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TRANSACTIONS OF THE AMERICAN MATHEMATICALSOCIETY Volume 175, January 1973 CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS^) BY CHARLES PINTER ABSTRACT. Several new formulations of the notion of cylindric algebra are presented. The class CA of all cylindric algebras of degree a is shown to be definitionally equivalent to a class of algebras in which only substitutions (together with the Boolean +, •, and —) are taken to be primitive operations. Then CA is shown to be definitionally equivalent to an equational class of algebras in which only substitutions and their taken to be primitive operations. 1. Introduction. substitutions In the theory ate Specifically, defined in (A) Just as (A) defines ments, fined in terms of algebraic certain operations of degree and diagonal a and k, À < a, in terms Thus, substitutions in [6] that (a) in every and diagonal and diagonal it is clearly in which elements possible are taken then the opera- if* ¿A- of cylindrifications that both cylindrifications called elements. by the formula S£x = cK(x - dKX) of substitutions. logic algebras/2) by S^, is defined itK=X; substitutions it is known with +, •, and —) are of cylindrifications algebra denoted S*x = x (together of cylindric terms if SI is a cylindric tion of k-for-K substitution, conjugates ele- may be de- to develop a theory to be the only primitive opera- tions. It is proved S\x = 1, and (b) in every K < a. It is noted in [6] that mension-complemented 0, 1, S\)K conditions. A simple cylindric algebras There tions. \<a in this The definition lindrifications Received only algebras wnere tne and elegant is more than d^ is the least as sense has been by the editors new treatment fot each of di- structures operations subject by Anne Preller cylindrifications to certain in [l0]. in terms has the disadvantage of dimension-complemented algebras; of substitu- of yielding cy- alternative October 4, 1971. AMS (MOS) subject classifications (1970). Primary 02J15. Key words algebra, and phrases. x such that „S^x of dimension-complemented obtained upon (b), above, in the case a possible ^X are unary element c x = 2. algebraic characterization one way of defining based CAa, (a) and (b) suggest cylindric (A, +,-,-, CAa, dimension-complemented Cylindric cylindrification, diagonal element, alge- braic logic. ( ) The work reported in this paper was done while the author held an NSF Faculty Fellowship. The author is indebted to the referee for valuable suggestions, especially regarding an important simplification of the material presented in §3. ( ) For notation and terminology, see Henkin, 167 Monk and Tarski [6], Copyright © 1973. American Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use CHARLES PINTER 168 definitions are available, is not needed. algebras Using in terms In this these, any one of these, we describe without which somewhat weaker In a recent every take complementedness to characterize an especially paper, of dimension all cylindric William Using these is definitionally ideas, simple Craig complementedness. algebras Using in terms of sub- form in the presence Í2] has shown of a condition that in a cylindric algebra ) and that cylindrifications in terms of substitutions a set of axioms equivalent (together in terms complementedness. entirely we derive cylindrifications of cylindric S y has a conjugate/ which for defining characterizations may be defined conjugates two methods than dimension substitution elements of dimension it is possible the assumption we get two simple stitutions, the assumption of substitutions. paper, of substitutions where [January to CAa, with the Boolean and diagonal and their for an equational