J. A. Kalman, Applications of Subdirect Products in General Algebra, P 45

APPLICATIONS OF SUBDIRECT PRODUCTS IN GENERAL ALGEBRA*

J. A. Kalman

(received 20 November, 1973)

1. Introduction

I very much appreciate the Colloquium Committee's invitation to give a general paper in algebra to be delivered to the whole Colloquium. I hope this talk will be suitable for a general mathemati cal audience, but it will not be a survey of algebra or even of a major branch of algebra. Instead I have chosen a much easier task. I shall begin by discussing a concrete example and a very easily formula ted problem concerning it - this problem could, in fact, have been formulated 200 years ago, in the time of Euler, or even 300 years ago, when Newton was still in his twenties. The problem itself is quite special, but it belongs to a class of problems on which considerable research has been done recently (of. [26]), and it leads to a conjec ture concerning certain related problems in this class.

The problem in question can be solved by applying a theorem of Garrett Birkhoff [3, Theorem 2]. Birkhoff's theorem is not at all Eulerian or Newtonian, inasmuch as its proof uses an equivalent of the axiom of choice. However, the statement of this theorem is not very difficult to understand, and its application to the problem mentioned is quite easy to follow.

After stating Birkhoff's theorem, and indicating how it may be used in solving the problem referred to above, I shall conclude by discussing briefly two of the standard applications of Birkhoff's theorem, namely Stone's theorem [23; 24, Theorem III] that every Boolean algebra is isomorphic to a field of sets, and an earlier

*Invited address delivered at the Sixth New Zealand Mathematics Colloquium, held at Wellington, 17-19 May, 1971.

Math. Chronicle 3(1974), 45-62.

45 theorem of Birkhoff [1, Theorem 25.2] that every distributive lattice is isomorphic to a ring of sets.

2. An example, a problem, and a conjecture

Example.The table

0 1 2

0 0 0 2 1 1 0 2 2 2 2 2 completely describes a binary operation, which we call 'multiplication' and denote by a dot or by juxtaposition, on the set {0,1,2} = G say; in fact we have 0.1 = 1.1 = 0 and, for all x in G , x.O = x and x.2 = 2.x = 2 . The pair G = <£,.> consisting of the set G and the operation . on G is an example of an 'algebra'.

In general, by an algebra A we mean (cf. [4, p. 132; 6, p. 41; 9, p. 8; 22, p. 16]) a non-empty set A (called the universe (of. [26, p. 275]) of A) together with a family [9, p. 4] of operations (not necessarily binary, but required to be 'finitary', i.e. n-ary for some n - 0,1,2,•••) on A . A finite algebra is one whose universe is finite. A groupoid is an algebra with a single binary operation (thus G is a groupoid). In this paper we shall use cursive letters A , B , •• for algebras and the corresponding Latin letters A ,B, ••• for their universes.

'Universal algebra' has been described as 'the study of features common to familiar algebraic systems such as groups, rings, lattices, etc.' [6, p. xiii]; the purpose of research in universal algebra is 'to find and develop the properties which such diverse algebras as rings, fields, Boolean algebras, lattices, and groups may have in common' [9, p. v]. According to Pierce [22, p. v], familiarity with the theory of universal algebras ’should be standard equipment for all mathematicians' but a remark of G. Higman [11] is not quite so complimentary: 'universal algebra is something everyone ought to know about, though nobody should

46 specialise in it.’ I shall refer to this remark of Higman's again later.

Some of the results about algebras which we shall be considering below are concerned not so much with common properties of groups, rings, lattices, etc. as with properties of classes of algebras defined in other ways. In view of this it seemed desirable to use some expression other than 'universal algebra' in the title of the talk - hence the use of 'general algebra', which is intended to convey a wider meaning.

Given an algebra A and denumerably many variables x, y, z, •••, we can consider the set ThA of all those 'laws' (also called 'identities' or 'equations') in the operations of A and the given variables which are satisfied by A ; ThA is called the equational theory of A [26, p. 276]. Well-known laws for a binary operation are the commutative and associative laws; in G , 1.0 = 1 but 0.1 = 0 , hence the commutative law xy = yx is not in T h G ; also (1.0).1 = 1.1 = 0 but 1.(0.1) = 1.0 = 1 , hence G is not associative.

