Equationally Complete Rings and Relation Algebras

MATHEMATICS

EQUATIONALLY COMPLETE RINGS AND RELATION ALGEBRAS BY

ALFRED TARSKI (Communicated by Prof. A.

HEYTING

at the meeting of September 24, 1955)

The aim of this note is to give a simple mathematical characterization of the class of all equationally complete rings and of that of all equationally complete relation algebras. 1 ) The problem of providing such a characterization for these two classes of algebras was communicated to the author by JAN KALICKI. § l. GENERAL REMARKS. We shall use the terminology of KALICKIScoTT [4 ], without following the symbolism of that paper in all details. Two algebras 2{ and 5.8 which are similar (i.e., belong to the same species in the sense of [ 1]) will be called equationally equivalent if their sets of identities coincide. The following three simple theorems of general nature supplement the results stated in Section 1 of [4].

Theorem 1.1. The class of algebras generated by a given algebra 2{ coincides with the class of all algebras similar to 2{ in which every identity of 2£ holds. Proof: From Definition 1.13 of [4] it is easily seen that, if 5.8 is in the class of algebras generated by 2£, then every identity of 2{ holds in 5.8. Assume now, conversely, that 5.8 is in the class K of all algebras in which every identity of 2{ holds. Clearly, K is an equational class of algebras, and 2{ is a functionally free K-algebra in the sense of [8]. Hence, by the main theorem of [8] (which is a consequence of the results in [1]), every algebra in K, and in particular the algebra 5.8, belongs to the class generated by 2£. This completes the proof. As an immediate corollary of Theorem l.l. we obtain Theorem l. 2. For· any two (similar) algebras 2{ and 5.8 the following three conditions are equivalent: (i) 2{ and 5.8 are equationally equivalent; 1J The main results of this note are stated without proof in [9]. The note was prepared for publication while the author was working on a research project in me foundations of mathematics sponsored by the National Science Foundation, US.A.

40 (ii) each of the algebras W and ~ belongs to the class generated by the other algebra; (iii) the class of algebras generated by W coincides with that generated by~.

Theorem 1. 3. Let K be any equational class of algebras in which at least one algebra has more than one element. Then the following three conditions are equivalent: (i) K is equationally complete; (ii) every algebra in K with more than one element is equationally complete; (iii) any two algebras in K, each with more than one element, are equationally equivalent. Proof: If (i) holds, then, by Definition 1.12 of (4], K is definable by an equationally complete set X of equations. Let W and ~ be any two algebras in K with more than one element. W and ~ are models of X. Hence, by Theorem 1.3 of [4 ], X is the set of .identities for each of these two algebras, and therefore W and ~ are equationally equivalent. Thus (i) implies (iii). Assume now that (iii) holds. By the hypothesis and Theorem 1.4 of (4], K has a member W which is equationally complete; clearly, W has more than one element. By (iii), every algebra in K with more than one element is equationally equivalent to W and hence is equationally complete as well. Thus (iii) implies (ii). Finally, assume that (ii) holds. By the hypothesis and Theorem 1.5 of (4], K is generated by some algebra W. Since K contains algebras with more than one element, W must have more than one element. Therefore, by (ii), W is equationally complete, and hence, by Theorem l. 7 of [4 ], K is equationally complete. This shows that (ii) implies (i), and the proof is complete. § 2. RINGS. 2 ) By a ring we understand, as usual, an algebraic system with two binary operations, + and ·, assumed to satisfy certain familiar postulates. In this conception of rings, the class of rings is not equational. It becomes, however, equational if, e.g., the fundamental operation of addition, + , is replaced by that of subtraction, -, or if the two binary operations + and · are supplemented by the unary operation of forming negatives (additive inverses). It is easily seen that all the results stated below remain valid if the conception of a ring is modified in either of the two ways just indicated. For every ring 'iR =
pa= 0 and aP=a for every element a (cf. (6]); it is called a p-zero-ring if

pa=O=a·b 2)

For the notions involved in .this section consult, e.g., [10].

41

for all elements a and b. As opposed to the class of all rings, the class R11 of all p-rings and the class ZR11 of all p-zero-rings are equationaL For instance, as a system of equations defining R11 we can choose the set consisting of the commutative and associative laws for +, the commutative and associative laws for · , one of the two distributive laws for · under + , and the following two equations:

px+y=y, x 11 =x; if the latter equation is replaced by

X·y+y=y, we obtain a system of equations defining ZR11 (and in fact a system containing several redundant equations). By ffi11 we denote the ring
a·b=O

for all elements a, b of the ring. Here pis again a positive integer; however, we extend this notation to the case p = 0 by stipulating that ffi0 is the ring of all integers with ordinary addition and multiplication while 3ffi0 is the ring of all integers with ordinary addition and with multiplication defined as in the case of a 3ffi11 , p > 0. The following result was obtained by KALICKI and without proof in [5]; we shall supply a proof here.