and in which conjugates. class of algebras substitutions and their + , • , and - ) are the only primitive opera- tions. All the cylindric to be of degree sented a> elsewhere, tions we shall and related 2. For the case considered examine 2. Substitution here mulation is as follows: Definition. and cylindric algebra are assumed hold which For the cases algebras. will be pre- <x= 0, 1, the ques- By a "substitution with "substitution" By a substitution algebra mean a system 21 = (A, + , • , - , 0, 1, S^ ) Boolean algebra and the S1^ ate unary operations ditions results paper are trivial. algebras mean a Boolean in this ct= 2, simpler in the form of an abstract. we shall 2.1. algebras operations. of degree a, briefly for- an SAa, we x^a where (A, +,.,-, on A which algebra" Our precise 0, l) is a satisfy the following con- for all x, y £ A and all k, X, p., v < a : (Sj) SKxix + y) = S$x + S^y, = -S£x, (52) S£(-x) (53) S£x = x, (s4) SCS%x= SÏSÏx, (S,) S^Sfx = S^x, if k4K (S6) S$ S$x = S^S'ix, Our notion in the latter, algebras, algebras formations tions itK4ri,v of substitution some additional algebras conditions as we have just defined invented S* may be compared 0~) For the notion of conjugate, to that of Preller are assumed. In addition, related [3], with the difference in the sense with is similar them, are closely by P. R. Halmos to replacements andzi 4 A. of Halmos the operations due to Jonsson to the transformation that we restrict [5, p. 289]. Sir) of [3] where and Tarski License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use in LIO], but, substitution [7], Thus, trans- our opera- r is restricted see our Definition to 2.5. 1973] CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS replacements. Then that the operations tially the same scends three our axioms S(t) as Halmos' the conditions immediate (S,)—(S,). In every substitution vation, we shall lindric (and in every write observation Indeed, S Now it follows in every the range cylindric Combining Axiom S2, whichtran- algebra, we adopt algebra c = rg k of S\, if operations of À. Indeed, In view of this obser- k / X. of SA to the range algebra, 5A it S\cK= c of cK in cy- and cKSx = 5A for Consequently, for every from the elementary properties cKx is the least y £ range (**) cylindric rg K for the range relates range in [3] is essen- to replacements—namely tot all X, p /= k. k / X (see [6, 1.5.8 and 1.5.9]). (*) of Halmos' of substitution of S* , for k / X, is independent S* = range henceforth algebras. Instead (S,) for our purposes. algebra that range An analogous SI. to the condition and our axiom of this axiom restricted by (A)), the range from (S,) endomorphisms, in our notion is all we need follows k, À < a, Axiom assumed consequences This are defined (S.) and (S2) correspond be Boolean 169 k < a. of cylindrification cK such (*) and (**), we get the following that that y > x. important property of cylindric alge- bras: (2.2) ex (2.2) clearly stitutions is the least shows in an algebra y £ rg k such that y > x. that cylindrifications which does can be expressed not need to be dimension in terms of sub- complemented. The proposition (2.3) ¿ shows that diagonal we consider elements substitution (77I) for each (77-2) for every and introduce (B) cKx is the least y such that S£y = 1 likewise algebras can be so expressed. having This x £ A and K < a, {y £ rg k: y > xj has a least k, X < a, {x: SAx = lj has a least that element, and element, for them operations cK and dKX by means is the least of \y £ rg k: y > xj, and element suggests the properties of the definitions (C) dKX is the least element of [*: ^xx = llThe notion algebras has shown similar Before to introduce to introduce clearly similar to (B) holds quantifiers cylindrifications in the work of P.