Possibly you are wondering whether G satisfies any laws, but, given a minute or two, you could easily find some. For instance, it is easily checked that the laws (1) x(xx) = x , xy = (xy)y , and (xy)z = (xz)y hold in G . From these laws we can derive other laws which hold in G by the logical properties of equality (i.e. that x = x is a law, and that from laws of the form t = u and u = v we may derive the laws u = t and t = v ) , by replacement (e.g. if we replace xy in the third law of (1) according to the second law of (1) we obtain the law ((xy)y)z = (xz)y ), and by uniform substitution of terms for variables (e.g. if we substitute the term xz for the variable y throughout the second law of (1) we obtain the law x(xz) = (x(xz))(xz) ) (cf. [2, Theorem 7]).

The question now arises whether we can derive all the laws in

47 ThG from those in (1) by repeated use of the logical properties of equality, replacement, and substitution. The answer to this question is 'no', as may be seen by noting that the law x(xy) = y(yx) is in ThG , and that the multiplicative groupoid with universe {0,1} and multiplication table 0 1

0 0 0 1 1 1 satisfies the laws in (1) but does not satisfy the law x(xy) = y{yx) .

The above reasoning depends on the elementary fact that if a law X can be derived from the laws in a set E by repeated use of the logical properties of equality, replacement, and substitution, then X is a consequence of E in the sense that every algebra which satisfies all the laws in E will also satisfy X . The converse of this fact is also true; in other words the syntactic notion of being derivable and the semantic notion of being a consequence are equivalent (Birkhoff's completeness theorem (cf. [2, Theorem 9; 6, Theorem IV.1.2; 9, Theorem 26.2; 26, p. 276])). However, by framing our definitions in terms of the notion of consequence rather than that of derivability, we are able to avoid requiring this converse below.

Given a set E of laws in denumerably many given variables for a given system of operations, we shall denote by Cn E the set of all those laws X in the given variables for the given operations such that X is a consequence of E in the sense of the previous paragraph. Now let A be any algebra with the given system of operations. Then we shall call E a base for Th A if CnE = ThA , and we shall say that the equational theory of A is finitely based if there exists a finite base for ThA [26, p. 277]. Of course, Birkhoff's completeness theorem gives equivalent formulations of these concepts in terms of derivability.

We have just seen that the laws (1) do not form a base for ThG . However, it may be that some other finite subset of ThG is a base

48 for T h G . Thus we are led to consider the following problem.

Problem.Is the equational theory of G finitely based?

One might guess that the equational theory of any finite algebra would be finitely based. If that were the case the above problem would, of course, have an affirmative solution. In fact Lyndon [17] proved that if an algebra has only two elements then its equational theory is finitely based. However, Lyndon's proof essentially depended on considering each two-element algebra separately, and thus did not throw any light on the situation for algebras with more than two elements. Then, three years later, Lyndon [18] found a seven- element algebra whose equational theory is not finitely based, and more v recently Murskii [21] has given an example of a three-element groupoid whose equational theory is not finitely based. A recent survey article by Evans on laws for semigroups includes a discussion of further results on finitely based equational theories, particularly for semigroups, and further references to the literature [8, pp. 21-24] .

It would be interesting to obtain an idea of which three-element algebras have a finitely based equational theory, but no general method is known for attacking this question. However, Birkhoff's theorem [3, Theorem 2] referred to in the Introduction can sometimes be applied to produce a finite base for the equational theory of a given algebra, and in particular does this for the groupoid G . Thus the problem stated above does indeed have an affirmative solution.

