Geodesic
Multiplicatively idempotent semirings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 4, pp. 41-70.

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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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E. M. Vechtomov; A. A. Petrov. Multiplicatively idempotent semirings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 4, pp. 41-70. http://geodesic.mathdoc.fr/item/FPM_2013_18_4_a3/

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