Abstract:

Green's relations on a semiring satisfying additional identities xⁿ≈x, (2ⁿ-1)x≈x, (x+y)^n-1≈x^n-1+y^n-1 and (xy)^n-1≈x^n-1y^n-1 are studied. Equivalent characterizations of $\stackrel{+}{\mathscr{H}}$∧$\dot{\mathscr{H}}$, $\stackrel{+}{\mathscr{H}}$∧$\dot{\mathscr{L}}$, $\stackrel{+}{\mathscr{H}}$∧$\dot{\mathscr{R}}$, $\stackrel{+}{\mathscr{H}}$∧$\dot{\mathscr{D}}$, $\stackrel{+}{\mathscr{H}}$∨$\dot{\mathscr{L}}$, $\stackrel{+}{\mathscr{H}}$∨$\dot{\mathscr{R}}$, $\stackrel{+}{\mathscr{H}}$∨$\dot{\mathscr{D}}$ and $\stackrel{+}{\mathscr{H}}$∨$\dot{\mathscr{H}}$ are obtained. The necessary and sufficient conditions for the above relations to be congruent are characterized, and then it is proved that the semiring classes determined by the above congruences are varieties.

Key words: semiring, variety, completely regular semigroup, Green's relation, congruence