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Fork Algebras in Usual and in Non-Well-Founded Set Theories1 Part I

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Fork Algebras in Usual and in Non-Well-Founded Set Theories1 Part I Ildik¶oSain Istv¶anN¶emeti FORK ALGEBRAS IN USUAL AND IN NON-WELL-FOUNDED SET THEORIES1 PART I Due to their high expressive power and applicability in computer sci- ence, fork algebras have intensively been studied lately. In particular, they have been fruitfully applied e.g. in the theory of programming (speci¯ca- tion, semantics etc.). The literature of fork algebras has been alive and active for at least ¯ve years by now. Some references are: [34], [35], [18], [36], [10], [11], [14], [8], [9], [15], [16] and [17]. Analogously to the situation with Boolean algebras, groups, semi- groups, polyadic algebras etc., it is considered desirable to develop a rep- resentation theory for fork algebras, too (cf. e.g. [14, [10]). This would consist of ² de¯ning a \concrete" class of, say, \set (or proper) fork algebras" (analogously to Boolean set algebras, transformation semigroups, polyadic set algebras), and ² an axiomatic class FA, de¯ned by ¯nitely many equations (or perhaps quasi-equations), of \abstract fork algebras"; and then ² a \representation" theorem stating that every member of FA is iso- morphic to a set fork algebra. Here, among others, we look at various possible choices for the con- crete class that could play the r^oleof set or proper fork algebras in such a representation theorem. Some of the candidate classes for this r^olewere investigated in [26], [29], [23]. In [26], [29] and [23], the following results were proved in usual set theory (ZFC). The equational theory Eq(TPA) of True Pairing Algebras (TPA's) and the equational theory Eq(SFA) of Proper (or Set) Fork Alge- 1 bras (SFA's) are not recursively enumerable, moreover, they both are ¦1- hard. The Axiom of Foundation was mentioned in the proofs. Therefore, 1Research supported by Hungarian National Foundation for scienti¯c research grants No's T16448, T7567, F17452. 158 the question comes up that, perhaps, in some of the currently investigated non-well-founded set theories, these classes of algebras might show a better behaviour. We will look into this question below. Most of the theorems below are true in usual set theory as well as in most of the set theories without the Axiom of Foundation proposed so far in the literature.2 Throughout, ZF denotes the usual Zermelo-Fraenkel set theory, ZFC denotes ZF plus the Axiom of Choice. ZFC¡ denotes the set theory ob- tained from ZFC by dropping the Axiom of Foundation. We use the nota- tion ZF¡ analogously. To unify the various non-well-founded set theories, Aczel [1, section 4 \Variants. ", p.41] introduces the flexible axiom AFA», where the equivalence relation » is a parameter of the axiom. For various choices of », we obtain di®erent set theories, e.g., Scott's non-well-founded set theory, the basic variant AFA of [1] etc. For all choices of » per- mitted in [1], our theorems labelled by \without Foundation" remain true for ZFC¡+AFA» (and also for ZFC). Summing up, we use the following convention. Convention 0.1. Our theorems below marked as \ (without Founda- tion)" (after the number of the theorem) are true in both usual set theory (ZFC) and in the non-well-founded set theories found in [7] or in [1, section 1{4]. We will also attempt a thorough search for the right choice of the notion of concrete (or proper) fork algebra (such that it could support a convincing representation theory). In sections 3, 4 we will compare and study the various candidates, with an emphasis on the competing ones available in the literature. 1. Proper Fork Algebras On binary relations, say R and S, there are the well known set theoretic operations, e.g. R [ S, R \ S,..., R ± S (relation composition), R¡1 (converse or inverse), Id (the identity relation). From the above mentioned ¯ve-year old literature of Fork Algebras, in De¯nition 1.2 (ii) below, we recall a new binary set theoretic operation 5 , called \fork", de¯ned on binary relations. So, R 5 S is a new binary relation derived from R and S. 2In passing we note that, by Andr¶eka-Kurucz-N¶emeti[5], Birkho®'s celebrated char- acterization of equational hulls depends on the Axiom of Foundation. 