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 $\$\; \backslash mathfrak\; M\; \$$ of all commutative multiplicatively idempotent semirings. We obtain necessary conditions when semirings from $\$\; \backslash mathfrak\; M\; \$$ are subdirectly irreducible. We consider some properties of the variety $\$\; \backslash mathfrak\; M\; \$$. In particular, we show that $\$\; \backslash mathfrak\; M\; \$$ is generated by two of its subvarieties, defined by the identities $3x\; =\; x$ and $3x\; =\; 2x$. We explore the variety $\$\; \backslash mathfrak\; N\; \$$ generated by two-element commutative multiplicatively idempotent semirings. It is proved that the lattice of all subvarieties of $\$\; \backslash mathfrak\; N\; \$$ is a $16$-element Boolean lattice.

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Last modified: May 13, 2014