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

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.