Scholarly article on topic 'How Iterative are Iterative Algebras?'

How Iterative are Iterative Algebras? Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — Jiří Adámek, Stefan Milius, Jiří Velebil

Abstract Iterative algebras are defined by the property that every guarded system of recursive equations has a unique solution. We prove that they have a much stronger property: every system of recursive equations has a unique strict solution. And we characterize those systems that have a unique solution in every iterative algebra.

Academic research paper on topic "How Iterative are Iterative Algebras?"

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Electronic Notes in Theoretical Computer Science 164 (2006) 157-175

www.elsevier.com/locate/entcs

How Iterative are Iterative Algebras?

Jiri Adamek1 '3 Stefan Milius1

Institute of Theoretical Computer Science, Technical University, Braunschweig, Germany

Jiri Velebil2 '3

Faculty of Electrical Engineering, Technical University, Praha, Czech Republic

Abstract

Iterative algebras are defined by the property that every guarded system of recursive equations has a unique solution. We prove that they have a much stronger property: every system of recursive equations has a unique strict solution. And we characterize those systems that have a unique solution in every iterative algebra.

Keywords: iterative algebra, guarded equation, strict solution, extensive category 1991 MSC: 68Q65, 18A15

1 Introduction

The aim of the present paper is to show that iterative algebras, i.e. algebras with unique solutions of all guarded systems of recursive equations, have solutions of unguarded systems as well. In fact, we introduce a natural concept of a"strict" solution (which is one that assigns to every ungrounded variable the result and prove that iterative algebras have unique strict solutions of all systems of recursive equations.

The motivation for our paper is two-fold. Firstly, in the paper of Evelyn Nelson [15] which introduced iterative algebras as a means to study the iterative theories of Calvin Elgot [10] (see also a very similar concept of Jerzy Tiuryn [16]) a complete

1 Email:{adamek, milius}@iti.cs.tu-bs.de

2 Email:velebil@math.feld.cvut.cz

3 Support by the grant MSM 6840770014 of the Ministry of Education of the Czech Republic is acknowledged.

1571-0661/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.entcs.2006.06.010

characterization of all uniquely solvable systems is provided. We show in our paper a categorical generalization of this: we introduce the concept of a a preguarded system of equations, and prove that these are precisely the systems with a unique solution in every iterative algebra. Secondly, our paper is the first step in a "reconciliation" of iterative algebras and iteration algebras of Stephen Bloom and Zoltan Esik [8]. The latter are algebras where all systems of recursive equations have solutions, and a choice of solutions subject to axioms is performed; the motivation stems from continuous algebras on CPO's, where recursive equations always have the least solution. The "reconciliation" mentioned above has two steps: one, the subject of the present paper, is to show that every iterative algebra has a "canonical" solution of every system of recursive equations. The other step, which we attend to in the paper [2] under preparation, is to show that these canonical solutions satisfy the axioms of iteration algebras. Observe that for ungrounded variables which are those where the given system of equations contains a cycle of length 1:

X « y y « x

or 3, etc., the least solution always assigns the value ±. And, on the other hand, ungrounded variables obviously force us, when considering unique solutions in iterative algebras, to restrict ourselves to systems that are (in a specified sense) guarded because one cannot require that for example x « x has a unique solution! Based on ideas of [8] we work with algebras having a global constant ±, and then we define a strict solution of a system of recursive equations as a solution assigning ± to every ungrounded variable. Our main result is:

iterative algebras have unique strict solutions

(of arbitrary recursive systems). This holds for H-algebras where H is a finitary endofunctor of a suitable category (such as Set or Set1 or Pos). Recall that free H-algebras form a monad F so that every algebra A can be described as a monadic

algebra a: FA -A. Recursive systems of equations can be represented by

morphisms

e: X-» F(X + A) (1)

where X is a finitely presentable object (of variables). An equation morphism e is called guarded if it is disjoint from the injection of variables

i0 = X -isU. X + A Vx+A > F{X + A).