This result is quite special, but the proof suggests that a more general result may be true. We observe that the groupoid G may be obtained from the two-element groupoid GQ with multiplication table

0 1

0 0 0 1 1 0 by 'bordering' GQ , i.e. by adjoining to GQ = {0,1} an element 2, and extending the multiplication of GQ to G by setting x .2 = 2.x = 2

49 for all a in G . In general, we say that an algebra A is obtained by bordering an algebra A Q if A is obtained from A Q by adjoining to the universe AQ of AQ an element <» not in 4 0 , and extending the operations of A Q to A by setting f(xl, • • • ,x^) = °° whenever f is an n-ary operation (n > 1) of AQ and x^ = ® for at least one i . (This construction has recently been used in [16, p. 486], where the notation A* is introduced for the algebra A .) In [12] it was shown that the equational theory of the above groupoid

G q is finitely based; in [13] it was shown that the equational theory of the three-element algebra obtained by bordering the two-element distributive lattice is finitely based; and the proof that the equational theory of G is finitely based depends mainly on ideas from the proofs in [12] and [13]. Accordingly the following conjecture seems plausible.

Conjecture.If a three-element algebra A may be obtained by border ing a two-element algebra A Q 3 then the equational theory of A is finitely based.

More generally, it would be interesting to know whether the property of having a finitely based equational theory is always preserved under bordering.

3. Birkhoff's theorem

In order to state Birkhoff's theorem, we first need to define some basic algebraic concepts in the context of universal algebra.

The notion of a 'congruence relation1 on an algebra is an abstrac tion of the well-known congruence relations on the integers (Example 1 below). A (binary) relation ~ on the universe A of an algebra A is called a congruence relation on A if (i) it is an equivalence relation on A [i.e. it is reflexive (x rsJ X for all x ) , symmetric (if x ~ y then y ~ x ) , and transitive (if x ~ y and y ~ z then x z )); and (ii) it preserves the operations of A (i.e. if x. ~ y .

50 (i = 1,••*,«) and f is an n-ary operation of A then /(*!»•••»*„) ~ f(y1 *••%!/„)) •

Example1. Let Z be the set of all integers, let n be a positive integer, and given x,y in 7L set x ~ y if n divides x - y . Then ~ is a congruence relation on each of the groupoids (jZ,+y , < 2Z,-> , and .

Example2. On any algebra A there are two special congruence relations, namely equality (=) which we shall also denote by 0 , and the relation I which identifies all elements of A (i.e. xly for all x,y in A).

Given a family (^j^ACA congruence relations on an algebra A , we may define a congruence relation C on A by setting xCy if and only if for all X in A . C is called the intersection of the family ( C ) . and is denoted by D C . This terminology A A € A X € A A and notation are used since if the C^ are regarded as sets of ordered pairs, i.e. as subsets of A x A , then fl C. is the set-theoretic A intersection of the C^ . We note that 0 is the smallest set of ordered pairs which is a congruence relation, and that I = Ay-A .

We can now make the following definition.

Definition1. An algebra A is said tobe subdivectly irreducible if, whenever (^A€A a family of congruence relations on A such that fl C, = 0 , it follows that C = 0 for some \i in A . X £ A x P

The reason for the name will be explained later.

Example The 3. three-element groupoid G is subdirectly irreducible. Indeed, apart from 0 and I , the only congruence relation ~ on G is the equivalence relation which identifies 0 and 1 but no other distinct elements of G , for if 0 ~ 2 then 1 = 1.0~1.2 = 2 whence ~ is I , and if 1 ~ 2 then 0 = 0.1~0.2 = 2 whence ~ i s i .

51 Given an algebra A , a subset B of A is called a subuniverse of A if B is closed under the operations of A . For example, the non-empty subuniverses of G are {0}, {2}, {0,1}, {0,2}, and G . Any non-empty subuniverse B of A determines an algebra B whose universe is B and whose operations are the restrictions to B of the operations of A ; such an algebra B is called a subalgebra of A .

Given two algebras A and A r of the same type {i.e. roughly such that there is a one-one arity-preserving correspondence between their operations {cf. [9, pp. 33-34; 22, pp. 16-17])), their direct product is the algebra A x A ' of the same type with universe A * A ’ and operations defined co-ordinatewise. For example, if R = K.R, +, is the real numbers under addition and multiplication then R x R = Kp,+,i^ is the complex numbers under the usual addition and with multiplication * defined by

{x,y) * {x’,y') = {xx’,yy') .

More generally, given a family (A^)^ ^ of algebras of the same type,

their direct product is the algebra A = n A. of the same type X £ A x with universe A = II A {cf. [6, p. 9; 9, p. 4; 22, p. 2]) and X € A A operations defined co-ordinatewise.

