A Stone Type Representation Theorem for Algebras of Relations of Higher Rank

The Stone representation theorem for Boolean algebras gives us a finite set of equations axiomatizing the class of Boolean set algebras. Boolean set algebras can be considered to be algebras of unary relations. As a contrast here we investigate algebras of n-ary relations (originating with Tarski). The new algebras have more operations since there are more natural set theoretic operations on n-ary relations than on unary ones. E.g. the identity relation appears as a new constant. The Resek-Thompson theorem we prove here gives a finite set of equations axiomatizing the class of algebras of n-ary relations (for every ordinal n). The (Resek-Thompson) theorem we are going to prove here is a "geometric" representation theorem for cylindric algebras. It provides an apparently satisfactory positive solution to the representation problem of cylindric algebras (summed up, e.g., in the introduction of [HMTI] and in, e.g., Henkin-Monk [74]). The theorem represents every "abstract" algebra satisfying the cylindric axioms (eight schemes of equations; cf. the remarks on the choice of the axioms at the end of the paper) by a "concrete" algebra of sets of sequences. The representing algebra is concrete in the sense that we do not have to know the operations of the algebra, it is enough to know its elements. I.e. if we know the elements of the algebra, we can "compute" the operations on them by using their concrete set theoretic structure. (This is similar to the Boolean case where if x, y are elements of a concrete algebra 93 then their meet must be the set theoretic x fl y independently of the choice of 93. Already in the Boolean case we have to know the greatest element of 93 in order to be able to compute the complement — x of x in 93.) The first version of the theorem was obtained by Diane Resek and is proved as Theorem 5.27 on p. 285 of Resek [75]. Resek's result is also announced in [HMTII, p. vi, p. 101 (item 3.2.88)] and Henkin-Resek [75, Theorem 4.3], and is mentioned, e.g., in Maddux [82] preceding Problem 5.21; but no proof has appeared in print for this important theorem so far (for reasons indicated below). Using the techniques of Thompson [79], Richard J. Thompson generalized Resek's theorem to the form in which it appears below. Thompson's result is (partially) quoted in [HMTII, 3.2.88] without proof, and otherwise is unpublished. Thompson's proof is of a proof theoretic nature and proves more than the theorem stated below. Further discussion of that proof is found at the end of this paper. In the introduction of Received by the editors February 18, 1987 and, in revised form, August 5, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 03G15, 03C95; Secondary 03G25, 03C75. Research supported by Hungarian National Foundation for Scientific Research grant No. 1810. The second author is visiting the Mathematics Institute Budapest on an IREX fellowship. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 671 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 672 H. ANDREKA AND R. J. THOMPSON [HMTII, p. vig], Resek's result is said to be one of the "primary advancements" of the theory after the first publication of [HMTI]. At the same time, the proof in Resek [75] is so long (more than 100 pages) that they could not include it in the book [HMTII]. Therefore, in [HMTII, p. 101] the problem of finding a shorter proof arises. The present note is aimed at solving this problem. The proof in this note originates with H. Andreka and is a generalization of her proof with I. Nemeti mentioned on pp. 834 and 794 of [HMTII] (cf. also pp. 245-247 of [HMTII]). Andreka presented the proof in this paper for the diagonal-free case (a arbitrary) at the Universal Algebra Colloquium at Szeged in the summer of 1985. The present proof of Lemma 1 (of this paper) was presented in 1984 at the logic seminar of the University of Colorado at Boulder by Andreka and Nemeti (it is due to Andreka but the basic idea comes from [HMTII, 3.2.52]). The first version of the full proof in this paper is in Andreka [86]. The relation algebraic analog of the Resek-Thompson theorem is Theorem 5.20(2) in Maddux [82]. We discuss the connections between the two theorems (and proofs) at the end of this paper. Of the axioms (Co)-(Cy), MGR used below, (Co)-(C7) are