On the Relational Basis of Cayley's Theorem and of Similar Representations for Algebras

Considering a binary operation as a ternary relation permits certain sections of the latter (which are functions) to be used in representing an abstract semigroup as a family of the self-maps of its underlying set under function composition. The idea is thus seen to be entirely similar to the way that the sections of a partial ordering under set inclusion represent the (abstract) partially ordered set. An extension of this procedure yields a uniform set of representation theorems for a large class of associative algebras. Consider the following theorems from basic mathematics: Cayley's Theorem. Every group G is isomorphic to a subgroup of the group (formed under function composition) of all bijections of G. Theorem. Every partially ordered set (S, <) is order-isomorphic to the set {ordered by set inclusion) of all subsets of S of the form {t € S: t < s}, s e S. What do the above two theorems have in common? Clearly both are representation theorems. Cayley's theorem implies that every abstract group operation is really function composition in disguise, while in the second we see that all partial orderings are actually different manifestations of the set inclusion. Beyond this, there appears to be very little in common between the two theorems (considering the very different structures involved), and the various known generalizations of Cayley's theorem are all algebraic in nature and offer no evidence of any further similarity. In this paper, however, we demonstrate the existence of a strong, though apparently unexplored relational link between the two theorems. As a natural application, we then use this new relational formalism to obtain a uniform set of representation theorems for a large class of associative algebras. By a relation in a set S we mean, as usual, a nonempty subset of the Cartesian product Sn , where « is a postive integer called the order of the relation. If n = 2 the relation is called binary and if n = 3 the relation is called ternary. Clearly, every partial ordering is a binary relation. On the other hand, every binary operation (such as a group operation) is a function of two variables and hence a ternary relation. Specifically, let : S2 —» S be the binary operation (f)(x, y) = xy on S (we use juxtapositions to denote the operation on S when Received by the editors September 2, 1992; originally communicated to the Proceedings of the AMS by Lance W. Small. 1991 Mathematics Subject Classification. Primary 08A99; Secondary 08A02, 08A05, 06A12, 06B15, 06F05, 06F25, 16G99. ©1995 American Mathematical Society