-F. in every in transformation in substitution Jurie. In [8], Jurie polyadic algebra, algebras satisfying and that a con- to (77I). going 2.4. Lemma. subject (77I) appears that a condition it may be used dition that (B) may be used satisfying to condition on, we consider some consequences Let 21 = (A, + , . , -, (nl). If c of (77I) and (rr2). 0, 1, SA)K x<a be a substitution is defined by (B), then License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use the following algebra statements 170 CHARLES PINTER [January hold for all x £ A and all k, X, p < a: (ii) ^xck (iii) = CK' ^V^ cK0=0, (iv) Proof. (vii) x<cKx, Furthermore, cK5^x < S*cKx ifn4X,p. from (77I) and (B) that if y = cKx, then that if y e rg k, then y = cKy. c cKx < cKy, S^x<cKx, (viii) It is clear (a) range * - y implies y e rg k, and also Thus, = rg k. it follows from (SA and (B), respectively, that (b) S'{SKX= SKXand cKcK=cK. Now (i) and (ii) follow immediately By (S.) and (S,), algebra S\ of 21. In fact, Halmos quantifier of 2I( and therefore [4, §4]). Now (vi) follows least Thus, by Halmos complete [4, Proof from (iii)—(v) by [6, 1.2.7], y £ ig k such rg k is a Boolean Boolean of Theorem sub- subalgebra 5], c is a (iii)—(v) hold. of (iv), (Sj) and (ii). element of 21, hence by (77I), rg k is a relatively of 21 (see sequence from (a) and (b). is an endomorphism It remains that y > Sx. and (vii) is an immediate to prove (viii): Now S*cKx by (ttI), con- c^S^x is the > S*x by (iv) and (S,); furthermore, by (ii) and (Sg), if v 4 A, then SlCKX- thus, cKS*x<S*cKx. The next SaSvcKx=Sv^cKxetgK; a definition introduces a concept which plays an important role in the next section. 2.5. algebra Definition and /, g £ (Jónsson and Tarski [7, Definition A, we say that g is the conjugate 1.11]). If A isa Boolean of / if, for any x, y £ A, fix) . y = 0 « X - g(y) = 0. 2.6. ject Lemma. to conditions spectively, ii) (iii) Let \ A, +,.,-, (77I) and in2). If c and fl^ \<a be a substitution are defined algebra sub- by (B) aW (C), re- then the following hold for all x £ A and all k, X, p. < a; S^x . dKX= x ■ dKX, (ii) S*x = cKix • dKX) S^dKfl= S\dKX (iv) Sx admits a conjugate.^) Proof, (i) Clearly which 0, 1, S\'K is the same as (4) Theorem 2.6(iv) if k4 u, ifx4X, S^iS^x + -x) = 1, hence by (tt2) and (C), dKX < S^x + -x, x • dKX < S\x. is a special case Analogously, S xx ■ dKX < x, so we get (i). of a more general Lemma 27]. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use result proved in Craig [2, 1973] CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS (ii) By 2.4(i) and 2.6(i), cKS$x . dKX= Sxx . d^ = x . rf^. ck{x •dKX) = ck(ckSAx • ¿KX)> so by 2.4(i) Sxx . cKdKX. But by (C) and 2.4(vii), cpSdu\' . Therefore dp.K) = Sidp.\- Now by (S,) and (S4), S^S^x if À /= p, then by 2.6(ii), = S^x = 1 =» S*x = 1; thus, StxldKfl= cfx(dKfJL. d^y) = Thus- = 1 by (C), (Sj)and(S2). It follows by (C) that dKX< S^d , and analogously, d (S4), SxdKX < SXS&K^ S^dKX S\dKfl< (iv) If k = À, we define is the conjugate . cKdKX= cK¿KA = 1, so cK(x • dKX) = Sxx • 1 = Sxx. hence Sh\x = 1 =» S^x = 1; symmetrically, by (C), dK/JL=d Thus, and (v), cK(x . dKX) = cKSxx (iii) If X = p, there is nothing to prove. SkSkx = S?x' 171 of Sx. If k/ T{x ■ y = 0 _ (applying S£) = 5idKX. Tx by Txx = x; in this (a) TKxx=cKx.dKX, and we proceed to show that < S¡¡_dKX . Thus, by (Sj)— X, we define case, it is obvious that Tx Tx by Tx is the conjugate cKx ■ dKX • y = 0 of Sx: by (a), =» cKx • 1 • S$y = 0 by (SA-(S2), 2.4(ii) and (C), =>x.S*y=0 by 2.4(iv). Conversely, x • S£y = 0 =*cK(x • SAy)= 0 by 2.4(iii), =*cKx • S£y = 0 ^cKx-dK\-sZy In our first initionally theorem, equivalent (t73) Sxc Note that by 2.6(i), =>TAx-y=0 by (a). we will show additional = c Sx that the theory of substitution □ of cylindric algebras which algebras satisfy is def(ttI), condition: provided p / k, X, where c only half of this (v), =° => cKx • ¿KX • y = 0 to the theory (772) and the following by 2.4(i)and condition is really is defined by (B). needed, as the other half is de- rived in 2.4(viii). 