J. Adâmek et al. /Electronic Notes in Theoretical Computer Science 164 (2006) 157-175 159

A solution is a morphism et : X-s- A such that the

F(X + A)-

F [et ,A]

commutes. By definition, an algebra A is iterative if and only if for every guarded equation morphism there exists a unique solution. In order to formulate, in the present generality, the idea of ungrounded variables, we compute the "first derived"

subobject ¿1 : X\ >—X as a pullback of the above embedding ¿0 : X-F(X+A)

along e. In the category of sets X1 C X represents the variables that e maps to X. And ei is the restriction of e. Then we form the "second derived" subobject X2 (representing variables that e maps to X1) as a pullback of X1 along e1, etc:

-Xo = X—^F(X + A)

il 0 io v '

Each in is easily seen to be a coproduct injection, and thus Xn = Xn+i + Xn+i where in+\: Xn+i >—Xn is the complementary coproduct injection of in+\: Xn+i >—Xn. In the category of sets X\ C X are the variables that e maps outside of X, then X2 are the variables that need two steps to be mapped outside of X, etc.

Definition. An equation morphism with X = X± + X2 + X3 + • • • is called pre-guarded.

In order to prove our theorem above, we demonstrate that in an iterative algebra

(i) every pre-guarded equation morphism has a unique solution, and

(ii) every equation morphism e: X-F(X+A) can be modified to a pre-guarded

equation morphism /: X->■ F(X + A) such that solutions of / are precisely

the strict solutions of e.

We work at the beginning with cia's (completely iterative algebras), where the restriction that the object X of variables be finitely presentable is lifted. This makes the theory of pre-guardedness and strictness simpler. Iterative algebras are then treated in the last section.

Related Work. For endofunctors of Set the unique existence of strict solutions has been proved by Larry Moss [14] and Stephen Bloom et al. [6], [7]. Our purely categorical proof is independent.

2 Extensive Categories, cia's and Iterative Algebras

The aim of this section is to shortly recall the three concepts in the title as a preparation for the theory presented further. Given an endofunctor H of a category A

160 J. Adàmek et al. / Electronic Notes in Theoretical Computer Science 164 (2006) 157-175

with finite coproducts, an H-algebra consists of an object A of A and a morphism

a : HA-A. A flat equation morphism in A is a morphism of the form

e : X-9- HX + A (2)

and a solution of e is a morphism et : X->■ A such that the square

e [a,A]

HX + A--—9- HA + A

commutes. The algebra A is called completely iterative (or, shortly cia), see [13], if every flat equation morphism has a unique solution. Example: let TZ be the terminal coalgebra of H(—) + Z. Then the coalgebra structure is invertible, whence TZ is a coproduct of HTZ and Z

TZ = HTZ + Z (3)

with injections

Tz : HTZ-TZ ( UTZ is an ii-algebra" )

r¡z : Z->■ TZ ( "embedding of variables" ).

In fact, TZ is a free cia on Z with nz as the universal arrow. We denote by T the monad of free cias for H. Its unit is n and the multiplication ^ is given by the unique homomorphism ¡j,z TTZ-TZ extending identity on TZ.

2.1 Definition [1]. An endofunctor H is called iteratable if TZ, a terminal coalgebra of H(—) + Z, exists for every Z.

2.2 Example. Let £ be a signature, i.e., a sequence of sets (Sn)neN. X-algebras in Set are H-algebras for the polynomial functor

HEZ = £0 + £1 x X + £2 x X2 + •••

is iteratable, and T^Z can be described as the algebra of all E-trees on Z, i.e., trees with leaves labelled in Z + and nodes with n > 0 successors labelled in En. Recall that a free H^-algebra on a set Z is the algebra F^Z of all finite E-trees on Z. Thus, equations in the sense of the introduction, see (1), are a special case of the following concept:

2.3 Definition. Let H be an iteratable endofunctor. An equation morphism in

a cia A is a morphism of the form,

e : X-9- T(X + A).

It is called guarded if it factors through the right-hand injection of T(X + A) X + [A + HT(X + A)],

X--2—*~T(X + A)

A + HT (X + A)

2.4 Notation. If A is a cia, we denote by 5 : TA-A the unique homomorphism

a-^A = id.

The proof of the following theorem is a straightforward adaptation of Theorem 3.9 in [13].

