation of S.

Proof. Let e E S be the identity of the group and put Y = Im(e) C X. Then S acts on Y since for all g E S, Im(g) = Im(eg) C Im(e) = Y. In fact, we will show that the representation of S as transformations of Y is a faithful permutation representation. Since e is idempotent, it acts identically on Y. If g E S, then g: Y —Y is surjeetive, since for each y E Y, y = e(y) = gg^l(y). Similarly, g is injeetive on Y since its action is invertible.

It only remains to show that the representation of S as permutations of Y is faithful. Suppose gi and g2 are such that gi(y) = Wi{y) f°r aH V £ Y- Then 9ie = ifee since Im(e) = Y. But since Mx is a faithful representation, this implies that gi = g±e = g2e = g2 as required.

We now give an arrow in CDiv(Set) from Sy to M\. There are two cases to distinguish. The first case is when 1M E S so that e = 1 m and S is actually a submonoid of M. In this case let ¿Mon: S —M and ¿get: Y X be the natural inclusions, then i = (¿Mon,?Set) is clearly an arrow of Reps(Set), Thus (lsY,Sy, i) is a concrete division of Mx by Sy.

On the other hand, if 1 m £ S, let S1 be the monoid formed by adding a new, disjoint identity element 1 to S, and let Sy be the monoid representation of S1 as endomorphisms of Y induced by the action of S on Y. Define ¿Mon^ S1 —M to be the unique monoid morphism which restricts to the natural inclusion of S into M. Let iset - Y —X also be the natural inclusion. Let fwon- act identically on S and map 1 to e, and set /set = 1 y- Then

it is easy to see that i: Sy —>■ M\. and /: Sy —>■ Sy are arrows of Reps(Set), making (f,Sy,i) the required concrete division. Thus, in either case, ¿get is an arrow from SY —Mx in Conc(Set), □

12 The Theory of Automata

A nice result of the developments in Part A of this paper is that we are able to make functorial the so called 'action monoid' construction in automaton theory in a more satisfying way than has been done before.

To give an account of this new development we begin in Section 12,1 by recapitulating the elements of the theory of automata. The new construction of the action monoid functor is then given in Section 12,2, To the author's knowledge, the only other functorial approach to the action monoid construction is given in [8]. We sketch this approach in Section 12,3 to allow the reader to draw their own conclusions about the relative merits of the two approaches,

12.1 Introduction to the theory of automata

In [8] Ehrig gives a categorical treatment of C-automata whose inputs, outputs and states are objects of any closed monoidal category C, For completeness, we give a brief account of this theory, but, in order to avoid alienating the reader and to simplify the exposition, we restrict our attention to the case C = Set, It should, however, be borne in mind that all of the developments in this section hold for any closed monoidal category. The interested reader may refer to [10] for a similar exposition where the automata have the additional (but for our purposes, unnecessary) feature of an initial state.

Fix sets X and Y, referred to as the input and output alphabets respectively, An (X,Y) automaton (or 'automaton' for short) is a pair ($S,A), where <&s'-X* —Set is a monoid representation landing on some set S, and A: S —Y is any function. The set S is referred to as the set of states, and A is called the output function. This corresponds to the well established notion of a Moore automaton without initial state.

Remark 12.1 If S and X are finite sets, an automaton may be represented by a directed graph with vertex set S, each vertex s labelled by A(s) of Y and for each .r e A' an edge labelled x from s to $s(a;)(s). So for each word w = Xi... xn of X*, and each s E S there is a path s 4 x^(s) • • • 4 w(s). If, in addition, Y is the set {0,1}, then our definition corresponds to the

familiar notion of a finite state acceptor, where the state s is terminal if and only if A(s) = 1,

For example, consider the automaton below with input alphabet X = {a, 6}, output alphabet Y = {0,1,2} and state set S = {s, t, u}.