2.7. Theorem, (i) // \A, +,■,-, 0, 1, c , c^KA'KA<a is a cylindric and Sx is defined by (A), then \A, +, • , - , 0, 1, Sx ) gebra satisfying (ii) A<a is a substitution al- (tt1)—(tt3), together with (B) and (C). Conversely, which conditions algebra if \A, +,-,-, 0, 1, S A) (77I)—(?73)hold, and if c . a is a substitution algebra in and d . are defined by (B) and (C) re- License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 172 CHARLES PINTER spectively, then [j anuary i A, + , . , - , O, 1, c K, dK>) K x<a is a cylindric algebra where (A) holds. Proof, (i) Let (A, +,•,-, 0, 1, cK, dKX)K x<a be a cylindric algebra, and de- fine S$ by (A). Then (Sj )-(S6) hold by [6, 1.5.3 and 1.5.10], and (tt2) holds by [6, 1.5.7]. Next, range cK = rg k by [6, 1.5.8(i) and 1.5.9(ii)L hence by [6. 1.2.4 and 1.2.9], we have (ttI ) and (B). Finally, (jt3) holds by [6, 1.5.8(ii)], and (C) holds by L6, 1.5.7]. (ii) Let (A, +,-,-, 0, 1, S*) x<abea (773) hold, and let cK and dKX be defined that conditions substitution is nothing ckc\ckx' to prove; hence, 1.1.l], assume are satisfied. We will show Well, (C0) holds We prove (C4) as follows: k 4 X. By 2.4(iv), li k = x < c x, so cc.x N°w if P 4=X-1A, then CKC\cKX This in which (77I)— by (B) and (C) respectively. (Cn)—(Cy) of [6, Definition by 2.1, and (Cj)— (CA are given by 2.4(iii)—(v). À there algebra shows that cKcxx < cxc (CA follows immediately To prove (CA, let = CKC\Sp\CKX by 2.4(H), = ck5mcXckx by (?73)' = sK^Xckx by2.4(i), = cXS>ßcKX bV <»3), = cxcKx by 2.4(ii). x; symmetrically, c c x < c cxx, proving (CA from (772) and (C). c^x = -ic — x). It easily follows from (773) and (S2) that *A^ = cfö tot p4K,X. Thus, S$c*dKX = cfiSKxdKX=I, so by (n2), dKX< cfLdKX. On the other hand, it follows from 2.4(iv) that cfidKX < dKX, so dKX= c^dKX £ ig p, hence, in particular, for k 4 p- Consequently, dKX = S^dKy Thus, by 2.6(iii), S^dKß= dKX if p 4 k> A, then cpSdK p-• dp-x) =s xdKu hy2-6^' = dKX' which proves (CA. Lastly, (C7) follows immediately (A) follows from 2.6(ii) and (Sj). Remark. stitution then there family of elements dric algebra; A similar D It follows from 2.7(Uthat algebra, (^^ indeed, remark was the We will show of either ness. We begin if \A, +,•,-, is at most one family 0, 1, S^)K x<a is a subof functions icK'K<a and one ^<a such that \A, +, ■, 0, 1, cK, dKXIK x<a is a cylinc made and dKX ate completely for polyadic for locally finite polyadic algebras sence from 2.6(a) and (S2), while next that by P.-F. Jurie by (B) and (C). [8], and, earlier, by L. LeBlanc L9J. Theorem of two conditions algebras determined 2.7 can be improved somewhat weaker than with some definitions: License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use significantly dimension in the precomplemented- < 1973] CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS 2.8. Definition. bra. For each k < a such 173 Let 21 = \A, + , • , - , 0, 1, 5A)K A<a be a substitution x £ A, the dimension set of x, in symbols that cKx -/ x. 21 is said to be dimension alge- Ax, is the set of all complemented if, for every x £ A, Ax 4 a. 2.9. Definition. Let 21 = (A, +,-,-, 0, 1, SA) K x<n be a substitution algebra. Z(k, X) = \x £ A: Sx x = 1 j. Then 21 is said to be normal if We write D , , Z(k, X) = {lj for each k < a. A cylindric algebra If 21 satisfies a cylindric each normal (7r2) and dKX satisfies algebra), k < a.