Congruence openings of additive Green’s relations on a semiring

Abstract

In a series of papers, Green’s relations on the additive and multiplicative reducts of a semiring proved to be a very useful tool in the study of semirings. However, in the vast majority of cases, Green’s relations are not congruences, and we show that in such cases it is much more convenient to use the congruence openings of Green’s relations, instead of the Green’s relations themselves. By means of these congruence openings we define and study several very interesting operators on the lattices of varieties of semirings and additively idempotent semirings, and, in particular, we establish order embeddings of the lattice of varieties of additively 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 that we introduced.