2.5 Theorem. In a cia every guarded equation morphism e: X-T(X + A) has

a unique solution, i.e., there exists a unique e^ : X-A such that the square

T(X + A) -

T [et ,A]

commutes.

2.6 Remark. Recall that a category A is called locally finitely presentable, see [12] or [4], if it has colimits and a set Afp of finitely presentable objects (i.e., objects A such that hom(A, —) preserves filtered colimits) such that every object is a filtered colimit of objects in Afp. Examples: Set, Set1, Pos, Vec are finitely presentable

categories. A functor // : A-A is called finitary if it preserves filtered colimits.

Every finitary functor has free algebras, and as proved by Michael Barr in [5], this yields a monad F of free H-algebras. Analogously as for the cia's we have FZ = HFZ + Z, where the coproduct injections are the H-algebra structure and the universal arrow.

2.7 Definition. Let H be a finitary endofunctor. A finitary equation morphism is a morphism of the form

e: X-*F(X +A),

where X is finitely presentable. It is called guarded if it factors through the right-hand coproduct injection of F(X + A) = X + [A + HF(X + A)].

2.8 Definition. An H-algebra is called iterative if every finitary flat equation morphism, i.e., (2) with X finitely presentable, has a unique solution.

2.9 Remark. In every iterative algebra A every finitary, guarded equation morphism e: X->■ F(X + A) has a unique solution e^ = A]-e (where a: FA

-A is the unique homomorphism extending id^)- See [3].

2.10 Example [15]. For H = H^ the subalgebra R^Z C T^Z of the E-tree algebra

formed by all rational trees, i.e., trees which have up to isomorphism only finitely many subtrees, is iterative. This is a free iterative E-algebra on Z.

2.11 Notation. We denote by R the monad of free iterative H-algebras. It exists for every finitary functor H, and we have RZ = HRZ + Z, similarly as for free algebras and free cias. See [3]. This allows us to define, in analogy to Definition 2.3, rational equation morphisms as morphisms e: X-R(X + A), X finitely presentable, and call them guarded provided that they factor through the right-hand coproduct injection of R(X + A) = X + [A + HR(X + A)]. Every iterative algebra has a unique solution e^ of every rational, guarded equation morphism e: X

-i?(X + A), i.e., a unique morphism e^ = a-R[e^, A]-e where 5: RA-A is

the unique homomorphism extending id^.

2.12 Definition [9]. A category is called extensive if it has finite coproducts which are

(a) disjoint, i.e., coproduct injections are monomorphisms and the intersection of coproduct injections of A + B is always 0 (initial object), and

(b) universal, i.e., for every morphism f: C-A1+A2 pullbacks of the coproduct

injections along f exist and turn C into the corresponding coproduct:

A'^-= + --lA'2

2.13 Notation. We denote, for every coproduct injection i: A-C, by i: A

-s- C the complementary coproduct injection, i.e., C = A + A with injections i

and i.

2.14 Definition. A category is called u-extensive if it has countable coproducts

which are (a) disjoint and (b) universal, i.e., for every morphism f: C-UneN A„

pullbacks of coproduct injections along f exist and turn C into the corresponding coproduct.

2.15 Examples. (1) Set is u-extensive. The category of finite sets is an example of an extensive category that is not u-extensive.

(2) Posets, graphs, and unary algebras form u-extensive categories.

(3) Free completions under countable coproducts are always u-extensive.

(4) If K is u-extensive then so is each functor category [A, K], A small.

3 Pre-Guarded Equation Morphisms

3.1 Assumption. Throughout this section H denotes an iteratable endofunctor of an u-extensive category, see Definitions 2.1 and 2.14. Coproduct injections of binary coproducts are called inl and inr.

3.2 Definition. Given an equation morphism e: X->■ T(X + A) the derived

subobjects Xn

■ X, n = 1,2, 3,... are defined by the following pullbacks

¿Q = inl

T (X + A)

where ¿0 is the left-hand coproduct injection of T(X + A) = X + [A + HT(X + A)], see (3) above.

