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Contributions to the theory of models. III
MATHEMATICS
CONTRIBUTIONS TO THE THEORY OF MODELS. III 1 ) BY
ALFRED TARSKI (Communicated by Prof. A.
HEYTING
at the meeting of September 25, 1954)
Equational classes of algebras; the class of representable relation algebras A relation 0 of the rank v + 1 is referred to as an operation and, specifically, a v-ary operation (or a function with v arguments) if for every ordered v-tuple (x0 , .•• , xv_1) there is at most one element xv such that (x0 , •.• , xv-v xv) belongs to 0; this element xv (if it exists) is denoted by O(x0 , ••• , xv_1 ). The set of all ordered v-tuples (x0 , •.• , xv_1 ) for which such an element xv exists is called the domain of 0, and the set of the correlated elements xv is called the range of 0. We shall consider relational systems. W=(A, 0 0 , ••. , 0;, ... ) of an arbitrary order eX in which all the relations 0; are operations; 0; being a ,";:'ary operation, we assume that the domain of 0~ is A• and that the range is included in A. Such relational systems are referred to as algebraic systems or simply as algebras. Let R be a fixed similarity class and A the class of all algebras in R. Clearly, I(A) C A, U(A) C A, and P(A) C A; § 2.
i.e., isomorphic images of algebras, unions of chains of algebras, and cardinal products of algebras are always algebras. On the other hand, a subsystem or a homomorphic image of an algebra is not necessarily an algebra. Usually, however, when speaking of homomorphic images of an algebra 2!, we have in mind only those homomorphic images which are algebras. A subsystem of an algebra 2! which is itself an algebra is called a subalgebra of 2!; thus, S(W) n A is the class of all subalgebras of 2!. Let G be a set of elements of 2!; we say that 2! is generated by G if there is no subalgebra of 2! which is different from 2! and contains all elements of G. An algebra 2! which is generated by a finite set of elements is said to be finitely generated. Given a cardinal y, the class of all subalgebras of 2! which are generated by a set with less than y elements is denoted by Gy(W). Thus G"'(W) is the class of all finitely generated subalgebras of 2!. In case y>w and the order of 2! is smaller than y, we have Gy(W) = Sy(W)
n A.
As is easily seen, the class A is in AC 11 and, in case all algebras of A are 1) Parts I and II of the present paper were published in these Proceedings, A 57, 572-588 ( 1954).
57
of finite order, it is in AC. If, for instance, the similarity class R consists of all relational systems with one ternary relation represented by the predicate P in the formalism T( R), then A can be characterized as the class of all systems of R which are models of the sentence
Av0 Av1 Vv2 {P(v0 , Vv v2 ) A Av3 [P(v 0, Vv v3)--+ v2 =v3 ]}, and hence A E AC. The algebras of A can be studied either within the formalism T(R) outlined in§ 1 or within a modified formalism T(A). In this new formalism all the non-logical predicates are replaced by operation symbols (and, in fact, P+ 1-placed predicates by P-placed operation symbols); appropriate combinations of variables and operation symbols are called terms; the only atomic formulas are equations, i.e., expressions tX={3 where IX and {3 are terms. A sentence of T(A) of the form Av,., ... Av,.,(q;)
where q; is an equation is referred to as an identity. A class K of algebras (i.e., a subclass of A) is called an equational class, K E EC, if it consists of all models of a finite set of identities. K is called an equational class in the wider sense, K E EC"', if it consists of all 'models of an arbitrary (finite or infinite) set of identities or, what amounts to the same, if .it is an intersection of arbitrarily many equational classes 2 ). It can easily be shown that EC C UC(A) and EC"' C UCAA). In other words, every equational class of algebras is a universal class relative to A, and similarly for equational and universal classes in the wider sense. Instead of giving a formal proof of these inclusions, we shall explain the idea on a particular example. Assume, as before, that our similarity class R consists of all relational systems with one ternary relation and that consequently A consists of all algebras with one binary operation. Let + be the operation symbol which represents this operation. Consider the class K of all associative algebras (groupoids), i.e., of all models of the identity
Av0 Av1 Av2 [(v0 +v1 ) + v2 = v0 + (v1 + v2 )]. Obviously, K E EC. On the other hand, returning to the formalism T(R) and using the predicate P to represent the ternary relations of systems in R, we can characterize K as the class of all systems in A which are models 2) Equational classes of algebras were first discussed in [1], where, however, no special term to denote such classes was introduced. In [13], p. 63, and [14], p. 190, they are referred to as equationally definable classes. A purely mathematical definition of equational classes is given in [17].
58 of the following universal sentence:
/\v 0
•••
/\v 6 [P(v 0 , Vv v3)
A
P(v3 , v 2 , v 4 )
A
P(vv v2 , v 5 )
A
P(v0 , v 5 , v6 ) -+V4 = v6 ].