ScoTT,

and stated

Theorem 2.1. For any given prime number p, every p-ring with more than one element is equationally complete, and so is every p-zero-ring. Proof: The class R11 of all p-rings is equational and has members with more than one element since e.g., ffi11 belongs to this class. Let ffi be an arbitrary p-ring with more than one element. It is known that ffi is isomorphic to a subalgebra of a direct power of ffi11 (cf. [6]); hence ffi belongs to the class of algebras generated by ffi11 • On the other hand, it is easily seen that every subring of ffi generated by an element a =F 0 is isomorphic to ffi11 , so that ffi11 belongs to the class generated by ffi. Consequently, by Theorem 1.2, ffi and ffiv are equationally equivalent. Hence any two p-rings with more than one element are equationally equivalent and therefore, by Theorem 1.3, every such p-ring is equationally complete. The proof for p-zero-rings is entirely analogous; the fact that every p-zero-ring is isomorphic to a subalgebra of a direct power of 3ffi11 can easily be derived from the well-known theorem which states that every Abelian group in which all non-zero elements are of order pis isomorphic to a subalgebra of a direct power-and, in fact, to what is sometimes called a weak direct power-- of the additive group of integers modulo p tcf., e.g., [7], p. 53).

42 The remarks in [4] at the end of Section 2 apply as well to the proof just outlined. Theorem 2.2. Every ring~ with more than one element has a subring which is homomorphic, for some prime number p, either to 9l, or to .8~11 • Proof: We distinguish the following two cases: (1) there is an element a in ~ such that a 2 =a*O; (2) there is no element a in ~ such that a2 =a*O. In case (1) the subring of~ generated by an element a with a 2 =a*O is easily seen to be isomorphic to ~n for some non-negative integer n* 1 and hence homomorphic to ~~~ for some prime number p. In case (2) we consider the subring ~' generated by an arbitrary element b 0, and we construct the principal ideal (b 2 ) in ~'. Obviously ~' is a commutative ring. If b belonged to (b 2 ), we would have for some element z in ~' and some integer n

*

b = z · b2 + nb 2 = (z · b + nb) · b; hence we could easily conclude that so that the element

(z · b + nb )2 = z. b +nb * 0, a=z·b+nb

would satisfy conditions which contradict our assumption (2). Thus b is not in (b 2 ). The quotient ring· ~'/(b 2 ) is clearly generated by the residue class 5 of b modulo (b 2 ); since b is not in (b 2 ) while b2 obviously is in (b 2 ), we have 5*0 and [j2=0 where 0 is the zero element of ~' /(b 2 ), i.e., simply coincides with (b 2 ). Hence, as is easily seen, the ring ~' /(b 2 ) is isomorphic to .8~n for some non-negative integer n* 1 and therefore homomorphic to .8~~~ for some prime number p. Consequently, the subring ~'of~ is itself homomorphic to gm,, which completes the proof. We can now establish the first principal result of this note: Theorem 2. 3. For a ring ~ with more than one element to be equationally complete it is necessary and sufficient that ~ be either a p-ring or a p-zero-ring for some prime number p. Proof: Assume that ~ is equationally complete. By Theorem 2.2, the class of algebras generated by ~ contains either ~~~or .8~11 for some prime number p; hence, by Theorem 1.6 of [4 ], ~is equationally equivalent either with ~~~ or with .8~11 , and therefore it is either a p-ring or a p-zeroring. Thus the condition of our theorem is necessary for~ to be equationally complete; by Theorem 2.1, it is also sufficient. Theorem 2.3 implies that the only equationally complete rings with a non-zero idempotent element are the p-rings, and the only equationally

43 complete rings with a non-zero nilpotent element are the p-zero-rings (where p is a prime). The second of these conclusions was previously found by KALICKI and ScoTT, and is stated in [5]. RELATION ALGEBRAS. 3 ) A relation algebra is an algebraic system Sf= (R, +, ;, -, ~, I') formed by a set R of arbitrary elements, two binary operations, + (Boolean addition) and ; (Peircean multiplication), two unary operations, -(complementation) and ~(conversion), and a distinguished element I' (the identity element) of R. The system Sf is assumed to satisfy certain postulates, all of which can be given the form of equations. A set of equations defining the class of relation algebras can be obtained, e.g., from Definition l.I in [2], p. 344, where (iv) is to be replaced by

§ 3.

a ; I'=a.