The representation of X* as endomorphisms of S which this describes is given as follows: the single object • of the category X* is mapped to S, the letter a is mapped to the 3-eyele (s t u) and b is mapped to Is,

Let A = ($s, A) and B = (^T, p) be automata, A morphism <t>\ A —y B is a function (f)\S^rT such that (1 x*,4>) ^s ^t is an arrow of Reps(Set) and A = p<j). If (j> is onto, then we say that B is a quotient of A.

Definition 12.2 The category of (X, F)-automata and their morphisms will be denoted Aut(X, Y).

Let A be an automaton as above, and let s E S be a state of A. Define a function fs: X* Y by fs(w) = A($s(w)(s)), Call fs the behaviour of A at s. As a heuristic, fs is the 'function computed by A when it is started in state s\ In the example above, fs is the number of times (modulo 3) the letter a occurs in the input. The set E(A) = {,/',. -v e is called the external behaviour of A and we say that A realizes the behaviour E(A).

Let / be any function X* —Y and let u E X*. Define the left shift of / by u to be the function

<?uf: X* y Y

U) I—> f(wu).

Denote by Yx* the set of all functions X* —Y. The operator a is clearly a left action of X* on Yx* since ouov = ouv.

It is easy to see that for any automaton A, the external behaviour E(A) is closed under left shift — if s is a state of A, and x EX then ax(fs) = fx(s)- In general we define a behaviour to be a subset P C Yx* which is closed under

left shift by words of X*. In the sequel we will see that every behaviour is realized by some automaton.

Definition 12.3 The category Beh( A. Y) has behaviours as objects with an arrow I' —(} if I' C Q. One can think of an arrow P —Q as the canonical inclusion function.

Let A and B be automata as above. The construction of E(A) from A is made functorial by defining, for any arrow o: A —H. the arrow E((/>): E(A) — E(B) by fs i—Y f(i>(s)- To show that E((f) is a well-defined arrow of Beh( A. Y) we merely show that fs and f^s) describe the same function X* —Y. Let w e X* then

f8(w) = \($s(w)(s)) = p<t>($s(w)(s))

= P^t(w)((/>(S)) = U(s)(W)

as required. It is now immediate that E is functorial and it is referred to as the external behaviour functor A lit, (A-. Y) —Beh( A. Y).

Let P be a behaviour. Then define an automaton N(P), the minimal realization, of P to have state set P, and action of A* on P given by left shift. Define the output function A by A (p) = p( 1) for each p e P. Suppose P and Q are behaviours with P C Q. Let the canonical inclusion function be i: P Q. Then define N(i):N(P) N(Q) by p i(p). That N(i) is a well-defined morphism of automata and that N is a functor Beli( A . Y) —Aut,( A'. Y) are both easy to verify.

The following theorem is proved in [8] and [10],

Theorem 12.4 The external behaviour functor E is left adjoint to the minimal realization N. □

Remark 12.5 Let A = ($s, A) be an automaton. The unit of the adjunction //: 1 Ant; y.V! XE is given by

t]A: A—yNE(A)

SI->fs-

This definition makes each arrow r]A onto so that NE(A) is a quotient of A. The reader familiar with the theory of automata, will recognize r]A as the Nerode quotient and NE(A) as the minimal automaton of A. (That is, the minimal automaton with behaviour E(A).)

Theorem 12.6 EN is the identity functor o/Beh(A, Y). Consequently, if P is any behaviour, N(P) has behaviour P.

Proof. Since Beh( A. Y) is just a poset, it is sufficient to show that EN is the identity on objects. Let P be a behaviour. Then N{P) has state set P, with action of X* on P given by left shift and output function 7 which maps

p to p( 1), Then EN(P) is the set {fp | p E P} where fp(w) = y(<Jw(p)) = ow{p){ 1) = p(w) proving that EN(P) = P. □

We are now able to explain why N is called the minimal realization functor. Theorem 12,6 shows that N(P) is an automaton which realizes the behaviour P. If A is any automaton with the behaviour P, then N(P) = NE(A) which is a quotient of A, by Remark 12,5, Thus, if we partially order the automata with behaviour P by A y B if B is a quotient of A, then N(P) is the minimum element of this poset.