Thus K is the intersection of A and of a universal class, i.e., K E UC(A). The following mathematical characterization of equational classes (in the wider sense) was given by BIRKHOFF in [1], Th. 10, p. 441: Theorem 2.1. Let K be a class of algebras of arbitrary order. For K E ECLJ it is necessary and sufficient that the following three conditions be satisfied:
n A C K; H(K) n AC K; P(K) C K.
S(K)
(i)
(ii) (iii)
With the essential help of 2.1 the author has obtained m [17] the following result:
Let K be a class of algebras of arbitrary order. For K E ECLJ it is necessary and sufficient that there exist an algebra Ill such that
K= H [SP(Ill) n A] n A. By restricting ourselves to algebras of finite order and by applying some results of § 1, we shall now show that the set of conditions (i)-(iii) of 2.1 can be replaced by another equivalent set of conditions. (The new conditions can perhaps be regarded as somewhat simpler than the original ones since they do not involve cardinal products with arbitrarily many factors.) Theorem 2. 2. Let K be a class of algebras of finite order. For K E EC"' it is necessary and sufficient that the following three conditions be satisfied : (i) (ii) (iii)
for every Ill
E
A, if Sw(lll) C S( K), then Ill E K; H(K)nACK; P3 ( K) C K.
Condition (i) can be replaced by (i')
U[S(K)
n
A] C K.
Proof: We shall apply 1.12 and 1.14 with M replaced by A; the hypothesis P(M) C M of 1.14 is then obviously satisfied. As was noticed above, if K E ECLJ, we also have K E UCLJ(A); hence condition (i) of our theorem holds by 1.12. By 2.1, conditions (ii) and (iii) hold as well. Thus these three conditions are necessary for K E ECLJ. Assume now, conversely, that conditions (i)-(iii) are satisfied. Condition (i) clearly implies 2.1 (i); condition (ii) coincides with 2.1 (ii). Moreover, from (ii) we obtain I(K) C K and IS(K) C SI(K) C S(K). Hence (i) implies l.12(ii); since l.12(i) obviously holds, we conclude that K EUCLJ(A). Consequently, by 1.14, condition (iii) implies 2.1 (iii). Therefore
59 all the three conditions of 2.1 are satisfied and K E EC.a. Thus conditions (i)-(iii) of our theorem are sufficient for K E EC.a. To comp~te the proof we have to show that (i) can be replaced by (i'), This part of the argument is analogous to the last part of the proof of 1.2, but is somewhat more involved. In deriving (i) from (i') we first notice that (i') implies 2.1 (i). Then we consider an algebra m, with the smallest possible power, for which (i) may fail. We distinguish three possible cases dependent on whether m is finite, denumerable, or non-denumerable; and we make use of the fact that a non-denumerable algebra mcan be represented as the union of a chain of subalgebras (and not only of subsystems) each of which has a smaller power than m. Notice that 2.2 does not remain valid if 2.2 (i) is replaced by the following weaker condition: (i")
for every
m: E A,
m: E
if Gw(m) C S(K), then
K.
In fact, the class K of all torsion groups (i.e., groups without elements of infinite order) satisfies this condition (i") as well as conditions 2.2 (ii), (iii), but does not satisfy 2.2 (i) and 2.1 (iii}, and hence is not a member of EC.a. Thus it appears essential that 2.2 (i) involves arbitrary subsystems (and not only subalgebras) of an algebra m. CHANG, RUBIN, and the author have jointly obtained the following result which extends 2.2 to algebras of arbitrary order (cf. [5]): Theorem 2.3. Let y be an infinite cardinal and let K be a class of algebras of order ~
for every
m: E
A, if S11 (m) C S(K), then H(K) II A C K; P11 (K) C K.
m: E
K;
Several variants of this theorem are known. In particular, under the assumption that the cardinal y is larger than w, it is possible to replace 2.3 (i) by a condition which involves only subalgebras, and not arbitrary subsystems, of m. More specifically, (i) can be then replaced by the following condition: (i')
for every
m: E A,
if S11 (m) II A C S( K), then
m: E K;
or else by the following two conditions: (iv) (v)
for every
m: E A,
S(K) II A C K; if S11 (m:) II A C K, then
m: E
K.
The theorem also remains true (for y>w) if, in (i') and (v), S11 (m) II A is replaced by Gw(m). Finally, (v) can be replaced by the formula (v')
U(K) C K.
