THE ZELEZNIKOW PROBLEM ON A CLASS OF ADDITIVELY IDEMPOTENT SEMIRINGS

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.