On the Left Inversive Semiring Congruences on Additive Regular Semirings

Abstract

An additive regular semiring S is left inversive if the set E+(S) of all additive idempotents is left regular. The set LC(S) of all left inversive semiring congruences on an additive regular semiring S is a lattice. The relations µ and k (resp.), induced by tr and ker (resp.), are congruences on LC(S) and each µ-class ½µ (resp. each k-class ½k) is a complete modular sublattice with ½min and ½max (resp. with ½min and ½max), as the least and greatest elements. ½min, ½max, ½min and ½max, in particular ²max, the maximum additive idempotent separating congruence has been characterized explicitly. A semiring is quasi-inversive if and only if it is a subdirect product of a left inversive and a right inversive semiring.