60 The proof of 2.3 will not be given here 3 ). For y = w the theorem obviously coincides with 2.2; for y =F- w it seems to require a method of reasoning which is essentially different from the one applied above in the proof of 2.2. The original proof of 2.3 for y =F- w was based upon ideas related to those which were used by BIRKHOFF in [1], in his proof of 2.1. CHANG has pointed out that 2.3 can be proved in a uniform manner, i.e., without distinguishing the cases y=w and y=F-w, and also without using the results of § 1. His proof is based upon the following general algebraic lemma which he has established (cf. [4]): Let y be an infinite cardinal and K a class of algebras (or, more generally, relational systems) of order fX < y. Then S,P(K) C ISP,(K). Due to this lemma, conditions (i)-(iii) of 2.1 can be derived from conditions (i)-(iii) of 2.2, or 2.3, by a purely algebraic method. Hence, by 2.1, conditions (i)-(iii) of 2.2, or 2.3, are sufficient for K E EC.d. The proof that all the conditions of 2.2 and 2.3 are necessary forK E EC.d is rather obvious. Nevertheless, it would be interesting to know whether these conditionsand, specifically, 2.2(i)-can be derived from those of 2.1 by means of an elementary and purely algebraic method. It may be interesting to notice that 2.1 applies, not only to algebras with finitary operations, but also to algebras with infinitary operations (which are not discussed in this paper). This is not the case as far as 2.2 and 2.3 are concerned. However, CHANG has established a result which is closely related to 2.3 and applies to algebras with infinitary operations (cf. [3]). In the remaining part of this section we shall concern ourselves with special algebras, in fact, with representable relation algebras. By applying a result of § 1 we shall show that the class of these algebras is equational (in the wider sense). A relation algebra 2r is a system formed by a set A, two binary operations, + and ;, two unary operations, - and-, and a distinguished element 1' of A; the element 1' can of course be replaced by a constant unary operation, so that 2r becomes an algebra in the sense established at the beginning of this section. The relation algebras are assumed to satisfy certain postulates all of which can be given the form of identities; thus the class of all relation algebras is equational 4 ). 3) The proof of 2.3 will be included in a paper by CHANG which is now being prepared for publication and which will embody his results summarized in [3] and [4]. 4) For notions involved in our discussion of relation algebras see [6] and also [10], part II. It is well known that Postulate (i) in the definition of relation algebras given in [6], p. 340, can be replaced by a (finite) set of equations. Since 1' has been included here in the system of fundamental notions of relation algebras, Postulate (iv) in this definition can also be replaced by an equation, in fact, by a; l' = a. Hence it is seen that the class of relation algebras is indeed equational.
IH
By a proper relation algebra we understand a relation algebra min which (i) A is a family of binary relations included in a given unit relation I, (ii) + is set-theoretical addition (formation of unions), (iii) ; is relationtheoretical multiplication (composition of relations), (iv)- is set-theoretical complementation with respect to I, (v)- is relation-theoretical conversion, and (vi) I' is the identity relation restricted to I, i.e., the set of all ordered couples (x, x) which are members of I. The unit relation is in general an equivalence relation (i.e., a symmetric and transitive relation); it may be in particular a relation of the form X 2 , i.e., it may consist of all couples (y, z) where y and z are elements of a given set X. An algebra which is isomorphic to a proper relation algebra is referred to as a representable relation algebra. By a result in Lyndon [II], p. 7I7, Th. 3, not every relation algebra is representable. Theorem 2.4. Let L be the class of all representable relation algebras. Then L E EC.1 5). Proof: Let K be the class of all atomistic relation algebras in which every atom is a functional element. From the definitions of an atom, an atomistic relation algebra, and a functional element it follows directly that K E AC.1 and, in fact, K E AC.1(A). J6NSSON and the author have shown that L=S(K) n A; cf. [IO], part II, Th. 4.31. Hence, by Theorem l.I3 (with M= A), L E UC<~(A). In other words, L consists of all algebras which are models of a certain set :E of universal sentences. (It may be mentioned that the conclusion L E UC<~(A) can also be derived by means of an entirely different method which is based upon Theorem I. 9 and the definition of proper relation algebras, and in which no specific results concerning arbitrary relation algebras and their connections with proper relation algebras, like Th. 4.3I in [IO], are involved.) Let L' be the class of all representable relation algebras which are simple (in the general algebraic sense of this term). It is known that, with every universal sentence a (of the theory of relation algebras), a well-determined identity a* can be correlated in such a way that a and a* are equivalent for all simple relation algebras; i.e., every simple relation algebra which is a model of a is also a model of a*, and conversely. (This easily follows, e.g., from [IO], part II, Th. 4.IO.) Let :E* be the set of the identities a* thus correlated with all universal sentences a which belong to the set E mentioned above; we see to it that all identities which form a postulate system for arbitrary relation algebras are included in :E*. Then, as is easily 5) Ths. 2.4 and 2.5 directly contradict certain results stated in [11], pp. 723, 726 (Th. 4), and 729 (footnote 19), as well as a certain consequence of these results which was pointed out in [22]. The contradiction is explained by the fact that the argument in [11] which leads to the results involved contains an error. The nature of this error will be cleared up in a note by Lyndon, to appear in Annals of Mathematics. At any rate it should be emphasized that the main result of [11] - the existence of a finite non-representable relation algebra - is not affected by this error and remains valid.