3.3 Remark. Since ¿0 is a coproduct injection, so is ¿1, and ei is a domain-codomain restriction of e. Analogously, since ¿1 is a coproduct injection, so is ¿2, and e2 is a domain-codomain restriction of e1, etc. We denote by

in. Xn

■X„_1 (n = 1, 2, 3,...)

the complementary coproduct injection, thus, Xn-\ = Xn + Xn for n = 1,2,3,... We consider X n as a subobject of X via

n- 1 -^ -^-i

3.4 Definition. An equation morphism e: X-s- T(X+A) is called pre-guarded

provided that X is a coproduct of the above subobjects Xn; shortly

X = X1+X2 + X3 + --- .

3.5 Example. If A = Set and H = He, then e represents, for X = {xi,x2,x3,... }, equations

Xi ~ ii(xi,X2,X3,...,ai,a2,a3,...) where the right-hand sides ti are (possibly infinite) E-trees on X + A. The variables of X1 = e-1(X0) are precisely those xi where ti is a single variable in X. That is, those xi where the corresponding equation has the form xi « xi'. We conclude that X\ are precisely the unguarded variables. To put it positively, X\ consists of all

the guarded variables. Here we have e\: X\->■ X, and thus x7- lies in

X2 = e-1(X1) if and only if xi' is unguarded. Consequently, for every xi £ X2 we have equations xi « xi' and

In other words, X"2 consists of all variables reaching a guarded variable in one step (of applying e). Analogously, xi £ X3 if and only if we have equations

Xi ^^ Xi', Xi' ^^ Xi"

and Xi» « Xi'» or, equivalently, X3 consists of all variables reaching a guarded variable in two steps, etc. To say

X = X1 + X2 + X3 + --means that every variable reaches a guarded variable in finitely many steps.

3.6 Remark. As demonstrated in Example 3.5, the intuition behind the subobjects X\, X2, X"3,... is such that X\ consists of all guarded variables. If e is a guarded equation morphism, then X = X\. If e is pre-guarded, we always have

a passage Xn

X i, for all n > 1, which to every variable assigns the guarded

variable eventually reached by applying e finitely many times. To formulate this categorically, we need the following

3.7 Notation We form a pullback of en: Xn-s- Xn-\ along the complement in

of in, see Remark 3.3; for i = in this gives us pullbacks

in+1 _

(n > 1)

n—1 *

The canonical passage from Xn to X\ is the composite morphism

u = [id, e2, e2-e3,... ] : X^ + X2 + X?J H---- -

■X -,

This defines a

3.8 Construction. Let A be a cia. For every pre-guarded equation morphism

e: X -T(X + A), X = Un>1X"n, we define, using (6), a guarded equation

morphism as follows

f = X,

■ T(X + A) T(w+A) ; T(X 1 + A).

Solutions of e and f are closely related:

3.9 Theorem. The equation morphism f is guarded and fulfils

(a) if e^ a solution of e, then e^-ii : X\-A is a solution of f, and

(b) tf f^ a solution of f, then p-u: X-A is a solution of e.

Proof. (1) We verify that f is guarded. Put

Jo = inl : Xi-T(X1 + A) = Xi + A + HT(X1 + A)

and compute a pullback of f along j0:

jo=inl

T (X + A)

T{u+A)=u+[A+HT{u+A)]

->T(Xi + A)

(2) Proof of (b). Given a solution /t; X\ — ->■ A is a solution of e, i.e., f t-u = 5-T[f t-u, Aj-e: X

A of f, we prove that f t-u : X ■>■ A. This equation will

be proved by considering the individual components of X = Xn, see (5). For n = 1 we use the definition (7) of f and obtain the commutative diagram

T{X + A) -

T (u+A)

T [f t,A]

For n = 2, the coproduct injection is i\-i,2 X2-^ X] thus we consider the diagram

All the inner parts except the one denoted by (*) clearly commute. The part (*)

commutes when composed with the passage to A, a-T[p, A]: T(X\ + A)-A,

i.e., this morphism merges the parallel pair /, inl: Xi ->■ T{X\ + A). In fact,

by the commutativity of the right-hand square in the above diagram it suffices to observe that f t = a-T[f inl:

T(Xi + A)-

T [f t,A]

The cases n = 3,4,... are analogous to the case n = 2.

