Semigroups Having Zeroid Elements

Abstract

An element u of S will be called a zeroid element of S if, for each element a of S, there exist x and y in S such that ax = ya = u. According to Huntington's set of group axioms,1 S is a group if and only if every element of S is a zeroid. If S has a zero element, e. g. if S is the multiplicative semigroup of a ring, then zero is the only zeroid element of S. In the present paper we show that the set U of zeroid elements of any semigroup S is either vacuous or else is a subgroup of S. U is a two-sided ideal contained in every left, right or two-sided ideal of S. It is therefore the Kerngruppe of S in the sense of Suschkewitsch.2 The identity element z of U commutes with every element of S, and the mapping a -> za (= az) is a homomorphism of S onto U. We define the core J of S to be the set of elements mapped into z. J is a subsemigroup of S containing z as zero element. If, on the other hand, we start with a group U and a semigroup J with zero, we can construct at least one semigroup S (e. g. the direct product of U and J) such that the group of zeroid elements of S is isomorphic with U. A few properties of subgroups and subsemigroups of a semigroup S having zeroid elements are discussed, and it is noted that any semigroup homomorphic with a subsemigroup of a group can be embedded in such a