FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2013, VOLUME 18, NUMBER 4, PAGES 41-70

Multiplicatively idempotent semirings

E. M. Vechtomov
A. A. Petrov

Abstract

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The article is devoted to the investigation of semirings with idempotent multiplication. General structure theorems for such semirings are proved. We focus on the study of the class $ \mathfrak M $ of all commutative multiplicatively idempotent semirings. We obtain necessary conditions when semirings from $ \mathfrak M $ are subdirectly irreducible. We consider some properties of the variety $ \mathfrak M $. In particular, we show that $ \mathfrak M $ is generated by two of its subvarieties, defined by the identities 3x = x and 3x = 2x. We explore the variety $ \mathfrak N $ generated by two-element commutative multiplicatively idempotent semirings. It is proved that the lattice of all subvarieties of $ \mathfrak N $ is a 16-element Boolean lattice.

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