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 <TEX>$\theta$</TEX> and k (resp.), induced by tr. and ker (resp.), are congruences on LC(S) and each <TEX>$\theta$</TEX>-class p<TEX>$\theta$</TEX> (resp. each k-class pk) is a complete modular sublattice with <TEX>$p_{min}$</TEX> and <TEX>$p_{max}$</TEX> (resp. With <TEX>$p^{min}$</TEX> and <TEX>$p^{max}$</TEX>), as the least and greatest elements. <TEX>$p_{min},\;p_{max},\;p^{min}$</TEX> and <TEX>$p^{max}$</TEX>, in particular <TEX>${\epsilon}_{max}$</TEX>, 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.