(J) will be called then clearly if it has the same property. (C) for all A, X < a (tot example, if 21 is 21 is normal if and only if 1 * d x = 1 for \¿K;\<a 2.10. Definition. bra satisfying (nl), Let 21 = \A, + , . , - , 0, 1, Sx)K x<a be a substitution and for each k < a, let cK be defined by (B). alge- We say that 21 has the RS property if (2.11) for all cKx= Z K k < a and x £ A. erty if it satisfies Equation 2.12. Six x¿K;X<a A cylindric algebra A is likewise said to have the RS prop- (2.11). (2.11) is due to Rasiowa and Sikorski Lll]. Theorem, (a) Every dimension complemented substitution algebra is normal. (b) Every normal substitution algebra satisfying (nl) and (tt2) has the RS prop- erty. Proof, such that (a) If 21 is not normal, Sxx = 1 for every for some p < a. x = S^x = S^S 7Ç f\ (b) then there À ?¿ k. L-.\<a s\x = A X¿k; = 1;' in either case, substitution Kfx<aCK^-¿Kx) X¿K;X< = CK (X ' XéK; thus, p £ Ax Lemma. easily that (77I) and (,7r2). Then, by [6,1. 2. 6], \<adK\) by the remark following 2.9. // (A, + , ■ , -, the RS property satisfying satisfying by2.6(ii), - cKx ( ) It follows complemented; we have a contradiction. algebra -^x^x<Jx-dK^ a CAa is normal x / 1 in A and some k < a If p = k, then for any X ^ p, x = S^x = Sxx = 1; if p ¿ k, then x = S^l fJ. IS. Let ** be a normal 2.I3. is some But 21 is dimension 0, 1, Sx) x<a is a substitution G algebra with (ttI) and (tt2), then (n3) holds. every normal CA for a<a>, iff it is dimension-complemented. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use is discrete. Thus, for a<(0, 174 CHARLES PINTER Proof. We have proved, Lemma 2.6(iv), follows by Jónsson ditive. Thus, if K 4 X, p, we have and Tarski XucKx Z K = S* My¿K;V<a = SKSxx+ r Z ^ Consequently, cKS x. Combining that each a conjugate. It S1^ is completely Z V é K,\\ ad- S*aSKv, ," v V <a. by (S4) and (S6), ^ S"vS*x. ^ S>cKx + S^x We conclude 1.14] S^ admits S*S£x=S*S£*+ M v ,"• A S«S*x v¿K,X-,v<a = V ¿K,X;V<a. Z that each [7, Theorem SKvx= Z v v¿K;V<a fi I anuary = 2^ ^J^x this with 2.4(viii) this section gives = c^x, hence Sfax us our result. with the following < D result: 2.14. Theorem, (i) // 21 = (A, + , • , -, 0, 1, cK, dKX)K x<a is a cylindric gebra with the RS property complemented cylindric '^> +i • > -, a normal substitution algebra, Conversely, resp. if 21 =(A, (resp. (resp. algebra, resp. by (A), then algebra a dimension +,-,-, a normal (nl) and (772), and if c then 21 = (A, +,.,-, erty cylindric 5^ is defined a dimension 21 = with the RS property complemented (resp. substitution alge- (nl) and (tt2), as well as (B) and (C). with the RS property isfying a normal and 0> 1> $)) K x<a z5 a substitution bra) satisfying (ii) (resp. algebra) al- 0, 1, Sx)K x<a is a substitution SA, resp. a dimension and d . are defined cylindric algebra, resp. SA) sat- by (B) and (C), respectively, 0, 1, cK, dKX> x<a is a cylindric a normal algebra complemented algebra with the RS prop- a dimension complemented cylindric algebra) in which (A) holds. Thus, mented, in the special or have (773). Theorem the theorem a problem 2.14, posed namely raised which Theorem In fact, complemented Theorem in the theorem possess a property are normal, dimension comple- without condition 2.7 may be re-stated algebras, 2.14 provides of Preller stronger [l0], is germane a partial it is required than dimension to solution that to the complemented- the property x, y there the question (T) is replaced by dimension given an affirmative answer by (B), and, correspondingly, 3. Some equivalences Craig L2] showed every substitution diagonal of algebras for dimension by Preller; algebra (T) For every Preller stated in Pre Her [lOj. substitution ness, case the