(3) Proof of (a). Let. e^ : X-A be a solution of e. We are to prove that the

outward square of the following diagram

T (X + A)

T (u+A)

(*) T(X + A)

T (i1+A) T [et,AT

commutes. All the inner parts except that denoted by (*) commute. For (*) it is sufficient to prove that T[et,A] merges id and T(¿1 + A)-T(u + A). Therefore, the proof of (a) will be finished by proving

e] =e]-iyu: X-A. (8)

We consider the individual components Xn of X = Xi + X2 + X3 + • • •, see (5): For n = 1 use u-ii = id to obtain e^-i1 = (e^-i1 •u)-i1.

For n = 2 we are to prove the equation et'i1-i2 = (et-ii-u)-ii -i2. Consider the diagram

from which the right-hand side of the desired equation is expressed as e^-er^ • It remains to verify e^ -¿i = e^-ei which follows from the next diagram

Cases n = 3,4,... are analogous. □

3.10 Corollary. In every cia all pre-guarded equation morphisms have unique solutions.

In fact, the morphism u is an epimorphism, due to u-ii = id, thus the unique existence of et follows from the unique existence of f t via (a) and (b) above.

3.11 Remark. How about the converse: if e: X->T(X + A) has unique solutions in all cia's, is e pre-guarded? The answer is affirmative whenever T satisfies mild side conditions: see Proposition 4.11 below.

4 Strict Solutions

4.1 Assumption. Throughout this section A denotes a category which

(a) is w-extensive

(b) has a terminal object, 1, and

(c) has the property that given pairwise disjoint subobjects An >—B (n <E N) each of which is a coproduct injection, then the induced morphism neN An >—B as also a coproduct injection.

Moreover, H denotes an iteratable functor for which a morphism

_L : 1-^ HO

has been chosen.

4.2 Notation. For every equation morphism an intersection of the derived subobjects Xn >—>■ X (see Definition 3.2) is denoted by

'¿oo ■ \ ^ A.

4.3 Remark. For every equation morphism e: X-T(X + A) we see that

(a) an intersection of all derived subobjects exists, and

(b) X = XtXl + Wn> Xn (with ¿oo and (5) as injections).

In fact, using Assumption 4.1(c), where An = Xn+i, we see that for y: Y =

Wn>iXn -X with components (5) there is a complement y: Y-X. It

is easy to verify that this is the desired intersection.

4.4 Notation. _L is a global constant of H, i.e., every ii-algebra HA — obtains the corresponding global element

±A = 1 HO —^ HA A.

All homomorphisms h: A

B preserve this global constant: h-±A = ^B. In

fact, consider the commutative diagram below:

■HO m ? HA-

In particular for any object Y we have a global element of TY which we denote by ± for short:

_L = 1-^ HO —iU HTY TY

4.5 Definition. Let A be a cia and e: X

T(X + A) an equation morphism

■with a solution et: X-A. We call et strict if its restriction to XtXj is ±a-

4.6 Construction. Let A be a cia. For every equation morphism

e: X-s- T(X + A)

we define a pre-guarded equation morphism

f: X-T(X + A)

by changing the left-hand component of e: XtXl + X%

/• inl = Xoo -f ■ inr = e • inr: JJ X

1 T{X + A)

T(X + A) to ±:

■ T (X + A)

where inl and inr are the coproduct injections of X = Iœ + Xn.

4.7 Theorem. The equation morphism f is pre-guarded and fulfils

(a) every strict solution of e is a solution of f, and

(b) every solution of f is a strict solution of e.

Proof. (1) / is pre-guarded. Let Zo = and denote by jo = inr: Zo-X

the coproduct injection. Let jk : Zk-Zk- 1, k > 1, denote the derived subobjects

of f. We will prove that

Zk = Xk+1 + Xk+2 H----, and jk= inr : Zk-Xk + Zk,

and that the corresponding morphism opposite fk_i is

fk = efe+i + ek+2 H----: Zk-Zk_ 1 (k > 1).

This proves obviously that f is pre-guarded since P|keN Zfe = 0.

