Endomorphism regular Ockham algebras of finite boolean type

If (L;f) is an Ockham algebra with dual space (X;g), then it is known that the semigroup of Ockham endomorphisms on L is (anti-)isomorphic to the semigroup Λ(X; g) of continuous order-preserving mappings on X that commute with g. Here we consider the case where L is a finite boolean lattice and f is a bijection. We begin by determining the size of Λ(X;g), and obtain necessary and sufficient conditions for this semigroup to be regular or orthodox. We also describe its structure when it is a group, or an inverse semigroup that is not a group. In the former case it is a cartesian product of cyclic groups and in the latter a cartesian product of cyclic groups each with a zero adjoined. An Ockham algebra (L;f) is a bounded distributive lattice L on which there is defined a dual endomorphism f. For the basic properties of Ockham algebras we refer the reader to [1]. The most obvious example of an Ockham algebra is, of course, a boolean algebra (B; ′). In general, a boolean lattice can be made into an Ockham algebra in many different ways. Throughout what follows we shall assume that the Ockham algebra (L;f) is of finite boolean type, in the sense that L is a finite boolean lattice and f is a dual automorphism. Then (L;f) necessarily belongs to the Berman class K_p,o for some p. If L ≃ 2^k then, by [1, Chapter 4], the dual space (X;g) is such that X is discretely ordered with |X| = k, and g is a permutation on X such that g^2p = id_x. We shall denote by X₁,... , X_m the orbits of g. For each i we choose and fix a representative x_i ε. X_i. Defining c_i = |X_i|, for each i, we therefore have X_i = {x_i,g(x_i),g²(x_i),...,g ^ci-1(x_i)}. Consider the set Λ(X;g) consisting of those mappings v:X→X that commute with g. By duality [2] we know that (Λ(X;g), °) is a semigroup that is (anti-)isomorphic to the semigroup End((L;f), °) of Ockham endomorphisms on L. By considering Λ(X;g) we can therefore obtain properties of End(L;f). Our principal objective is to determine precisely when this semigroup is regular. For this purpose, we begin by observing the following results.