RS property, elements exists k such that S^x = x and S*y = y tot allA< whether the theorem in [10] holds complementedness to tfcis question, that Preller's in substitution that in a polyadic admits a conjugate, may be defined 2.8). We have (our Definition that (SA be replaced In a recent with generalized of substitutions License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use cK be defined by (nl). paper, diagonal and that cylindrifications in terms condition with the proviso algebras. algebra in case a. William elements and generalized and their conjugates. 1973] CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS In this section conditions we shall which are shown characterization tions use the ideas introduced to be equivalent of cylindric algebras by Craig to formulate to (tt1)—(tt3), in terms 175 and thus of substitutions several obtain another and Boolean no- alone. In this element, and the succeeding then the least lemma, we record 3.1. pages, element of B will be designated a few elementary Lemma. if B is an ordered properties Let A be a Boolean (i) // / has a conjugate fg(x) = miniy Suppose g(x) = miniy £ tange Proof, for each x £ A, lj. x £ A, then g is the conjugate (a) is an immediate consequence For (b ), we have of Definition element. just seen that If of f. 2.5 and the fact that g(x) < g(x), so by (a), x < fg(x), hence g(x) £ \y £ A: f(y) > xj; now if f(y) > x, then by (a), g(x) < y, which For (c), we have of A. for each x £ A, x £ A, \y £ A: f(y) > xj has a least £ A: f(y) > xj for each (i) and let f be an endomorphism f: y > xj, £A:/(y)= that for each / is an endomorphism. In our next g, then (d) g(l)=miniy (ii) by min B. of conjugates. algebra (a) x < f(y) ~ g(x) < y, (b) g(x) = min \y £ A : f(y) > xj, (c) set and B has a least x < fg(x), proves (b). so fg(x) £ {y £ tan f: y > xj; now if y £ ran /, say y = f(z), and y = f(z) > x, then by (a), g(x) < z, so /g(x) < ¡(z) = y, which proves (c). (ii) Finally, Clearly (d) is a special fg(x) < /(y) =» * < f(y). a Boolean case f(y) > x => g(x) < y. Thus, endomorphism, We introduce of (b). By hypothesis, x < /(y) <=>g(x) < y; from this we immediately two new conditions (7t4) for all K, X < a, From Lemma 3.1, with deduce and the fact that g is the conjugate for substitution Sx admits (775) for all K, X < a and every fg(x) > x, so g(x) < y => that / is of /. G algebras: a conjugate; x £ A, {y: Sxy > xj has a least Sx replacing /, together element. with Lemma 2.6(iv), we imme- diately get the following result: 3.2. Theorem. /t2 every substitution algebra, the following conditions are equivalent: (i) (77I) and (tt2) hold; (ii) (774) holds; (iii) (775) holds. A substitution gebra admitting From 3.1(i)b, lowing algebra satisfying these conditions is called a substitution al- conjugates. with two conditions Sx replacing are equivalent (D) Tx is the conjugate / and Tx replacing for every g, we deduce SAa admitting of Sx, tot all k, X < a, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use that the fol- conjugates: 176 CHARLES PINTER (E) [j anuary Txx = miniy: Sx y > xj, for ail k, X < a and every x £ A. Similarly, in every by 3.1(i)c, SAa admitting (F) with Sx replacing conjugates, / and Tx replacing (B) is equivalent cKx = S^TKxx if À / k, where TX is defined and by 3.1(i)d, (C) is equivalent (G) dKX= Tfl, equivalent by (D) (resp. (E)), to remarks, (775)) replacing The purpose that where T$ is defined by (D) (resp. (E)). By 3.2 and our last (?74) (resp. g, it follows to Theorems 2.7 and 2.14 can be re-stated (77I) and (772), (F) replacing of our next result is to show that with (B), and (G) replacing (773) may likewise (C). be replaced by conditions. We introduce three new conditions for SA's