Semiring varieties related to multiplicative Green’s relations on a semiring

Abstract

In this paper, by means of congruence openings of multiplicative Green’s relations on a semiring we define and study several varieties of semirings, obtain the relationship between these varieties and give Mal’cev product decompositions of some varieties of idempotent semirings. In particular, we establish order embeddings of the lattice of all subvarieties of the variety of multiplicatively idempotent semirings (resp. idempotent semirings) into the direct products of the lattices of open (resp. closed) varieties with respect to two opening (resp. closure) operators on this lattice what we introduced.