Definition 12.7 Denote by Autmjn(JY", Y) the full subcategory of Aut (X, Y) whose objects are minimal realizations of behaviours.

Theorem 12.8 The minimal realization functor N is full and faithful, allowing us to regard Beh( A. V) as isomorphic to Autmjn(X,Y). Since the inclusion N has a left adjoint, Autmjn(X, Y) is a reflective subcategory of Aut(X,y) (see [16, p.89]).

Proof. The faithfulness of N follows immediately from the definition. To see that N is full, suppose <f>: N(P) —N(Q) is an arrow of A lit, (A'. Y) for behaviours P and Q. Then as proved earlier when showing E is functorial, for each state s of N(P), the behaviour fs of N(P) at s and the behaviour f^ of N(Q) at 4>(s) are equal. But a simple calculation shows that the behaviour of N(P) at s is s itself. Thus o(.s) = f^ = fs = s, showing that (j> is the canonical inclusion of the state set of N(P) into the state set of N(Q). That is, 4> is also an arrow of Beh( A . Y). □

Corollary 12.9 Every arrow of Autmjn(X,Y) is injective as a function on state sets. □

12.2 The action monoid construction

Now, to the construction of the action monoid as a functor O: Autmjn(JY", Y) —y Conc(Set),

Recall that the action monoid 0(A) of an automaton A = ($s, A) is the concrete monoid of endomorphisms of the set S consisting of elements {$s(w) | w E X*}. In the example above, the action monoid would be the cyclic group generated by the 3-evele (s t u).

Let A = ($s, A) and B = p) be minimal automata, and <t>\ A—y B an automaton morphism. Then 0(A) is a faithful representation of the monoid M = X*/ landing on the object S of Set, The action of 0(A) is given by = $(w) where is the class of w. (Refer to Section 4 for the definition of Let N be the monoid X*/ Then Q(B) is the

corresponding faithful representation of N by endomorphisms of T.

Then since every arrow of Autmjn is injective, S embeds in T, so that Define the representation Ns: N Set by the action iV5([ii;]^)(s) = Then Ns embeds in fi(B) by (ljv, 4>). The quotient map ip: N M

defined by [w]^ gives a strong epimorphism (ip,ls)'.Ns —0(A) in

Reps(Set), Then the triple ((ip, Is), Ns, (In, 4>)) '1s an arrow from 0(A) to Q(B) in CDiv(Set).

Since x(('1I>,1s),Ns,(1n,(I>)) = 4> we nia.v now define i>(o) = o: i>(. I) — fl(B) in Conc(Set), This completes the definition of the action monoid functor fi: Autmjn(JY", Y) —Conc(Set),

Remark 12.10 The reader will note that O may be composed with NE to give a functor from the category Aut(A, Y) to Conc(Set) rather than from Autmjn(A, Y). However, for any (X,Y) automaton A this functor computes the action monoid of its minimal automaton NE(A).

12.3 Ehrig's action monoid functor

In [8, §5,6], Ehrig defines a functor from a category Aut(S') of automata to a category of transformation monoids, whose object function is precisely the action monoid of the automaton (cf. Remark 12,10),

Fix some set S. An S-automaton is simply an object —Set of

Reps(Set) landing on S, where X is any set. In other words, an ¿^-automaton can have any input set but it has state set S. Let B = X'* —Set be another ¿^-automaton, An ¿^-automaton morphism from $ to ^ is a homo-morphism a: X* —X'* such that for each w E X*, $(w) = xifa(w). Therefore the action monoid of $ is contained in the action monoid of Notice that this definition of morphism is just a strict morphism of Reps(Set), Denote by Aut(S') the category of 5-automata and their morphisms.

