Quasi-multiplicative maps on Baer semigroups

Abstract

A Baer semigroup is a semigroup 5 with 0 and 1 in which, for each x∈S, the left annihilator L(x) = {y∈S: yx = 0} of x is a principal left ideal generated by an idempotent and the right annihilator R(x) = {y∈S: xy = 0} of x is a principal right ideal generated by an idempotent. Baer semigroups are of interest because (see [5]) the left annihilators of the elements of a Baer semigroup S, ℒ(S) = {L(x): x∈S}, form a bounded lattice and (see [4]) every bounded lattice arises in this manner. In this note we look at a type of map φ on a Baer semigroup S which has the property that Sφ is a Baer semigroup. (The homomorphic image of a Baer semigroup need not be a Baer semigroup. For a case where it is, see [7].) When the Baer semigroup is specialized to a Boolean algebra, this type of map generalizes Halmos's notion of a quantifier.