Case k = 1: To find a pullback of / = [_L!,e-jo] along i0: X-T(X + A),

we just compute a pullback of e-jo along ¿0: in fact the component contributes nothing to the pullback because it factors through ¿0, the complement of ¿0, and A is extensive. Here is the pullback of e-jo along ¿0:

X2 + X3 + • • • = Zi mr + x2 + x3 +

J0 = inr

+ X2 + X3 + • • • = X1 zl-mT,X = x00 + x1+x2 + x3 +

X^rT(X + A)

Consequently we have Z\ = X2 + X3 + • • • with j\ = inr : Z\-X = Iœ + X\ +

Z\, and the corresponding morphism f\ : Z\-X is

f1 = Z1^*X00 + Z1-2+X.

Case k = 2: We compute a pullback of / = ei ■ inr along ji:

II Pn-

n> 2 -I

II Xn = Zv

by computing first, a pullback Pn of e\ along the n-t.h component Xn-X, n > 2,

of ji, see (5)

D _ V «ri.+ l v in

fn — A „.4-1 -JL -

n-1 ■

The connecting maps are en : Pn-Xn and ¿2.....in'in+i'■ Pn-X\. Thus,

due to extensivity, a pullback of e\ along j\ is II„>2 Xn+i = Z2 with the connecting

maps 0n>2 1 • -^ find inr: Z2-X\ = XlXl + X2 + Z2. The pullback

of fi = ei - inr along j is thus

X3 + x4 + X5 + • • • = Z2 mr > Zi = x2 + x3 + x4 +

Xi = + X 2 + X 3 + X4 +

x3 + x4 + x5 + • • • = z2

II §n+1

We obtain Z2 = X3 + X4 + X5 H----, j'2 = inr, and /2 = II„>2 e«+i-

Case k > 3: Here we use the obvious pullbacks

II en +1 II eri. + l II eri. + l

ri >4 ri >3 ri >2

■Z2-

J. Adämek et al. /Electronic Notes in Theoretical Computer Science 164 (2006) 157-175 171

(2) Proof of (b). If f t is a solution of f, then f t is strict:

HO T(X + A) -

T [f t,A]

HT{X + A) -

HT[f t,A]

■HTA

We see that the passage from HO to HA is //! (because a-T[f\ A]-! = !: 0 ■ thus f= a-H= as required. And f t is a solution of e, i.e., the equation

•A),

5-T[/t, A]-e = /t : X^ + Xn-- A

holds (see (4) in the introduction): for the right-hand component jo : Xn-X

this follows from ejo = fjo. For the left-hand one form a limit of the pullbacks defining in and en:

X0 —*T{X + A)

to conclude = . Thus, the diagram

T (X + A)

T [f t,A]

commutes, proving the left-hand component of (9).

(3) Proof of (a). If et is a strict solution of e, then we are to prove that the equation a-T[e^A]-/ = et holds (cf. (4)): for the right-hand component with domain Xn this follows from the fact that /jo = e jo- For the left-hand component use the fact that both et and / yield ± (in A and T(X + A), respectively) and that a-T[et,A] preserves being a homomorphism (see Notation 4.4). □

4.8 Corollary. In every cia every equation morphism has a unique strict solution.

4.9 Remark. We will now turn our attention to the question of whether an equation having a unique solution in every cia must be pre-guarded. In the case of A = Set, the answer is affirmative whenever H1 has at least two elements. In general categories we need the following

4.10 Definition. We say that the free cia monad T is nontrivial if it preserves monomorphisms and has at least two global constants,

4.11 Proposition. Suppose that morphisms from non-initial objects to 1 are epi-morphisms. If the free cia monad is nontrivial, then every equation morphism e : X ->■ T(X + A) with a unique solution in TA is pre-guarded.

card A(1,T0) > 2.

Remark. We consider e as an equation in TA via X T (X + TA).

T(X + A)

Proof. Suppose that e is not pre-guarded. For every global element b: 1-TO

we can find a solution el: X->■ TA such that

1 —^ TO TA.