62
seen, L' coincides with the class of all simple algebras which are models of E*. Let now 2! be an arbitrary algebra which is a model of E*. Thus 2! is a relation algebra. Hence, as is known, 2! is a subalgebra of the cardinal product of simple relation algebras 2!i such that each of these algebras 2ii is a homomorphic image of 2!; cf. [10], part II, Th. 4.15 (a consequence of a general algebraic result in [2]). Therefore, by 2.1, all these algebras 2ii are models of E*, and hence they are representable relation algebras. As is easily seen, every cardinal product of representable relation algebras and every subalgebra of a representable relation algebra is itself representable. Consequently, 2! is a representable relation algebra. We want now to show that, conversely, every representable relation algebra 2! is a model of E*. Without loss of generality we can obviously assume that 2! is a proper relation algebra. The unit relation 1 of 2{ is an equivalence relation and hence, by the partition theorem, it can be represented as the union of mutually exclusive relations of the form Xi where i ranges over elements of a set I. For every i E I and for every relation R in 2! we put: fi(R)=R n
x;.
As is easily seen, the function fi thus defined maps homomorphically 2!
onto a proper relation algebra ~. The unit relation of 2!i is Xf; hence, 2ii is a simple representable relation algebra (cf. [10], part II, Th. 4.28) and therefore a model of E*. Moreover, we notice that 2! is isomorphic to a subalgebra of the cardinal product ~ie 1 (2ii) and hence 2! itself is a model of E* (cf. 2.1). Thus we have shown that the representable relation algebras coincide with the algebras which are models of a certain set E* of identities. Consequently, L E EC.a, and this is what was to be proved. It is interesting to notice that no direct and constructive proof of Th. 2.4 is known at present; i.e., we cannot exhibit any set of identities which would form an adequate postulate system for representable relation algebras. We do not know whether such a postulate system can be finite (i.e., whether L is in EC and not only in ECLJ).
Theorem 2.5. (i) Every algebra which is a homomorphic image of a representable relq,tion algebra is a representable relation algebra. (ii) If every finitely generated subalgebra of an algebra 2! is a representable relation algebra (or, more generally, if every finite subsystem of an algebra 2! is isomorphically embeddable in a representable relation algebra), then 2! is itself a representable relation algebra. Proof: This is an immediate consequence of 2.2 and 2.4. We do not know how to prove Th. 2.5 directly, without the help of 2.4; this is the reason why we have been unable to base the proof of 2.4 upon one of the general theorems of this section, 2.1, 2.2, or 2.3.
63 Th. 2.4 can be partly extended to the class L of those relation algebras which are isomorphically representable as subalgebras of complex algebras of groups (cf. [10], part I, Def. 3.8). By applying Th. 1.13 of this paper and Ths. 5.10 and 5.11 of [10], part II, we easily show that L E UC.1(A) and, in fact, that L consists of all simple relation algebras which are models of a certain set of identities. Hence 2.5 also applies to algebras of L; of course, this is trivial as far as 2.5 (i) is concerned since all algebras of L are simple. Ths. 2.4 and 2.5 can be fully extended to representable finitely-dimensional cylindric algebras (cf. [18] and [26]). The proof will not be given here; generally speaking, it is analogous to that applied above in the discussion of representable relation algebras. BIBLIOGRAPHY l.
2. 3.
4. 5.
6.
7.
8.
9. 10.
11. 12.
13. 14. 15.
BIRKHOFF, G., On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31, 433-454 (1935). , Subdirect unions in universal algebras. Bulletin of the American Mathematical Society, 50, 764-768 (1944). CHANG, C. C., A characterization of equational classes of algebras with arbitrary operations. Bulletin of the American Mathematical Society, 60, 76 (1954). , Two general theorems on direct products of algebras. Bulletin of the American Mathematical Society, 60, 76 (1954). , H. RuBIN and A. TARSKI, A characterization of equational classes of algebras with finitary operations. Bulletin of the American Mathematical Society, 60, 76-77 (1954). CHIN, L. H. and A. TARSKI, Distributive and modular laws in the arithmetic of relation algebras. University of California Publications in Mathematics, new series, 1, 341-384 (1951). FRA'iss:E, R., Sur une classification des systemes de relations faisant intervenir les ordinaux transfinis. Comptes Rendus des Seances de l'Academie des Sciences (Paris), 228, 1682-1684 (1949). HENKIN, L., Some interconnections between modern algebra and mathematical logic. Transactions of the American Mathematical Society, 74, 410-427 (1953). HoRN, A., On sentences which are true of direct unions of algebras. Journal of Symbolic Logic, 16, 14-21 (1951). J6NSSON, B. and A. TARSKI, Boolean algebras with operators. Part I, American Journal of Mathematics, 73, 891-939 (1951); part II, ibid., 74, 127-162 (1952). . LYNDON, R. C., The representation of relational algebras. Annals of Mathematics, 51, 707-729 (1950). MALOEV, A., Uber die Einbettung von assoziativen Systemen in Gruppen. Part I, Matematicheskii Sbornik, 6, 331-336 (1939); part II, ibid., 8, 251-264 (1940). McKINSEY, J. C. C., The decision problem for some classes of sentences without quantifiers. Journal of Symbolic Logic, 8, 61-76 (1943). and A. TARSKI, The algebra of topology. Annals of Mathematics, 45, 141-191 (1944). RASIOWA, M., A proof of the compactness theorem for arithmetical classes. Fundamenta Mathematicae, 39, 8-14 (1952).
