Multiplicatively idempotent semirings

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.