The proof is precisely the proof of Theorem 4.7 where a: HA-A is the replaced

by ta '■ HTA -TA (with ta = Ha) and _L is replaced by b. We will prove

that e has more than one solution by showing that e\ determines b; for that we

just observe that T\: TO ->■ TA is a monomorphism. In fact, !: 0 ->■ A is

a monomorphism since in every extensive category initial objects are strict, and T preserves monomorphisms. □

4.12 Example. Suppose that our base category is A = Set.

(1) Whenever H1 has more than one element then H has a nontrivial free cia monad. In fact, T preserves monomorphisms: see Proposition 6.1 in [3]. And to prove card T0 > 2, we decompose H = H' + H" with H'1 = 0 and H"1 = 0. This can be done by chosing any a € H1 and defining H'X and H''X as the inverse

images of {a} and HI — {a}, respectively, under HI = HX -HI. Consider

coalgebras

A = 1 consta= ff'l<—»-#1 and B = 1 constfc; H"l<—>H1

(a € H'l, b € H" 1). It is clear that the unique homomorphism A->■ TO is disjoint

with the unique homomorphism B-^T0. Therefore, card TO > 2.

(2) Conversely, whenever for every equation morphism e the implication

e has unique solution e is pre-guarded

holds, then H1 must have more than one element. In fact, card H1 = 1 implies that T0, a terminal H-coalgebra, has a unique element. Then the equation x ~ x has a unique solution in T0.

5 Iterative Algebras

5.1 Assumption. In this section A is a locally finitely presentable, w-extensive category such that every finitely presentable object is a finite coproduct of indecomposable, finitely presentable objects. And H is a finitary endofunctor for which a morphism

_L: 1-HO

has been chosen.

5.2 Definition. For a rational equation morphism e: X->■ R(X+A), (see 2.11),

we define derived subobjects Xn >—X precisely as in Definition 3.2, just replacing T by R everywhere.

5.3 Remark. We thus have pullbacks

■X . . ?R(X + A)

We also use the remaining notation in: Xn-Xn-i and en : Xn->■ Xn-i as in

Section 3.

5.4 Lemma. Every rational equation morphism e has a least derived subobject, i.e., there exists n with Xn = Xn-1 (more precisely: such that in is an isomorphism).

Proof. Let e: X->■ R{X + A) be a rational equation morphism. By assumption,

X is a coproduct of k indecomposable objects, X = Y1 + ••• + Yk. For every

coproduct injection г: Z ->■ X we obtain the corresponding morphisms : Zi

-Y with Z = Z1 + ••• + Zk and z = z1 + ••• + zk. Since each z¿ is a coproduct

injection of Y, either Zi = 0 or Zi = Y. Consequently, there are (in case Y¿ ^ 0 for every i) precisely 2k subjects of X which are coproduct injections. Since the subobjects Xn >—X, n £ N, are pairwise disjoint, it follows that there exists an m G N such that ~Xm = 0. Thus Xrn ^ Xrn+1 + ~Xrn+1 ^ Xrn+1. □

5.5 Definition. A rational equation morphism e is called pre-guarded provided that it has a trivial derived subobject, i.e., Xn = 0 for some n.

Remark. This is equivalent to XTO = 0 (due to Lemma 5.4). Thus, e is pre-guarded iff X = ЦА"П, compare Definition 3.4.

5.6 Theorem. In every iterative algebra all pre-guarded rational equation mor-phisms have unique solutions.

Proof. This is completely analogous to the proof in Section 3, see Theorem 3.9

and Corollary 3.10. Given the pre-guarded rational equation morphism e: X-

R(X + A), we have Xn = 0, i.e., X = X± + • • • + Xn and we define a guarded equation morphism

f = Xi ^^ X R{X + A) R(u+A) = R(Xi + A)

where и: X-Xi has components id-^, ei, ёгёг,... ,ёгё2.....ё„. Observe that

since и is a split epimorphism and X is finitely presentable, so is Xi. Thus, / is a rational equation morphism. Since f is guarded, it has a unique solution f t: X ->■ A, see Remark 4.6. The rest is as in Section 3. □

5.7 Definition. Let e: X-R(X + A) be a rational equation morphism in an

iterative algebra A. A solution et: X-A of e is called strict if its restriction

to some derived subobject is ±a, i.e., there exists n for which the square

commutes.

5.8 Theorem. In every iterative algebra every finitary equation morphism has a unique strict solution.

Proof. This is completely analogous to Section 4, see Theorem 4.7 and Corollary 4.8: choose n such that Xn = Xn+i, see Lemma 5.4, then the role of in Section 4 is now played by Xn. □

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