admitting conjugates: (776) If x £ rg K and y £ rg À and x < y, then there is some such that 2 £ rg k O rg X x < z < y. (nl) &.dKX/ a for all k, X < a, where d x is given by (C). (778) Si T£= T^SÏ if K / p, v and X / p, where T^ is defined by (D) or (E). Condition 3.3. Theorem. admitting tively, (776) is a special conjugates, case of condition // 21 = \A, +,•,-, and then the following (Ind) in Jurie 0, 1, Sx)K x<a is a substitution c , d . and Tx are defined three L8]. conditions algebra by (B), (C) and (D), respec- are equivalent: (i) (tt3) holds; (ii) (776)and (nl) hold; (iii) (778) holds. Proof. (773) => (776) A (777). If (773)holds, then 21 is a cylindric 2.7. Thus, if x £ rg k and y £ rg À and x < y, then rg À and ex = c ,c x = c c . x £ rg k, so x < ex < c algebra by y = y; but c^x £ rgK n rg X. This proves c. x £ (776); (777)holds by [6, I.3.3]. (776) A (777) => (7r3). Assume (3.4) If a= ci 3, this that (776) and (777) hold. £ rg v is the same statement We have for all v / k, X. as (777); thus, we assume that a> 3. K, A; for any n / k, v, SxS^dKX = S^SXdKX = 1, so by (?72), dKX<s^KX- lows by 2.4(i) and (vi) that cvdRX < c^S^d^ 2.4(vii), = S^dKy Let v 4- It fol- But S^dKX < cvdKX by hence (*) 5Xx=cAx for any 77^ k, 7,. Now by (777), there is some K, X, tot otherwise dKX = 1, and therefore assume for otherwise that p/v, p < a such that dKX £ rg p; we may assume (3.4) certainly (3.4) holds immediately. holds; Now, that similarly, let p / we may n / k, p, v; by (S6), SpvndKX = SvnS»dKX= SldKX, so SyKX £ rg p. But by (*), S»dKX = SyKX, hence S^dKX £ rg p. Thus, by (S3) and <S4), SvßdKX= S^d^ dKX, so d . £ rg v. Next, we have License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use = S^kX = 1973] CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS (3.5) c„c.=c.c„ Note first that cKcxx By' (B), z = c..c,x, K A hence CKcxx £ rß ^' so for all k, X < a. £ rg X. Indeed, c c x e rg K, so by (776), there by 2.4(iv), is some ex < cKcxx; = ckc\Xm This cally, cKc x < c cKx, proving (3.5). but c^x £ rg À and z £ rg k Pi rg À such that c.x c^c.x < c.x, K A £ rg° A. Now x — A cxckcXx 177 shows Finally, so cc^x X K that c c x < c < z < c cxx. < c,c„c,x; — A K A but c x; symmetri- if p 4 k, X, then cuSXx=^\-CKix-dKX)] by 2.6(H), = cKcp.{x-dKX] by3-5- = cK^cp.x ' dKX^ by 2-4(v)and = Sxc¿A by 2.6(H). 3-4> (778) =» (?73). By (778), (S6) and (F), it p 4 «, X, then SKxCflx = SKxS^T>^x = S^S^x^S^Slx-c^x. (773) =»(778). First, we note that in every (3.6) conjugates, T*x-cKx.dKk. Indeed, (3.6). it is easily verified We have already that (3.4) holds. that proved ex that • d . = minly: We shall operations gebra (n3) ==>(776) A (777), hence together (of degree algebra and algebras whose with substitutions a) with conjugates, T^ is the conjugate primitive and their briefly 1, Sx, r^)KX<awhere operations conjugates. a SACa, (A, +,-,-, ample, SACamay (57) be axiomatized Indeed, It is obvious that class( together ) of algebras. For ex- with TKxix + y) = TKxx + TKxy, (58) x<SÏTÏx, (s9) al- of Theorem 3.2. it is an equational by (Sj)—(Sg) By a substitution we mean an algebra of S1^ for all k, X < a. of SACa is that are the Boolean 0, 1, Sty KX<a is a substitution \A, + , • , _ > 0, 1, Sy K x<a W*H satisfy the conditions The interest we may assume G now consider + , -,-,0, S^y > x\, so by (E) we get Thus, by (7r3), (3.4) and (3.6), SKX T£ x = SKx(c^x •dfjLV)= ^XC^-SUau-c^x.d^^T^x. ^. SAa admitting and x>tkxskXx. suppose rem 1.14]. 