due to Tarski, while MGR was discovered by Leon Henkin (see [HMTI, pp. 17, 194-195, 408]). Henkin proved (C0)-(C7) P MGR (refuting a conjecture of Tarski) (cf. [HMTII, 3.2.71, p. 89]). The ideas in Thompson [79] are not unrelated to the "transformational" approach of William R. Craig to algebraic logic (cf. Craig [74, 74a] and the notes at the end of this paper about works of Craig, Pinter and Howard). Resek's theorem says that (Co)-(C7)+ all MGR's axiomatize CrsQnCAQ. Thompson's improvement of this theorem is twofold: He replaced the infinitely many MGR-equations with just two of them, hence proved finite axiomatizability of CrsQ fl CAa; and further by weakening the axiom (C4) of commutativity of cylindrifications to the weaker (C4), he made it possible to replace the class CrsQ fl CAa (which has a mixed nature, namely CrsQ is a "concrete" class while CAa is "abstract") with the purely "concrete" class Da (the definitions of these notions can be found below). To avoid misunderstandings, we note that the first author did not contribute to the theorem in this paper while the second author did not contribute to the proof in this paper. ACKNOWLEDGMENT. H. Andreka is grateful to J. D. Monk, for bringing Resek's theorem to her attention, and for suggesting the project of searching out a "reasonably short" proof for this important theorem. Hajnal Andreka is also grateful to R. D. Maddux, for explaining the basic ideas of the step-by-step method, which he used in [M78] to prove SA C SR1RRA, and for pointing out that this method should be applicable for cylindric algebras, too. We use the notation of [HMTI, HMTII]. Let a be any ordinal. We recall from [HMTI] that an algebra 21 = (A,+, -, -,0, l,Ci,dij)ijea, where +, • are binary operations, -, Ci are unary operations and 0,1, dij are constants for every i, j E a, is a cylindric algebra (a CAa) if it satisfies the following identities for every i,j, k Ea. (Co)-(Cs) (A,+, -, -,0, l,Cj),eQ is a Boolean algebra with additive closure operators Ci such that the complements of enclosed elements are enclosed (i.e. x = Oi Jb r C*l ~~ Jb —JU J t (C5) dx% = 1, (C6) d^ = ck(dik ■ dkj) if k £ {i,j}, (C7) d%j ■ Ci(dij •x) CjCiX-djk if fc ̂ {i,j}, and let E = {Co,Ci,C2,C3,C4,C5,C6,C7, MGR}. Mod E denotes the class of all algebras that satisfy E (and which are similar to CAQ's). We recall from [HMTII] the following definition of Crsa. By a Crsa we shall understand a Boolean algebra of sets of a-sequences where the non-Boolean operations (a, dij) are derived from the "a-sequence structure" in a natural way. In more detail: If / is any a-sequence and i E a then f(i/u), or /£, denotes the sequence which agrees with / on a ~ {i} and which is u on its ith place. Crsa is defined to be the class of those algebras 21 = (A, +, •, —, 0, la, Ci, dij)ij€a for which la is a set of a-sequences such that (A, +, -, —, 0, la) is a Boolean set algebra, further Cl(x) = {fEl%:(3u)f(i/u)Ex}, d^ = {/ E la: fi = fj} for all i, j Ea and x E A, Da = {*E Crsa: (Vm E a)(V/ 6 la)/(t'//;) E la}, where la is the greatest element of 21. IDa denotes the class of all isomorphic copies of elements of Da. THEOREM 1 (RESEK-THOMPSON). IDa = Mod E for any a > 2. PROOF (ANDREKA). It is easy to check that Da f= E. The essential part of the proof is to show ModE C IDa. Let 21 E ModE. We will show 21 E IDa. We may assume that 2t is atomic, by Jonsson-Tarski [51, 2.15, 2.18] (see also [HMTI, 2.7.5, 2.7.13]); namely: every Boolean algebra with operators 21 can be embedded into an atomic one such that all the equations valid in 21, and in which "—" does not occur, continue to hold in the atomic one. (Notice that, in E, "—" occurs only in (Co)(C3), where Ci—CiX = —CiX can be replaced with Ci(x ■ Ciy) = ax ■ ay; cf. [HMTI, p. 177i5].) Thus from now on we assume that 2t is atomic and 21 f= E. Let At 21 denote the set of all atoms of 21. We want to "build" an isomorphism rep: 2l>-» 93, for some 93 E CrsQ, for which (*) below holds: (*) rep(z) = (^J{rep(a): a E At 21, a < x} for every x E A. Let V be a set of a-sequences and for every A C V and i,j E a let dX = {/ E V: (3u)f(i/u) E A}, TJi; = {/ € V: fi = fj}. Assume that rep: A -f {A: A C V} is a function for which (*) holds. Then it is easy to check that rep is an isomorphism onto a 93 € CrsQ with l58 C V if and only if conditions (i)-(v) below hold for every a, b E At 21 and i, j E a: (i) rep(a) n rep(6) = 0 if a ^ b, (ii) rep(o) C D%i if a < dfj and rep(a) n Dtj = 0 if a • da = 0, (iii) rep(a) C drep(b) if a < cfb, (iv) rep(a) n Ctrep(b) = 0 if a • cfb = 0, (v) rep(a) ^ 0. