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

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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.
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