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Title: | The Zeleznikow problem on a class of additively idempotent semirings | Authors: | Shao Y. Crvenković, Sinisa Mitrović, Melanija |
Issue Date: | 1-Dec-2013 | Journal: | Journal of the Australian Mathematical Society | 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. | URI: | https://open.uns.ac.rs/handle/123456789/8128 | ISSN: | 14467887 | DOI: | 10.1017/S1446788713000359 |
Appears in Collections: | PMF Publikacije/Publications |
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