64 16. 17. 18. 19. 20. 21. 22. 23.
24. 25. 26. 27. 28.
RoBINSON, A., On the metamathematics of algebra. X+195 pp. (Amsterdam, 1951). TARSKI, A., A remark on functionally free algebras. Annals of Mathematics, 47, 163-165 (1946). , A representation theorem for cylindric algebras. Bulletin of the American Mathematical Society, 58, 65-66 (1952). , Der Wahrheitsbegritf in den formalisierten Sprachen. Studia Philosophica, 1, 261-404 (1936). , Grundzuge des Systemenkalkuls. Part I, Fundamenta Mathematicae, 25, 503-526 (1935); part II, ibid., 26, 283-301 (1936). , On equational classes of algebras with finitary operations. Bulletin of the American Mathematical Society, 60, 78 (1954). , On representable relation algebras. Bulletin of the American Mathematical Society, 58, 172 (1952). , Some notions and methods on the borderline of algebra and metamathematics. Proceedings of the International Congress of Mathematicians, Providence, R. I., 1, 705-720 (1952). , Universal arithmetical classes of mathematical systems. Bulletin of the American Mathematical Society, 59, 390-391 (1953). , A. MosTOWSKI, and R. M. RoBINSON. Undecidable theories. XII+98 pp. (Amsterdam, 1953). and F. B. THOMPSON, Some general properties of cylindric algebras. Bulletin of the American Mathematical Society, 58, 65 (1952). VAUGHT, R. L., Remarks on universal arithmetical classes. Bulletin of the American Mathematical Society, 59, 391 (1953). WAERDEN, B. L. VAN DER, Moderne Algebra. Vol. 1, X+272 pp. (second edition, Berlin (1937}, reprinted New York (1943)).
University of California, Berkeley.
CONTRIBUTIONS TO THE THEORY OF MODELS. III 1 ) BY
ALFRED TARSKI (Communicated by Prof. A.
HEYTING
at the meeting of September 25, 1954)
Equational classes of algebras; the class of representable relation algebras A relation 0 of the rank v + 1 is referred to as an operation and, specifically, a v-ary operation (or a function with v arguments) if for every ordered v-tuple (x0 , .•• , xv_1) there is at most one element xv such that (x0 , •.• , xv-v xv) belongs to 0; this element xv (if it exists) is denoted by O(x0 , ••• , xv_1 ). The set of all ordered v-tuples (x0 , •.• , xv_1 ) for which such an element xv exists is called the domain of 0, and the set of the correlated elements xv is called the range of 0. We shall consider relational systems. W=(A, 0 0 , ••. , 0;, ... ) of an arbitrary order eX in which all the relations 0; are operations; 0; being a ,";:'ary operation, we assume that the domain of 0~ is A• and that the range is included in A. Such relational systems are referred to as algebraic systems or simply as algebras. Let R be a fixed similarity class and A the class of all algebras in R. Clearly, I(A) C A, U(A) C A, and P(A) C A; § 2.
i.e., isomorphic images of algebras, unions of chains of algebras, and cardinal products of algebras are always algebras. On the other hand, a subsystem or a homomorphic image of an algebra is not necessarily an algebra. Usually, however, when speaking of homomorphic images of an algebra 2!, we have in mind only those homomorphic images which are algebras. A subsystem of an algebra 2! which is itself an algebra is called a subalgebra of 2!; thus, S(W) n A is the class of all subalgebras of 2!. Let G be a set of elements of 2!; we say that 2! is generated by G if there is no subalgebra of 2! which is different from 2! and contains all elements of G. An algebra 2! which is generated by a finite set of elements is said to be finitely generated. Given a cardinal y, the class of all subalgebras of 2! which are generated by a set with less than y elements is denoted by Gy(W). Thus G"'(W) is the class of all finitely generated subalgebras of 2!. In case y>w and the order of 2! is smaller than y, we have Gy(W) = Sy(W)
n A.
As is easily seen, the class A is in AC 11 and, in case all algebras of A are 1) Parts I and II of the present paper were published in these Proceedings, A 57, 572-588 ( 1954).