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 674 H. ANDREKA AND R. J. THOMPSON We shall construct (a set V of a-sequences and) a function rep with the above properties, step by step. For every a-sequence / let ker(/) = {(i,j) E 2a: fi = fj} and for every a E At21 let Ker(a) = {(i,j) E 2a: a < da}. Then Ker(a) is an equivalence relation on a by our axioms (C5)-(C7). For every a E At 21 let fa be an a-sequence such that for every a, b E At 21 we have (a) ker(/„) = Ker(a), (b)Rg(fa)nRg(fb)=Oiia^b. Such a system (fa: a E At 21) of a-sequences does exist. Define rep0(a) = {/a}, for every a € At2t. Then the function rep0 satisfies conditions (i), (ii) and (iv), (v) but it does not satisfy condition (iii). Below, we shall make condition (iii) become true step by step, and later we shall check that conditions (i), (ii), (iv), (v) remain true in each step. Let R = At 21 x At 21 x a, p be an ordinal and let r: p —► R be an enumeration of R such that for all n E p and (a, b, i) E R there is m E p, m > n such that r(m) = (a,b,i). Such p and r clearly exist. Assume that nE p and rep„: At21 —> {A: A C V'} is already defined where V is a set of a-sequences. We define rep„+1: At21 —► {A: A C V"}, where V" is a set of a-sequences. Let r(n) = (a,b,i). If a ^ Cib then repn+1 = rep„. Assume a < ctb. Then repn+1(e) = repn(e) for all e 6 At 21, e ^ b. Further, Case 1. b < dij for some j E a, j ^ i. Then repn+1(6) = repn(b) U {f(i/fj): f E repn(a)}. Case 2. 6 ^ dtJ for all j E a, j ^ i. For every / E rep„(a) let uj be such that (c) uf <£ (J{Rg(9): 9 E U(repn(e): e E At21}}, (d) uf ytugii f / g, f,gE repn(a). Now rep„+i(6) = repn(b) U {f{i/uf): f E repn(a)}. Let n € p be a limit ordinal and assume that repm is defined for all m < n. Then repn(e) = M{repm(e): m < n} for all e E At 21. By this, (repn: n E p) is defined. Now we define rep(a) = M{rep„(a): n E p} for every a E At 21, and V = {J{rep(a): oeAt2f}. We are going to check that conditions (i)-(v) hold for the above rep and V. First we check that condition (iii) holds. Assume that a < Cib, a, b E At 21 and i E a. Let / E rep(a). Then / E rep„(a) for some n E p. Let m > n, m E p be such that r(m) = (a,b,i). Then by our construction, there is some u for which f(i/u) E repm+1(6) C rep(6), i.e. / E Ciiep(b). We have seen that rep(a) C Cirep(6). Thus condition (iii) is satisfied. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use A STONE TYPE REPRESENTATION THEOREM 675 Next we show that conditions (i), (ii), (iv), (v) hold, too. This we will show by induction. First we check condition (ii). It is easy to see that condition (ii) is equivalent to (ii) ker(/) = Ker (a) for all / 6 rep(a). Now (ii)' holds for rep0 (in place of rep, i.e. in (ii)' we replace "rep" everywhere with "rep0") by our condition (a). Assume that (ii) holds for rep„. We show that it holds for repn+1, too. Let r(n) = (a,b,i), and let e E At2l be arbitrary. If e ^ b or ii a £ c$ then repn+1(e) = repn(e), hence we are done by the inductive hypothesis. Assume (e = b and) a < Cib. By (Ce), this implies Ker(a) n 2(a ~ {i}) = Ker(fr) fl 2(a ~ {i}), therefore by our construction, and by the inductive hypothesis, we have (V/ E repn+1(b)) ker(f) = Ker(6). We have seen that (ii)' holds for repn+1, too. It is easy to see that if n E p is a limit ordinal and (ii)' holds for all repm, m < n, then it also holds for repn. For this same reason, if (ii) holds for all repn, n E p, then it also holds for rep. We have seen that condition (ii) holds. Next we check that conditions (i), (iv) hold. Instead of conditions (i), (iv) we shall prove a stronger condition (iv) . To formulate (iv) , we need some definitions. For all i,j E a, i ^ j, define t-x = d%3 ■ c,x and t\x = x. <*a denotes the termfunction defined by t'} in 21. Claim 1. tf: At 21 -♦ At 21 is a function. PROOF. Claim 1 follows directly from [HMTI, 1.10.4(h)] whose proof does not involve (C4). Q.E.D. (Claim 1) For all i,j E a let tbe a symbol and let Q be the set of all finite sequences of tljS, i.e. let Q = {ty. i,j E a}*, where for any set H, H* denotes the free monoid generated by H. Let a = tl> ■ ■ ■ ty. Then we define