We can now define the notion of a ’subdirect product’.

DefinitionLet 2. (A^)^ ^ be a family of algebras of the same type

with direct product A = ^ n ^ A^ . Then by a subdirect product of

the A^ we mean a subalgebra B of A with the property that, for each y in A and each a in ^ , there exists an element y y b = {b.'\. r . of B such that b = a . v A'X € A y y

In other words, a subdirect product of algebras A^ (X € A) is a subalgebra B of the direct product of the A^ such that, for each y in A , the projection ^ : b •—► b^ on B to A^ is surjective.

52 We also need a few notions related to homomorphisms. If A and A' are algebras of the same type, a mapping / : A -*■ A' is called a homomorphism on A to A' if it preserves all the operations (e.g. for multiplicative groupoids, if f(xy) = f(x)f(y) for all x,y in A). A surjective homomorphism is called an epimorphism, and an algebra A' is called an epimorphic image of A if there exists an epimorphism on A to A' . A bijective homomorphism is called an isomorphism, and two algebras A and A' are said to be isomorphic, and we write A = A' , if there exists an isomorphism on A to A' ; = is then an equivalence relation on any set of algebras of the same type. Isomorphic algebras have the same abstract properties. Close relationships exist between congruence relations and homomorphisms, but we do not need to discuss these here.

With these notions we can explain the reason for the terminology in Definition 1. It is easily shown that an algebra A is subdirectly irreducible in the sense of Definition 1 if and only if, whenever there is an isomorphism h on A to a subdirect product of algebras A^ , one of the composite mappings tt o h is an isomorphism on A to A . A A

We can now state Birkhoff's theorem.

THEOREM. Every algebra A is isomorphic to a subdirect product of subdirectly irreducible epimorphic images of A .

The proof is not very difficult. It uses various elementary results on congruence relations, isomorphism theorems, etc., together with an application of Zorn's lemma. Details may be found in Birkhoff's original paper [3], or in the books of Birkhoff [4, p. 193], Cohn [6, p. 100], Gratzer [9, p. 124], or Pierce [22, p. 51]. The second chapter of Pierce's book is devoted mainly to subdirect decompositions.

Typical applications of Birkhoff's theorem are not just for one algebra but for a whole 'variety' of algebras of the same type. Given a sequence x,y,z,••• of variables, and a set E of laws in these variables for algebras of a given type, we define the variety E* to

53 be the class of all those algebras of the given type which satisfy the given laws: E* = {A : E c ThA} .

Varieties are classes, not sets, but we do not need to concern ourselve with any set-theoretic difficulties. It is not difficult to show that

(1) (Cn E) * = E* for every set E of laws in given variables for algebras of a given type; this fact will be required only for the Remark following the proo: of Corollary 2 below.

Since the property of satisfying a law is preserved under the taking of epimorphic images, we have the following corollary of Birkhoff's theorem.

Corollary 1. Given a variety 1/ of algebras of the same typey every algebra in \J is isomorphic to a subdirect product of subdirectly irreducible algebras in V .

Such 'structure theorems', which to some extent reduce the study of a class of algebras to that of algebras of a simpler nature, are one of the main goals of modern algebra.

The applications of Birkhoff's theorem which we shall discuss are, in fact, applications of Corollary 1 to cases where it is possible to determine all the subdirectly irreducible algebras in a variety V .

From Corollary 1 we can easily derive a sufficient condition for a set of laws to be a base for the equational theory of an algebra. In order to formulate this condition we need one more concept. Let A be an algebra. Then an algebra B of the same type as A is said to be generated by A , and we write 8 € [A], if 8 can be constructed from A by taking direct products, subalgebras and epimorphic images. We note that if 8 is generated by A then Th A c Th B . Birkhoff [2, Theorem 10] (cf. [6, Theorem IV.3.1; 9, Theorem 26.3; 22, Theorem 5.2.5]) has given a characterisation of varieties by purely algebraic

54 conditions from which it follows that

(2) [A] = (ThA)* for every algebra A ; like (1), this result will be required only for the Remark following the proof of Corollary 2 below.

We can now state the following sufficient condition for a set E c ThA to be a base for ThA. This condition is also necessary, as noted in the Remark below.