57
of finite order, it is in AC. If, for instance, the similarity class R consists of all relational systems with one ternary relation represented by the predicate P in the formalism T( R), then A can be characterized as the class of all systems of R which are models of the sentence
Av0 Av1 Vv2 {P(v0 , Vv v2 ) A Av3 [P(v 0, Vv v3)--+ v2 =v3 ]}, and hence A E AC. The algebras of A can be studied either within the formalism T(R) outlined in§ 1 or within a modified formalism T(A). In this new formalism all the non-logical predicates are replaced by operation symbols (and, in fact, P+ 1-placed predicates by P-placed operation symbols); appropriate combinations of variables and operation symbols are called terms; the only atomic formulas are equations, i.e., expressions tX={3 where IX and {3 are terms. A sentence of T(A) of the form Av,., ... Av,.,(q;)
where q; is an equation is referred to as an identity. A class K of algebras (i.e., a subclass of A) is called an equational class, K E EC, if it consists of all models of a finite set of identities. K is called an equational class in the wider sense, K E EC"', if it consists of all 'models of an arbitrary (finite or infinite) set of identities or, what amounts to the same, if .it is an intersection of arbitrarily many equational classes 2 ). It can easily be shown that EC C UC(A) and EC"' C UCAA). In other words, every equational class of algebras is a universal class relative to A, and similarly for equational and universal classes in the wider sense. Instead of giving a formal proof of these inclusions, we shall explain the idea on a particular example. Assume, as before, that our similarity class R consists of all relational systems with one ternary relation and that consequently A consists of all algebras with one binary operation. Let + be the operation symbol which represents this operation. Consider the class K of all associative algebras (groupoids), i.e., of all models of the identity
Av0 Av1 Av2 [(v0 +v1 ) + v2 = v0 + (v1 + v2 )]. Obviously, K E EC. On the other hand, returning to the formalism T(R) and using the predicate P to represent the ternary relations of systems in R, we can characterize K as the class of all systems in A which are models 2) Equational classes of algebras were first discussed in [1], where, however, no special term to denote such classes was introduced. In [13], p. 63, and [14], p. 190, they are referred to as equationally definable classes. A purely mathematical definition of equational classes is given in [17].
58 of the following universal sentence:
/\v 0
•••
/\v 6 [P(v 0 , Vv v3)
A
P(v3 , v 2 , v 4 )
A
P(vv v2 , v 5 )
A
P(v0 , v 5 , v6 ) -+V4 = v6 ].
Thus K is the intersection of A and of a universal class, i.e., K E UC(A). The following mathematical characterization of equational classes (in the wider sense) was given by BIRKHOFF in [1], Th. 10, p. 441: Theorem 2.1. Let K be a class of algebras of arbitrary order. For K E ECLJ it is necessary and sufficient that the following three conditions be satisfied:
n A C K; H(K) n AC K; P(K) C K.
S(K)
(i)
(ii) (iii)
With the essential help of 2.1 the author has obtained m [17] the following result:
Let K be a class of algebras of arbitrary order. For K E ECLJ it is necessary and sufficient that there exist an algebra Ill such that
K= H [SP(Ill) n A] n A. By restricting ourselves to algebras of finite order and by applying some results of § 1, we shall now show that the set of conditions (i)-(iii) of 2.1 can be replaced by another equivalent set of conditions. (The new conditions can perhaps be regarded as somewhat simpler than the original ones since they do not involve cardinal products with arbitrarily many factors.) Theorem 2. 2. Let K be a class of algebras of finite order. For K E EC"' it is necessary and sufficient that the following three conditions be satisfied : (i) (ii) (iii)
for every Ill
E
A, if Sw(lll) C S( K), then Ill E K; H(K)nACK; P3 ( K) C K.
Condition (i) can be replaced by (i')
U[S(K)
n
A] C K.
Proof: We shall apply 1.12 and 1.14 with M replaced by A; the hypothesis P(M) C M of 1.14 is then obviously satisfied. As was noticed above, if K E ECLJ, we also have K E UCLJ(A); hence condition (i) of our theorem holds by 1.12. By 2.1, conditions (ii) and (iii) hold as well. Thus these three conditions are necessary for K E ECLJ. Assume now, conversely, that conditions (i)-(iii) are satisfied. Condition (i) clearly implies 2.1 (i); condition (ii) coincides with 2.1 (ii). Moreover, from (ii) we obtain I(K) C K and IS(K) C SI(K) C S(K). Hence (i) implies l.12(ii); since l.12(i) obviously holds, we conclude that K EUCLJ(A). Consequently, by 1.14, condition (iii) implies 2.1 (iii). Therefore
59 all the three conditions of 2.1 are satisfied and K E EC.a. Thus conditions (i)-(iii) of our theorem are sufficient for K E EC.a. To comp~te the proof we have to show that (i) can be replaced by (i'), This part of the argument is analogous to the last part of the proof of 1.2, but is somewhat more involved. In deriving (i) from (i') we first notice that (i') implies 2.1 (i). Then we consider an algebra m, with the smallest possible power, for which (i) may fail. We distinguish three possible cases dependent on whether m is finite, denumerable, or non-denumerable; and we make use of the fact that a non-denumerable algebra mcan be represented as the union of a chain of subalgebras (and not only of subsystems) each of which has a smaller power than m. Notice that 2.2 does not remain valid if 2.2 (i) is replaced by the following weaker condition: (i")
for every
m: E A,
m: E
if Gw(m) C S(K), then
K.
