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 (Formula presented.) of all commutative multiplicatively idempotent semirings. We obtain necessary conditions under which semirings from (Formula presented.) are subdirectly irreducible. We consider some properties of the variety (Formula presented.). In particular, we show that (Formula presented.) is generated by two of its subvarieties, defined by the identities 3x = x and 3x = 2x. We explore the variety (Formula presented.) generated by two-element commutative multiplicatively idempotent semirings. It is proved that the lattice of all subvarieties of (Formula presented.) is a 16-element Boolean lattice. © 2015, Springer Science+Business Media New York.