The monoid of all endomorphisms of the set S will be denoted End(S). Let End(S') be the poset of submonoids of End(S) regarded as a category where arrows are inclusions. It is now obvious by the definition of a morphism in Aut(S'), that mapping every A in Aut(S') to its action monoid in End (S) describes a functor,

13 An Introduction to Vines

In this section, we give an informal introduction to Lavers' theory of vines [15], and from this basis, develop the category Vine of vines and investigate the category of concrete vine monoids. The reader is assumed to be familiar with Artin's theory of braids [2], The category Vine is not regular (since it is not finite complete) and so provides us with an example of a category from which we can construct Conc(Vine) but not Div(Reps(Vine)),

In the following discussion [n] will refer to the set {1,2,..., n}, with the usual total order. Fix n > 0. An n-braid is a set of n arcs in B3 from the n initial nodes I(n) = {(¿,0,1) | i E [n]} to the n terminal nodes T(n) = {(¿,0,0) | i E [n]}, each of which is strictly decreasing in the ^-coordinate. Every braid may be regarded as a product of the 2(n — 1) generators, pictured below for 1 < i < n — 1.

1 i i + 1 II

Two braids are regarded as equivalent if thev can be deformed, one into the

Rather than give a formal definition for vines, we rely on the reader's intuition about braids and some illuminating examples, A vine from I(n) to T(n) may be written as a product of the generators Oi and a^1 as pictured above, together with the generators gi shown below, where 1 < i < n — 1,

1 i i + 1 II

As a result, the strings of a vine may merge, but not separate. Among the homotopv equivalence relations we have a^gi = = gt, meaning that

strings can twist and untwist about a join. Some examples of equivalent vines follow.

(Note: The word 'vine' may be taken to mean either the set of ares or, the homotopv equivalence class of the set of arcs, depending on the context.) As with braids, composition is given by concatenation and shrinkage, and is a well-defined and associative operation on homotopv equivalence classes of vines. Note that if a string in the concatenation is not connected to any initial node, it is homotopieally shrunk to a point. An example of composition of vines follows.

The set of equivalence classes of vines is a monoid with the same identity element as the braid group — a string connects the ¿th initial node to the ¿th terminal node with no intertwining or joining of the strings. Defining relations for the monoid of all vines on n nodes are given in [15].

Lavers' vines always go from n initial nodes to n terminal nodes. We make here the straightforward generalization that a vine may go from J(n) to T(m) where n is not necessarily equal to m — it consists of n strings connected to each of the n initial nodes, each string ending at a node in T(m). This will be known as an n m vine. Composition of vines is then only allowed when the number of terminal nodes of the first vine is equal to the number of initial nodes of the second. This generalization permits us to define the category Vine whose objects are the natural numbers 0,1,2,... and whose arrows from n to m, are precisely the (homotopv equivalence classes of) n —m, vines.

Definition 13.1 If v is an n —rn vine, let v denote the map [n] —[rn] that maps i to k if the ¿th string meets the A;th terminal node (A;, 0,0) e T(m). The map v will be called the function associated with v.

Proposition 13.2 There is a functor Vine —Set which takes n to [n] and the n —m-vine v to v. □

A simple vine is one which is homotopic to a vine in which the strings

never intertwine. Some simple vines are pictured below.

injeetive.

Proposition 13.3 The simple n n-vines form a monoid isomorphic to the 'monoid of order-preserving transformations of {1,2,... ,n}. The isomorphism is given by mapping v to v, the function associated with v. □

The following theorem is a direct consequence of [15, Theorem 2],

Theorem 13.4 Every vine u : m n can be written u = vb where b is a m —y m-braid and v:m n is a simple vine. □

Lemma 13.5 Every braid n p, n > 0 has a left inverse.