Thus the class of relation algebras is equational. For every relation algebra Sf= (R, +, ;, -, ~, I') we define the elements 0' (the diversity element), I (the unit element), 0 (the zero element), and the binary operation · (Boolean multiplication) by putting: O'=I',

I=I'+O',

-

O=I,

a·b=ii+b.

A relation algebra Sf is called Boolean if the equations

a ; b=a·b and a=a hold for arbitrary elements a, b of the algebra. The element l' can be defined (though not equationally) in terms of the operation ; and hence it can be removed from the system of fundamental notions of relation algebras (cf. [2]). In this case the class of relation algebras is no longer equational. The problem of characterizing mathematically those relation algebras (R, +, ;, -, ~) (with I' not regarded as a fundamental notion) which are equationally complete admits of a simple solution; in fact, by Theorem 3.5 of [4 ], a relation algebra in this sense is equationally complete if and only if it is Boolean. In the present note we shall concern ourselves exclusively with relation algebras (R, +, ;, -, ~, I'), for which the analogous problem is more complicated. It will be convenient to use an alternative notation for the elements 0, 1', I of a relation algebra, and in fact to denote them sometimes by C1 , C2 , C3 , respectively. If n is one of the three integers I, 2, 3, we shall denote by K the class of all relation algebras in which the following equations hold: (i,) 0' ; O'=C,., (ii) x; 1; x; I; (x·l' +x· O'); I; (x. O' +x·I') = o. (Since the operation ; is associative, most parentheses on the left side of (ii) have been omitted.) K<1 > proves to coincide with the class of all 3)

For the notions involved in this section see [2] and [3].

44

Boolean algebras; in defining K<1> we can omit equation (ii) and replace equation (i1 ) by a simpler one: 0'=0 (or 1'= 1). It is easily seen that no relation algebra with more than one element is a common member of any two of the three classes K<1 >, K< 2 >, K< 3>; in fact, in no such relation algebra do any two of the three equations (i1 ), (~), (is) hold. With every non-negative integer n we correlate a relation algebra m = (R, +, ;, -, ~, 1') constructed as follows. Let A be the set of all non-negative integers smaller than n. R consists of four (not necessarily distinct) binary relations - the empty relation 0, the unit relation U (the set of all ordered couples (a, b) with a and b in A), the identity relation I (the set of all ordered couples (a, a) with a in A), and the diversity relation D (the complement of I to U). For + and - we take the set-theoretical operations of addition (i.e., formation of unions) and complementation (with respect to U), for ; and~ the relation-theoretical operations of relative multiplication and conversion, and for 1' the identity relation I. It may be noticed that all the binary relations in R are symmetric, and hence a= a for every a in R. Clearly m has only one element, m has four distinct elements. We shall be interested exclusively in m(l)' m<2>' and m(3).

Theorem 3 .1. For n= l, 2, 3 and for every relation algebra m the following two conditions are equivalent: (i) m has more than one element, and the equation 0' ; 0' =c.. holds in m; (ii) mhas a subalgebra isomorphic to m. Proof: If (i) holds, we can easily check that the elements 1', 0', l, 0 of mform a subalgebra of misomorphic to m. Assmne, conversely, that (ii) holds, and let m' be a subalgebra of m isomorphic to m. Obviously m', and hence also m, has more than one element. The element l' of m must be in m' (since l' is one of the fundamental notions of relation algebras), and therefore the elements 0', 1, and 0 of m are also in m'; moreover, under the isomorphism which maps m' onto m, these four elements 1', 0', l, 0 must respectively be mapped onto the binary relations I, D, U, 0 which constitute m. Hence 0' ; 0' = C,. holds in m', as well as in the algebra m itself, and the proof is complete.

m

Theorem 3.2. For n= l, 2, 3 and for every relation algebra the following two conditions are equivalent: (i) mis simple and belongs to K; (ii) m is isomorphic to m. Proof: Assume that (i) holds. Hence, by the definition of K, the following two equations are identities of m: (1) (2)

0' ; 0' =C.. , x; 1; x; l; (x·1'+i·O'); l; (x·O' +i·1')=0.