Corollary2. Let A be an algebra, let E c ThA , and suppose that the subdireotly irreducible algebras in the variety E* are all generated by A . Then E is a base for Th A .

Proof.We wish to show that Cn E = ThA . Since E c Th A , it follows that CnE c ThA . To prove the opposite inclusion we must show that if B ( E* then Th A c Th B . But if B € E* then, by Corollary 1, B is isomorphic to a subdirect product of subdirectly irreducible algebras in E* ; hence, by the hypothesis, B is iso morphic to a subdirect product of algebras generated by A ; and thus B is itself generated by A . Thus ThA c Th B .

Remark.The sufficient condition in Corollary 2 is also necessary. In fact if A is an algebra and E is a base for Th A then every algebra in E* is generated by A since

E* = (Cn E)* = (ThA)* = [A] by (1) and (2) above.

4. A finite base for ThG

We now have all the general machinery needed for proving the result stated in the abstract [14], namely that the four laws

x(xx) = s , x{xy) = y(yx) , {xy)z = (xz) (z/s) , and (*) (xy)z = (xz)y form a base for ThG . I hope to publish separately a formal proof of

55 this result. What I would like to do here is to give a few steps of this proof, in a form which indicates how the base (*) for Th G was found. The proof will, of course, be based on Corollary 2 to Birkhoff’s theorem.

Lemma 1. Let E^ = {x^ , where X^ is the law (xy)z = (xz)(yz) ,

and suppose that A € E* . Then, for each a in A 3 we may define a congruence relation C on A by setting arC^z/if and only if xa = ya .

Proof.C is obviously an equivalence relation, and is a congruence relation since if x C ^ and UCQV then

(xu)a = (xa) (ua) = (ya) (va) = (yv)a ,

and hence (xu)Ca (yv) .

Lemma 2.Let l2 = * where X1 is as in Lemma ls X2 is the law a: (axe)= x t and X3 is the law (xy)z = (xz)y . Let A € I* , and define congruence relations Ca on A as in Lemma 1. Then (1 C = 0 . a Z A a

Proof.We wish to show that if x fl C \y then x = y . But the a £ A ar hypothesis implies that x C y and x C y , i.e. that xx yy x(xx) = y(xx) and x(yy) = y(yy) . Hence y = y(yy)= x(yy) = [x(xx)] (yy) = [y(xx)](yy) = [y(yy)](xx) = y (xx) = x (axr) = x .

Lemma 3.Let = {xi ,X ,X ,X^}, where Xj, X2, X3 are defined as in Lerrmas 1 and 2, and X^ is the law (xy)y = xy . Then if a groupoid A in E* is subdirectly irreducible> A contains an element 0 such that xO = x for all x in A .

Proof.By Lemma 2, A contains an element 0 such that CQ = 0 . Since (x0)0 = xO by X^ , we have (xO)CQx for all x , and hence xO = x .

56 We continue in this way, introducing finitely many additional laws X .X , • • •, X all of which are in Th G , and eventually find that, 5* 6 ’ n to within isomorphism, the only subdirectly irreducible groupoids in the variety are ^ itself, and its subgroupoids with universes {0}, {0,1}, and {0,2} . In view of Corollary 2 to Birkhoff’s theorem, this proves that a ^ase ^or Th G . Finally, we tidy up our result by showing that Xlf««*,X are all derivable from, and hence consequences of, the four laws (*) in ThG . Thus (*) forms a base for ThG , and in fact it can be shown that (*) is a minimal base, i.e. that the laws (*) are independent. Incidentally we have shown that a subdirectly irreducible member of the variety of groupoids satisfying the laws (*) has at most three elements, so that this variety is residually small in the sense of Taylor [27].

5. Representation theorems for Boolean algebras and distributive lattices

By a 'field of sets' we mean an algebra of the form ^F,D,U,~^ where F is a non-empty set of subsets of some set S which is closed under D (intersection), U (union) and ~ (complementation with respect to S ) . For present purposes we may define a 'Boolean algebra' to be an algebra A = 0 > A,v, where the binary operations A and v and the unary operation ' satisfy the laws which hold in every field of sets; in fact we need assume only a finite subset of these laws, namely those used below. Every Boolean algebra A contains elements 0, I such that a A a' = 0 and ava' = I for all a in A .