Proof. Let /: n —p be a braid. Consider the picture obtained by reflection of / in the x — y plane. This may not be a vine since there is not necessarily an arc emanating from every node in the top plane of the reflection. Construct a vine by adding arcs emanating from top nodes which are not already at the top of an arc. The bottom nodes at which they terminate may be chosen arbitrarily. Call the vine so obtained /', Then it is clear that f'f = 1„, □

An example of the construction of Lemma 13,5 is pictured below.

Proof. By Lemma 13,5 we see that braids n —y p, n > 0 are certainly monic. Further, since there are is only one arrow into 0, every braid 0 —^ p is monic, thus all braids are monic. Conversely, suppose the vine u \ m —y n is not a braid. Write (by Theorem 13,4) u = vb where b is a m, —m-braid and v: rn —n is a simple vine. Let bbe the (group) inverse of b. Since vb has joins, there must be consecutive nodes indexed by i, i + 1 in I(m) such that

v(i) = v(i + 1). Let s: 2 —rn be a braid whieh connects 1 E 1(2) to i E T(rn) and 2 E 1(2) to ¿ + 1 E T(m). Let i:2->mbea braid which connects 1 E 1(2) to i + 1 E T(m) and 2 E 1(2) to i E T(m). Then it is clear that

ub^ls = vbb^ls = vs = vt = nh '/.

But since b 1 is monic, b ls ^ b lt, proving that u is not monic, □

The category Vine has initial object 0 and terminal object 1, Thus we may construct the category Conc(Vine), An object of Conc(Vine) is called a vine 'monoid.

13.1 An arrow of Cone (Vine).

Consider the vine monoid M3 on 3 strings generated by p as in the following diagram.

• • •

The single generator p of M3.

This is clearly infinite, and does not contain the two element monoid N2 (shown below) as a submonoid.

The vine monoid N2 with elements {l,o}.

However N2 may certainly be found as a divisor by the division ((t|, 12), M2, (1m, where M2(p) is defined to be the 2-vine o, r] is af: 2 3 (shown in Figure 2), and t) : M ^ N is the natural quotient. To see that (1m, v) '1s an arrow of Reps (Vine) simply notice that r]M2(p) = M3(p)r] as in Figure 2. Note: The arrow r] is not the simplest arrow from N2 to M3 in Cone (Vine). We deliberately chose a less obvious arrow in order to show the robustness of the intuition with respect to the possibilities allowed by the definition.

Fig. 2. From left to right: 77, M${p)r) and r]M2(p).

13.2 From Vine to FreeGrp

Let {xi,x2, ■ ■ ■} be an infinite alphabet. Denote by Fn the free group on the generators {a^,.. .xn}. Let FreeGrp be the category whose objects are Fn,n > 0, and whose arrows are all group homomorphisms between them. The trivial group is free of rank 0, so that it is initial and terminal in both FreeGrp and FreeGrp We can therefore speak of the category Cone (FreeGrp0^), Firstly, it is easy to see that

Proposition 13.7 Surjective maps in FreeGrp are epimorphic. □

Artin's famous representation theorem states that every n —n-braid defines an automorphism on the free group of rank n. This proceeds by identifying the free group with generators {xi,... ,xn} with the homotopv group of the plane with a puncture at each of the coordinates of I(n). Then the braid acts on the fundamental group as in the following diagram.

Via the observation that this action can be reversed, we formulate the following representation theorem for vines. It is the first part of [15, Theorem 6].

Theorem 13.8 The monoid Vn of all n —n vines has a faithful contravariant representation as a monoid of endomorphisms of a free group Fn of rank n. The representation is induced by a mapping Vn End(Fn) defined by

C {si):xj

ifj i {M +1}

XjXjjj-xXj if j — i Xi ifj = i + l

C {g%)-Xj

Xj ifj£{i,i +1} 1 if j = i

XiXi+1 j = i + 1

Fig. 3. The contravariant action of a vine on a generator of the fundamental group of the 3-punctured plane.