159 Notation 1.1. hx; yi is the usual set theoretic ordered pair of x and y. That is, hx; yi = ffxg; fx; ygg. For any set X, P(X) denotes the powerset (set of all subsets) of X. 2 Definition 1.2. (i) By a Set Relation Algebra (SRA for short) we understand an algebra A = hA; [; \; ¡; ;;U £ U; ±; ¡1; Idi, where U is a set, A ⊆ P(U £ U), hA; [; \; ¡; ;;U £ Ui is a Boolean set algebra with greatest element U £ U, Id = fha; ai : a 2 Ug, and ±; ¡1 are the usual set theoretic operations of composition and converse (or inverse) of relations. (R ± S = fhx; yi : xRz & zRy for some z 2 Ug, R¡1 = fhx; yi : hy; xi 2 Rg for any R; S ⊆ U £ U.) The constant symbols 0, 1 denote the elements ; and U £ U, respec- tively. (ii) Let R and S be binary relations. Then R 5 S is a new binary relation de¯ned as follows. def R 5 S = fhx; hy; zii : xRy & xSzg: We call 5 the fork operation. (iii) A is called a Set (or Proper) Fork Algebra (an SFA) i® A = hB; 5 i, where B is an SRA closed under 5 , and the operation 5 of A is as de¯ned in (ii) above. 2 Throughout, RA abbreviates the expression \relation algebra" and FA abbreviates \fork algebra". (So we might write \A is a set RA" or \set FA" etc.) For a class K of algebras, I K, S K, P K and H K denote, respectively, isomorphic copies, subalgebras, isomorphic copies of direct products and homomorphic images of members of K. Remark 1.3. There are two ways of de¯ning set RA's. In one of the cases we require the greatest element 1A of a set RA A to be a square U £ U, while in the other case, 1A is required to be an equivalence relation E ⊆ U £ U only. Let us call the second class (with 1A = E) the class of generalized set RA's (GSRA's). 2 Theorem 1.3.1. (Tarski) (i) GSRA's form an equational class, while SRA's are the subdirectly irreducible members of this class. Therefore (by Birkho®'s theorem): (ii) I GSRA = SP SRA. 160 The de¯nition of GSFA is obtained from that of SFA in a completely analogous way (that is, requiring only 1A = E etc.). Proposition 1.3.2. I GSFA = SP SFA. For various reasons, it is GSRA and GSFA that we are interested in, cf. e.g. [19]. In particular, the classes RRA of representable RA's and def RFA of representable fork algebras are de¯ned by RRA = I GSRA and def RFA = I GSFA : Instead of introducing GSRA and GSFA directly (and then investigat- ing them), we introduced here SRA and SFA, and will investigate SP SRA and SP SFA. By Proposition 1.3.2 and Theorem 1.3.1 above, this will have the same e®ect (as if we introduced the \GS-versions" and investigated them). Remark 1.4. Some of the fork algebra papers (e.g. [10]) use free grupoids in de¯ning SFA (instead of explicitly using set theoretic pairing). This makes no di®erence because of the following. It is known from algebra that every free grupoid G is isomorphic to a free grupoid U = hU; +i such that U £ U ⊆ U and (8x; y 2 U)(x + y) = hx; yi, cf. e.g. [19, Part I, pp. 129{ 131] or [22, p.226, p.228, p.231]. Therefore the two de¯nitions of SFA are equivalent (in usual set theories3 ). We will discuss these grupoid versions of SFA in section 3. Instead of free grupoids, some other papers (e.g. [35]) use free semigroups for de¯ning 5 . We will discuss this version of SFA's in section 4. 2 In Theorems 1.7{1.10 below we will see that the class I SFA of alge- bras isomorphic to SFA's is not axiomatizable; that the equational theory Eq(SFA) of SFA's is not ¯nitely axiomatizable; and that there are further negative results in this line. Proposition 1.5. Assume ZF set theory is consistent. Then (i){(iv) below hold. (i) In ZFC¡, or in ZFC, it is not provable that Eq(SFA) would be re- cursively enumerable, hence no decidable (in particular, no ¯nite) axiomatization can be proved complete (and sound) for Eq(SFA). 3 In non-well-founded set theories the groupoid version gives us a smaller class SFAG ½ SFA which is even less axiomatizable than SFA. E.g., all our negative theorems proved in ZFC about SFA carry over trivially to SFAG in all set theories in which GÄodel's Incompleteness Theorem holds (e.g. in ZF¡), cf. section 3. 161 (ii) Statement (i) remains true for the quasi-equational theory as well as for the ¯rst order theory of SFA. (iii) In ZFC¡, or in ZFC, it is not provable that I SFA would be axiomatiz- able (or even pseudo-axiomatizable) by any set of ¯rst order formulas. Proof. This is an easy consequence of results in [29] or [23]. Remark 1.6. The concepts of ¯niteness, computability and recursive enumerability admit satisfactory de¯nitions in ZF¡. We are using these de¯nitions in the present paper. 2 Let us investigate whether Proposition 1.5 above forces us to give up all hope for obtaining positive results using the idea of permitting non- well-founded sets outlined in the introduction.
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