21 is an SACa; then (S7) holds by Jonsson Next, - TKxx. T$x = 0, hence - S^T^x Sxx • S'x - x = 0, so Tx Sx x • - x - 0, giving (SA. and Tarski [7, Theo- • x = 0, giving (Sg). Finally, Conversely, one verifies imme- diately that if (S7)-(Sg) hold, then Tx is the conjugate of Sy From our last remarks, of substitution algebras ( ) For the notion together with Theorem with conjugates of equational class, satisfying see, 2.7, it is clear that the theory (778) is definitionally for example, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use [6, p. 44]. equivalent 178 CHARLES PINTER to the theory the strong of cylindric sense tion whether it is possible operations tutions algebras. that all added to define are the Boolean equivalent each for a class algebra additional of algebras operations lindric algebra c this, is equational subclass 0, 1, S'y) class equivalent x<a subject which and is any set to CA , where to (S,)—(S,) and are not equations. 0, 1, S'x)K x<a£ SAa which \A, + , • , -, 0, 1, c , d x> . is a cy- of SAa, given in Birkhoff if and only if K is closed and arbitrary to give an example direct under [l]: a class of dimension then B' = (B,U,n,^, the formation products. of subalge- For our purposes, of a CA^ with a subalgebra a) with base <d (see which it will is not a CA*. [6, Definition 1.1.5]). If we define CK(X n DkX) tot each X £ B, 0, °>ù>,^>KtX<û,isa P £ B be the set of all strictly that is, P = \p £ "&>: k < X => p < pxj. CA*. increasing sequences of natural Let C = ¡0, wu>, P, ~Pj; (C, U, O, -v, 0, "to, StyK x<ù) is a subalgebra of B'. for P ^ 0 and Sx P = 0; but this numbers, it is easy to see that for all k, X < co, S^ 0 = 0, S^Cco) = Mû),Sx P = 0, and S^P not be discrete, K of similar Let B = (ß, U, n, ~, 0, "co, CK, DKA>KiA<Cl) be a full cylindric SlX= Let of substi- we will show that the class of all (A, +,.,-, images 3.7. Example. set algebra only primitive K is an equational is definitionally axioms we use the criterion homomorphic suffice whose in the ques- and Sxx = c (x • d' x) tot all x £ A and k ¡¿ X. We will prove that To prove bras, which and d . such that CA* is not an equational algebras that raises (the conjugates in the negative: (A, +, . , -, must include Let CA* designate admit such this question is a structure conditions, K of algebras and substitutions as primitive) equivalent This to CAa. We will now answer of axioms they are definitionally are equational. a class operations are not to be taken definitionally In fact, definitions [January = œco. Thus, S = If S were a CA*^, it would contradicts [6, 2.1.9(ii)]; thus S is not a CA*ÙJ . 3.8. we have Remark. shown In Example that rem 2.7, condition 3.7, one easily S is not a CA*^; this verifies It might be of some interest to consider algebras algebras substitution and (B) holds. algebra with quantifiers, for example, (7r5). it clear algebras are easily The class details). derived \A, +,-,-, that But in Theo- 0, 1, Sx, cK'K x<a to where \A, +,.,-, of these it may be axiomatized 2.4 (i)—(v); (we omit the simple and cylindric makes (?r3) is indispensable. be called substitution to be equational; that S satisfies observation 0, 1, Sy algebras is easily by (Sj)—(S6) The relationship from Theorem License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use between 2.7. A together is a shown with these algebras 1973] CYLINDRIC ALGEBRAS AND ALGEBRAS OF SUBSTITUTIONS 179 BIBLIOGRAPHY 1. G. Birkhoff, On the structure of abstract algebras, Proc. 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Preller, Substitution algebras in their relation to cylindric algebras, Arch. Math. Logik. Grundlagenforsch. 13 (1970), 91-96. 11. H. Rasiowa and R. Sikorski, Math. 37 (1950), 193-200. A proof of the completeness theorem of GödeI, Fund. MR 12, 661. DEPARTMENT OF MATHEMATICS, BUCKNELL UNIVERSITY, LEWISBURG, PENNSYLVANIA 17837 DEPARTMENT OF MATHEMATICS,UNIVERSITY OF CALIFORNIA, BERKELEY, CALIFORNIA 94720 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use