This representation theorem is now used to define a functor T: Vine — FreeGrp0^ which is faithful and preserves monos, and hence induces a faithful functor Cone (Vine) —Conc(FreeGrp°P),

Let m, < q be positive integers. Define pq,m: Fq —Fm to be the homomor-phism which acts identically on the generator Xi if i < m, and maps a^ 1 otherwise. Let im.q \ Fm Fq be the natural injection.

Let v:m ^ n be a vine and let q be greater than both m, and n. Define lv: Q Q to be the vine which is the same as v from the first m, starting nodes to the first n terminal nodes but which has a simple vine from the starting nodes numbered m +1 through q, to the qth terminal node, as in the following diagram.

The vines t;: 4 —3 and 6^6.

When there can be no confusion as to the value of q, we will often write 7„ instead of 7^,

Definition 13.9 For each object n of Vine define T(n) to be Fn in FreeGrp, Let v: m n be a vine, and let q be any integer greater than both m and n. Then r(w) is the homomorphism pq,mÇ,{lÎ)in,q'- Fn Fm.

This definition is independent of q and corresponds to the action described by Figure 4.

The following proposition is clear.

Fig. 4. The geometric interpretation of the action of a vine v: 3 —» 2 as a homomor-phism T(v) : F2 ^ Fs.

Proposition 13.10 Let k m, n be a diagram in Vine, and let q be greater than k, m and n. Then 7^7^ = □

Lemma 13.11 T is a functor Vine —FreeGrp0^.

Proof. It is clear that r(l„) = lFn. We just need to show that for each diagram k -A m, -A n in Vine, r(tm) = r(u)r(?;) in FreeGrp, Fix q > max{k,m,,n}. By definition r(vu) = pq,kC{lvu)in,q, while T^T^v) = Pq,kC{lu)im,qPq,mC{lv)in,q- Now the image of C(lv)in,q lies in the subgroup of Fq generated by (xi,..., xm) upon which pq.m acts identically, so that T(u)T(v) = Pq,kC{lv)C{lv%,,q-

By Theorem 13.8, C is a eontravariant monoid homomorphism on objects so that C(7«)C(7t>) = C(7w7«)- But by Proposition 13.10 jvju = jvu so that r(u)r(7;) = Pq^CilvuJi^q = T(vu) as required. □

Lemma 13.12 The functor T is faithful.

Proof. Let u, v. m —V n be vines, and let q be greater than both m and n. For the remainder of this proof p will denote pq,m and i will denote in,q. Suppose r(u) = r(w). Then for all generators Xj of Fn, p((jv)i(xj) = p((lv)i(xj), whence, for each generator Xj of Fq with j < n,

(4) PC(lv)(Xj) =PC(lu)(Xj)-

If C(t«j) were n°t equal to C(Tm); then by the construction of 7 they would only differ on some generators xjt,i < n. In combination with equation (4) this yields p((jv)(xj) = p((ju)(xj) for all j < q.

Suppose C(lv) # C(7«)> then for some xi}i < n, ,)(xi) # C(tu)(xi) while both C(lv)(xj) and ((jy)(xj) are elements of the subgroup (xi,... ,xm) of Fq, by the construction of 7. But p is injeetive on this subgroup, therefore C(tv){xi) = C(tu)(xi) for all generators Xi of Fq. By Theorem 13.8, this implies that 7„ = 7„ and hence, by the construction of 7, v = u. □

Lemma 13.13 If v is a monic vine then T(v) is a surjective homomorphism, and therefore a monic arrow of FreeGrp

Proof. Since a monic vine is a braid (b, say), any generating loop Xi at the top may be "pushed down" via Artin's representation to get a loop a at the bottom. When a is acted upon by T(b) we get xjt back again. Thus, every generating loop is in the image of T(b) as required. □

The preeeeding three lemmas in combination with Theorem 7,7 show that Theorem 13.14 The functor T induces a faithful functor

14 Closing Comments

The material in this paper leaves open possibilities for improvement and extension, two of which we discuss here.

