The Zeleznikow problem on a class of additively idempotent semirings

Abstract

A semiring is a set S with two binary operations + and · such that both the additive reduct S+ and the multiplicative reduct S• are semigroups which satisfy the distributive laws. If R is a ring, then, following Chaptal ['Anneaux dont le demi-groupe multiplicatif est inverse', C. R. Acad. Sci. Paris Ser. A-B 262 (1966), 274-277], R• is a union of groups if and only if R• is an inverse semigroup if and only if R• is a Clifford semigroup. In Zeleznikow ['Regular semirings', Semigroup Forum 23 (1981), 119-136], it is proved that if R is a regular ring then R• is orthodox if and only if R• is a union of groups if and only if R• is an inverse semigroup if and only if R• is a Clifford semigroup. The latter result, also known as Zeleznikow's theorem, does not hold in general even for semirings S with S+ a semilattice Zeleznikow ['Regular semirings', Semigroup Forum 23 (1981), 119-136]. The Zeleznikow problem on a certain class of semirings involves finding condition(s) such that Zeleznikow's theorem holds on that class. The main objective of this paper is to solve the Zeleznikow problem for those semirings S for which S+ is a semilattice. © 2013 Australian Mathematical Publishing Association Inc.