In fact, the class K of all torsion groups (i.e., groups without elements of infinite order) satisfies this condition (i") as well as conditions 2.2 (ii), (iii), but does not satisfy 2.2 (i) and 2.1 (iii}, and hence is not a member of EC.a. Thus it appears essential that 2.2 (i) involves arbitrary subsystems (and not only subalgebras) of an algebra m. CHANG, RUBIN, and the author have jointly obtained the following result which extends 2.2 to algebras of arbitrary order (cf. [5]): Theorem 2.3. Let y be an infinite cardinal and let K be a class of algebras of order ~
for every
m: E
A, if S11 (m) C S(K), then H(K) II A C K; P11 (K) C K.
m: E
K;
Several variants of this theorem are known. In particular, under the assumption that the cardinal y is larger than w, it is possible to replace 2.3 (i) by a condition which involves only subalgebras, and not arbitrary subsystems, of m. More specifically, (i) can be then replaced by the following condition: (i')
for every
m: E A,
if S11 (m) II A C S( K), then
m: E K;
or else by the following two conditions: (iv) (v)
for every
m: E A,
S(K) II A C K; if S11 (m:) II A C K, then
m: E
K.
The theorem also remains true (for y>w) if, in (i') and (v), S11 (m) II A is replaced by Gw(m). Finally, (v) can be replaced by the formula (v')
U(K) C K.
60 The proof of 2.3 will not be given here 3 ). For y = w the theorem obviously coincides with 2.2; for y =F- w it seems to require a method of reasoning which is essentially different from the one applied above in the proof of 2.2. The original proof of 2.3 for y =F- w was based upon ideas related to those which were used by BIRKHOFF in [1], in his proof of 2.1. CHANG has pointed out that 2.3 can be proved in a uniform manner, i.e., without distinguishing the cases y=w and y=F-w, and also without using the results of § 1. His proof is based upon the following general algebraic lemma which he has established (cf. [4]): Let y be an infinite cardinal and K a class of algebras (or, more generally, relational systems) of order fX < y. Then S,P(K) C ISP,(K). Due to this lemma, conditions (i)-(iii) of 2.1 can be derived from conditions (i)-(iii) of 2.2, or 2.3, by a purely algebraic method. Hence, by 2.1, conditions (i)-(iii) of 2.2, or 2.3, are sufficient for K E EC.d. The proof that all the conditions of 2.2 and 2.3 are necessary forK E EC.d is rather obvious. Nevertheless, it would be interesting to know whether these conditionsand, specifically, 2.2(i)-can be derived from those of 2.1 by means of an elementary and purely algebraic method. It may be interesting to notice that 2.1 applies, not only to algebras with finitary operations, but also to algebras with infinitary operations (which are not discussed in this paper). This is not the case as far as 2.2 and 2.3 are concerned. However, CHANG has established a result which is closely related to 2.3 and applies to algebras with infinitary operations (cf. [3]). In the remaining part of this section we shall concern ourselves with special algebras, in fact, with representable relation algebras. By applying a result of § 1 we shall show that the class of these algebras is equational (in the wider sense). A relation algebra 2r is a system formed by a set A, two binary operations, + and ;, two unary operations, - and-, and a distinguished element 1' of A; the element 1' can of course be replaced by a constant unary operation, so that 2r becomes an algebra in the sense established at the beginning of this section. The relation algebras are assumed to satisfy certain postulates all of which can be given the form of identities; thus the class of all relation algebras is equational 4 ). 3) The proof of 2.3 will be included in a paper by CHANG which is now being prepared for publication and which will embody his results summarized in [3] and [4]. 4) For notions involved in our discussion of relation algebras see [6] and also [10], part II. It is well known that Postulate (i) in the definition of relation algebras given in [6], p. 340, can be replaced by a (finite) set of equations. Since 1' has been included here in the system of fundamental notions of relation algebras, Postulate (iv) in this definition can also be replaced by an equation, in fact, by a; l' = a. Hence it is seen that the class of relation algebras is indeed equational.
