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