Abstract:
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.