Firstly, as the development above is expressed in terms of monoids, it poses difficulties for studying concrete semigroups. Despite the fact that any concrete semigroup which is not a monoid may be freely endowed with an identity to make it a concrete monoid, it is not obvious how to distinguish the concrete divisors of the original semigroup from those which divide only the concrete monoid. An interesting problem for further reseach would be to generalize our constructions to concrete semigroups (perhaps using the work of Tilson [22] on semigroupoids) and to algebras in general.

Two of the more notable theorems of universal algebra are the variety theorems of Birkhoff, and in the case of finite algebras, Reitermann, in which, roughly speaking, equational classes of (abstract) algebras of a particular type are characterized as classes closed under division and direct product.

The foregoing material in this paper gives rise to the question of whether there are 'nice' closure operations on concrete monoids giving rise to a 'concrete variety' with some simple description analogous to a set of equations.

Since we have defined concrete division, our analogy demands that we find some operation on concrete monoids corresponding to direct product. Once again, the examples are obvious but a general definition has as yet eluded the author.

Example 14.1 ['Product' of transformation monoids] Given transformation monoids M\ and NY we may represent the abstract monoid M x N by transformations of XUY faithfully by (m, n)(x) = m(x) and (m, n)(y) = n(y). So, in this way we can define a 'product' transformation monoid by Mx ® NY =

For the purposes of the following illustration, let a transformation / of the set {xi,... ,xn} be written

Cone (Vine) —Cone (FreeGrp(>l>).

(M x N)xoy-

f{xl) f{x2) • • • f{xn)

Suppose

and suppose

then by the definition above,

Mx <g> NY

12 3 4 12 3 4

12 3 4 2 13 4

12 3 4 12 3 3

12 3 4 2 13 3

The reason we quoted the word 'product' is that MX®NY is not generally the categorical product of Mx and NY in the category Cone (Set), This is obvious, since if X = 0 and Y ^ 0, then there can be no arrow (projection) of Conc(Set) from (M x N)Y to M®. In fact, the existence of products in Conc(Set) is yet to be determined.

Example 14.2 ['Product' of vine monoids] Let m, and n be non-negative integers. Given vine monoids Mm and Nn, we can form a concrete vine monoid Mm ® Nn as follows. Define a representation rm+„: N —Vine by

b I—y

for all b E N. That is, we make an m, + n m, + n-vine Tm+n(b) from the n n-vine Nn(b) by juxtaposing the m, identity vine on its left. In a similar way we can define Arn+n: M Vine by juxtaposing the n-string identity on the right.

We now define a faithful representation M,„ ® Nn of M x N landing on rn + n as the homomorphic extension of the map defined by

(a, A m+n(a)

(l,b)^rm+n(b)

for all a E M and b E N. By way of example, let

and let

Then .1 /•_> 0 N2 is the concrete vine monoid with the two generators

As in the previous example, M,„ ® Nn is not generally the categorical product of M,„ and Nn, and the existance of products in Cone (Vine) is yet to be determined.

One might be lead to conjecture that the constructions above are described for a general representing category C by a functor

whose object function is Mx x Ny (M x iV)xjjr for some representation of MxN as endomorphisms of X ]J Y. In the case that C = Set this is plausible since the disjoint union XUY is the coproduct X ]J Y. This conjecture is, however, seen to be untrue when C = Vine since coproducts do not, in general, exist in Vine, We leave the verification of this simple fact as an exercise to the reader.

Acknowledgement

Giulio Katis and Steve Lack greatly assisted the author with their explanations of the elements of category theory, Tim Lavers introduced the author to the theory of vines and suggested the method of proof of Lemma 13,11, The author's supervisor, David Easdown was unsparing of his time in proofreading this manuscript and suggested many improvements.

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