Right zero unions of N-semigroups

Abstract

=1 commutative semigroup is called archimedean if every element divides a positive integer power of each other element. An N-semigroup is one of four possible types of archimedean semigroups. Chrislock [l] extends the notion of archjmedeanness to noncommutative semigroups. An N*-semigroup is a noncommutative archimedean semigroup that can be thought of as a generalization of an N-semigroup. A basic reference on N-semigroups is [5]. Let F = {S a : u E R} be a disjoint family of semigroups. F has a right zero union, if there exists a semigroup T that is a disjoint union of the S, , where each S, is a left ideal of T. Thus, T is homomorphic onto the right zero semigroup R by mapping each S, to 01. It will be shown that an N*-semigroup is a right zero union of N-semigroups and conversely. Section 2 can be thought of as preliminary material. Section 3 examines right zero unions of N-semigroups. It will be shown in that section that a right zero union of N-semigroups is isomorphic to a subdirect product of an ,V-semigroup and a right zero semigroup. Conversely, any such subdirect product is a right zero union of N-semigroups. Section 4 proves, biter alia, the equivalence of the two concepts N*-semigroups and right zero unions of N-semigroups. An N-semigroup has a representation involving an abelian group G and a function I defined on G x G. Similarly, an N*-semigroup has a representation with an abelian right group H (if H = G x R then G is abelian) and an I* function defined on H x H. Section 4 will examine I* functions, and will conclude with a method for constructing them. Omitted proofs can be found in [4]. A g eneral reference to semigroup theory is [3]. The study of right zero unions is a special case of the study of “bands of semigroups.”