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Semirings embedded in a completely regular semiring

MK. SenS. K. Maity — 2004

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Recently, we have shown that a semiring S is completely regular if and only if S is a union of skew-rings. In this paper we show that a semiring S satisfying a 2 = n a can be embedded in a completely regular semiring if and only if S is additive separative.

A note on orthodox additive inverse semirings

MK. SenS. K. Maity — 2004

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

We show in an additive inverse regular semiring ( S , + , · ) with E ( S ) as the set of all multiplicative idempotents and E + ( S ) as the set of all additive idempotents, the following conditions are equivalent: (i) For all e , f E ( S ) , e f E + ( S ) implies f e E + ( S ) . (ii) ( S , · ) is orthodox. (iii) ( S , · ) is a semilattice of groups. This result generalizes the corresponding result of regular ring.

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