Completely Regular Semirings

Completely regular semigroup plays an important role in semigroup theory. Completely regular semigroups are nothing but union of its maximal subgroups. Similar to semigroup, one may ask whether we can extend completely regular semigroups to completely regular semirings so that completely regular semiring would have analogous properties as completely regular semigroup? In order to answer this question, we defined completely regular semiring and show that any completely regular semiring can be expressed as a union of skew-rings. Also here we have characterized completely simple semirings and finally established that a semiring is completely simple if and only if it isomorphic to a Rees matrix semiring. In this connection we defined Clifford semirings, generalized Clifford semirings, semisimple Clifford semirings and studied their properties.