Lemma1. Let A be a Boolean algebra and let a € A . Then we may define congruence relations Ca and on A by setting x C a y if and only if x a a = y ha , and x Da y if and only if x v a = y v a .

The proof for C depends on the laws (a: Aw) A a = (x A a) A (u A a) t

(x v u) A a = (x A a) v (w a a) , and x ' A a = (xAa)' A a , and the proof for D depends on the dual laws.

57 Lemma 2.A subdirectly irreducible Boolean algebra has at most two elements.

Proof.With the help of a few Boolean laws, it is easily shown that

^a ^®a ~ ® ^0r every a any Boolean algebra. Hence, for every a in a subdirectly irreducible Boolean algebra, either 0^=0 or D = 0 . If C - 0 then a is the element J , and if D = 0 a a a then a is the element 0 .

By Corollary 1 to Birkhoff's theorem it now follows that a given Boolean algebra either has just one element or is isomorphic to a subdirect product 8 of two-element Boolean algebras A indexed by A a set A . By using the idea of a characteristic function we easily see that the algebra 8 is isomorphic to a field of subsets of the set A . This proves the following theorem.

THEOREM 1 (Stone). Every Boolean algebra is isomorphic to a field of sets.

It follows from Theorem 1 that if an algebra satisfies the laws which hold in every field of sets then it essentially is a field of sets. Perhaps this result is not surprising, but it is not trivial as its known proofs all use the axiom of choice (cf. [10; 25; 7]), and it was some seventy years after Boole's death before it was clearly formulated and proved. An analogous result holds for groups (Cayley's theorem), but is more elementary as its proof does not use the axiom of choice.

By a 'ring of sets' we mean an algebra of the form Cl ,lC> where R is a non-empty set of subsets of some set which is closed under D and U . For present purposes we may define a 'distributive lattice' to be an algebra A = ^4,A,\^ where the binary operations A and V satisfy the laws which hold in every ring of sets. Analogues of Lemmas 1 and 2 above hold for distributive lattices, and an argument similar to that given above for Theorem 1 proves the following theorem.

58 THEOREM 2 (Birkhoff). Every distributive lattice is isanorphic to a ring of sets.

Birkhoff's subdirect decomposition theorem may be regarded as the culmination of a development for which Theorems 1 and 2 above provided the main initial impetus. For instance, in 1937 McCoy and Montgomery [20, Theorem 5] , starting from Stone's theorem and a remark of Stone on the close relation between the representation of Boolean rings and the theory of direct sums of rings, obtained the theorem that every p-ring is isomorphic to a subring of a direct sum of p-rings with p elements, and in 1938 McCoy [19, Theorem 4] went on to show that every ring is isomorphic to a subring of a direct sum of irreducible rings. These results are easy corollaries of Birkhoff1s subdirect decomposition theorem, and several further corollaries of this theorem are given in Birkhoff's original paper [3]. Developments of Theorems 1 and 2 in other directions continue to the present time (cf. [15]).

6. Conclusion

Earlier I mentioned G. Higman's remark that 'universal algebra is something everyone ought to know about, though nobody should specialise in it.' Reviewing Cohn's book [6] for the American Mathematical Monthly, Brainerd [5] gave the following explanation of why he large iy agreed with Higman. 'To develop a new argument or result in, say, group theory and then to generalise it to a wider class of universal algebras, puts a result of proven utility at the disposal of workers studying algebras of this wider class, and hence is a useful contribu tion to mathematics. On the other hand, to obtain a result about universal algebras as such, without reference to an application, may result in yet another hollow addition to the already overblown mathematical literature.'

I hope I have succeeded in showing that Birkhoff's theorem belongs to the useful part of universal algebra. It is useful in look ing for bases for the equational theory of an algebra, even though it does not provide an algorithm for finding a finite base if one exists,

59 and it gives the difficult representation theorems for Boolean algebras and distributive lattices of Stone and Birkhoff as easy corollaries. Because it is formulated in the context of universal algebra, it is an easily available tool for workers in any part of algebra.

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University of Auckland