IH
By a proper relation algebra we understand a relation algebra min which (i) A is a family of binary relations included in a given unit relation I, (ii) + is set-theoretical addition (formation of unions), (iii) ; is relationtheoretical multiplication (composition of relations), (iv)- is set-theoretical complementation with respect to I, (v)- is relation-theoretical conversion, and (vi) I' is the identity relation restricted to I, i.e., the set of all ordered couples (x, x) which are members of I. The unit relation is in general an equivalence relation (i.e., a symmetric and transitive relation); it may be in particular a relation of the form X 2 , i.e., it may consist of all couples (y, z) where y and z are elements of a given set X. An algebra which is isomorphic to a proper relation algebra is referred to as a representable relation algebra. By a result in Lyndon [II], p. 7I7, Th. 3, not every relation algebra is representable. Theorem 2.4. Let L be the class of all representable relation algebras. Then L E EC.1 5). Proof: Let K be the class of all atomistic relation algebras in which every atom is a functional element. From the definitions of an atom, an atomistic relation algebra, and a functional element it follows directly that K E AC.1 and, in fact, K E AC.1(A). J6NSSON and the author have shown that L=S(K) n A; cf. [IO], part II, Th. 4.31. Hence, by Theorem l.I3 (with M= A), L E UC<~(A). In other words, L consists of all algebras which are models of a certain set :E of universal sentences. (It may be mentioned that the conclusion L E UC<~(A) can also be derived by means of an entirely different method which is based upon Theorem I. 9 and the definition of proper relation algebras, and in which no specific results concerning arbitrary relation algebras and their connections with proper relation algebras, like Th. 4.3I in [IO], are involved.) Let L' be the class of all representable relation algebras which are simple (in the general algebraic sense of this term). It is known that, with every universal sentence a (of the theory of relation algebras), a well-determined identity a* can be correlated in such a way that a and a* are equivalent for all simple relation algebras; i.e., every simple relation algebra which is a model of a is also a model of a*, and conversely. (This easily follows, e.g., from [IO], part II, Th. 4.IO.) Let :E* be the set of the identities a* thus correlated with all universal sentences a which belong to the set E mentioned above; we see to it that all identities which form a postulate system for arbitrary relation algebras are included in :E*. Then, as is easily 5) Ths. 2.4 and 2.5 directly contradict certain results stated in [11], pp. 723, 726 (Th. 4), and 729 (footnote 19), as well as a certain consequence of these results which was pointed out in [22]. The contradiction is explained by the fact that the argument in [11] which leads to the results involved contains an error. The nature of this error will be cleared up in a note by Lyndon, to appear in Annals of Mathematics. At any rate it should be emphasized that the main result of [11] - the existence of a finite non-representable relation algebra - is not affected by this error and remains valid.
62
seen, L' coincides with the class of all simple algebras which are models of E*. Let now 2! be an arbitrary algebra which is a model of E*. Thus 2! is a relation algebra. Hence, as is known, 2! is a subalgebra of the cardinal product of simple relation algebras 2!i such that each of these algebras 2ii is a homomorphic image of 2!; cf. [10], part II, Th. 4.15 (a consequence of a general algebraic result in [2]). Therefore, by 2.1, all these algebras 2ii are models of E*, and hence they are representable relation algebras. As is easily seen, every cardinal product of representable relation algebras and every subalgebra of a representable relation algebra is itself representable. Consequently, 2! is a representable relation algebra. We want now to show that, conversely, every representable relation algebra 2! is a model of E*. Without loss of generality we can obviously assume that 2! is a proper relation algebra. The unit relation 1 of 2{ is an equivalence relation and hence, by the partition theorem, it can be represented as the union of mutually exclusive relations of the form Xi where i ranges over elements of a set I. For every i E I and for every relation R in 2! we put: fi(R)=R n
x;.
As is easily seen, the function fi thus defined maps homomorphically 2!
onto a proper relation algebra ~. The unit relation of 2!i is Xf; hence, 2ii is a simple representable relation algebra (cf. [10], part II, Th. 4.28) and therefore a model of E*. Moreover, we notice that 2! is isomorphic to a subalgebra of the cardinal product ~ie 1 (2ii) and hence 2! itself is a model of E* (cf. 2.1). Thus we have shown that the representable relation algebras coincide with the algebras which are models of a certain set E* of identities. Consequently, L E EC.a, and this is what was to be proved. It is interesting to notice that no direct and constructive proof of Th. 2.4 is known at present; i.e., we cannot exhibit any set of identities which would form an adequate postulate system for representable relation algebras. We do not know whether such a postulate system can be finite (i.e., whether L is in EC and not only in ECLJ).
Theorem 2.5. (i) Every algebra which is a homomorphic image of a representable relq,tion algebra is a representable relation algebra. (ii) If every finitely generated subalgebra of an algebra 2! is a representable relation algebra (or, more generally, if every finite subsystem of an algebra 2! is isomorphically embeddable in a representable relation algebra), then 2! is itself a representable relation algebra. Proof: This is an immediate consequence of 2.2 and 2.4. We do not know how to prove Th. 2.5 directly, without the help of 2.4; this is the reason why we have been unable to base the proof of 2.4 upon one of the general theorems of this section, 2.1, 2.2, or 2.3.
63 Th. 2.4 can be partly extended to the class L of those relation algebras which are isomorphically representable as subalgebras of complex algebras of groups (cf. [10], part I, Def. 3.8). By applying Th. 1.13 of this paper and Ths. 5.10 and 5.11 of [10], part II, we easily show that L E UC.1(A) and, in fact, that L consists of all simple relation algebras which are models of a certain set of identities. Hence 2.5 also applies to algebras of L; of course, this is trivial as far as 2.5 (i) is concerned since all algebras of L are simple. Ths. 2.4 and 2.5 can be fully extended to representable finitely-dimensional cylindric algebras (cf. [18] and [26]). The proof will not be given here; generally speaking, it is analogous to that applied above in the discussion of representable relation algebras. BIBLIOGRAPHY l.
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