Some subvarieties of semiring variety COS$ ^{+}_{3} $

Abstract

In this paper, we study some subvarieties of a semiring variety determined by certain additional identities. We first present alternative characterizations for equivalences $ \overset{+}{\mathcal{H}}{\cap}\overset{\cdot}{\mathcal{L}} $, $ \overset{+}{\mathcal{H}}{\cap}\overset{\cdot}{\mathcal{R}} $, $ \overset{+}{\mathcal{H}}{\cap}\overset{\cdot}{\mathcal{D}} $, $ \overset{+}{\mathcal{H}}{\vee}\overset{\cdot}{\mathcal{L}} $, $ \overset{+}{\mathcal{H}}{\vee}\overset{\cdot}{\mathcal{R}} $, $ \overset{+}{\mathcal{H}}{\vee}\overset{\cdot}{\mathcal{D}} $. Then we give the sufficient and necessary conditions for these equivalences to be congruence. Finally, we prove that semiring classes defined by these congruences are varieties and provide equational bases.

Keywords

• semiring
• green's relation
• congruence
• variety of semirings
• subvariety