Narrow Results

Green’s ∗-relations on semirings satisfying additional identities $x^n\approx x$ and $2(x+y)\approx 2x+2y$ are studied. Equivalent characterizations of ${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℒ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\wedge {{ℛ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\wedge {{𝒟}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\wedge {{\mathscr{H}}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\vee {{ℒ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\vee {{ℛ}}_{\ast }^{•}$, ${{\mathscr{H}}}_{\ast }^{+}\vee {{𝒟}}_{\ast }^{•}$ and ${{\mathscr{H}}}_{\ast }^{+}\vee {{\mathscr{H}}}_{\ast }^{•}$ are obtained and equivalent conditions are given for the above relations to be congruences. It is proved that semiring classes determined by these congruences are varieties.