Exponents of Finite Simple Lattices

In the late 1930's Garrett Birkhoff [3] pioneered the theory of distributive lattices. An important component in this theory is the concept of exponentiation of lattices [4]: for a lattice L and a partially ordered set P let L denote the set of all order-preserving maps of P to L partially ordered b y / ^ g if and only if/(;c) ^ g(x) for each x e P (see Figure 1). Indeed, If is a lattice. This concept, as it turns out, is a powerful tool in codifying finite distributive lattices. Let 2 denote the two-element chain. Evidently, 2 is a distributive lattice for every partially ordered set P. On the other hand, as Birkhoff observed, every finite distributive lattice D is isomorphic to 2 for some finite partially ordered set P; moreover, the dual of P is isomorphic to the partially ordered set of all join irreducible elements of D. In recent years, H. A. Priestley [10, 11] established a far-reaching categorical duality theory for distributive lattices (the finite version of which, incidentally, was already implicit in Birkhoff's early work on distributive lattices [3]). The purpose of this paper is to generalize both the representation theory and the categorical duality theory of distributive lattices to exponents of lattices. The starting point for our investigation is concerned with a natural consequence of the representation theorem for finite distributive lattices.