45 It is known that a relation algebra ffi is simple if and only if it has more than one element and, for all elements a, b of ffi, the formula

a;1;b=0 implies that a=O or b=O (cf. [3], pp. 132 f.). Hence we conclude from (2) that ffi has no elements different from 0, 1, 0', and 1'. By combining this conclusion with (1) we easily check that ffi is isomorphic to ffiC">. Thus (i) implies (ii). The proof that, conversely, (ii) implies (i) is automatic. In connection with this theorem it may be mentioned that in every simple relation algebra one of the equations 0' ; 0' =C,., n= 1, 2, 3, must hold (cf. [3], p. 150); hence every simple relation algebra with at _most four distinct elements is isomorphic to one of the algebras ffi(l>, ffiCZ>, and ffics>. There is, however, a relation algebra with exactlyfour distinct elements which is isomorphic to none of these three algebras; it belongs to KC 1>, but is not simple. Theorem 3. 3. For n"= 1, 2, 3 and for every relation algebra ffi with more than one element the following three conditions are equivalent: (i) ffi belongs to Ken>; (ii) ffi is isomorphic to a subalgebra of a direct power of fficn>; (iii) ffi is equationally equivalent with ffiCn>. Proof: Assume first that (i) holds. It is known that ffi is isomorphic to a subalgebra of a direct product of simple relation algebras, each of which is a homomorphic image of ffi (see [3], p. 135). Since the class Ken> is equationally definable, we conclude (e.g., by Theorem 1.1) that all these simple algebras belong to Ken> and that consequently, by Theorem 3.2, (ii) holds. Assume now that (ii) holds. Since the equation 0' ; 0' = C,. holds in ffiC">, it holds in ffi as well; hence, by Theorem 3.1, ffi has a subalgebra isomorphic to fficn>. Thus each of the two algebras ffi and ffiCn> belongs to the class generated by the other algebra; by Theorem 1.2, this implies (iii). Finally, we can automatically check that (iii) implies (i), which completes the proof. Theorem 3.4. Every relation algebra ffi with more than one element ha.s a subalgebra which is homomorphic to one of the algebras fficU, fficz>, and ffiCS>. Proof: By a known result (cf. [3], Theorem 3.46, p. 151), ffi is isomorphic to the direct product of three relation algebras ffi1, ffi2, ffi3 such that the equation 0' ; 0'=0,. holds in ffi,., n= 1, 2, 3. At least one of these three algebras, say, ffi,.,, has more than one element. Clearly ffi is homomorphic to ffi,.,, and, by Theorem 3.1, ffi,., has a subalgebra isomorphic to ffiCn'>. Hence, remembering that a subalgebra of a homomorphic image of an algebra is a homomorphic image of a subalgebra, we immediately obtain the conclusion. (Instead of applying Theorem 3.46 of [3] in this argument, we could

46 use the fact that ffi is homomorphic to a simple relation algebra; cf. the proof of Theorem 3.3 above.) The second principal result of this note can now be easily established: Theorem 3.5. For a relation algebra ffi with more than one element to be equationally complete it is necessary and sufficient that ffi belong to one of the three classes K<1>, K< 2>, and K. Proof: Each class K, n= l, 2, 3, is equational and contains some algebras (e.g., the algebra ffi) with more than one element; by Theorem 3.3, any two algebras in K with more than one element are equationally equivalent. Hence, by Theorem 1.3, the condition of our theorem is sufficient for ffi to be equationally complete. To prove that this condition is also necessary, we argue as in the proof of Theorem 2.3, by applying Theorem 3.4 (and 3.3) instead of 2.2.

BIBLIOGRAPHY 1. BmKHOFF, G., On the structure of abstract algebras, Proceedings of the Cambridge

Philosophical Society, 31, 433-454 (1935). 2. 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). 3. J6NssoN, B. and A. TARSKI, Boolean algebras with operators, Part II, American Journal of Mathematics, 74, 127-162 (1952). 4. KALICKI, J. and D. ScoTT, Equational completeness of abstract algebras, Indagationes Mathematicae, 17, 650--659 (1955). and , Some equationally complete algebras, Bulletin of the 5. American Mathematical Society, 59, 77 f. (1953). 6. McCoY, N. H. and D. MONTGOMERY, Representation of generalized Boolean rings, Duke Mathematical Journal, 3, 455---459 (1937). 7. PRUFER, H., untersuchungen uber die Zerlegbarkeit der absiihlbaren primiiren Abelschen Gruppen, Mathematische Zeitschrift, 17, 35-61 (1923). 8. TARSKI, A., A remark on functionally free algebras, Annals of Mathematics, 47, 163-165 (1946). 9. , On equationally complete rings and relation algebras, Bulletin of the American Mathematical Society, 60, 142 (1954). 10. WAERDEN, B. L. VAN DER, Moderne Algebra, 1-2, (2nd edition, reprinted in U.S.A., New York, 1943).

University of California, Berkeley