Abstract:In this paper, it was proved that, if S is a Right regular semiring, then (S, +) is a band under the following cases. 1. If S is multiplicatively subidempotent semiring. 2. If S is an almost idempotent semiring.
In this paper we study the conditions under which an additively inverse semiring is a band and commutative. Also in the case of totally ordered additive inverse semiring in which ( , )S is positively totally ordered, we prove the additive structure is max( , ) a b b a a b for all , abin . S We have also framed an example for this result by considering 4 elements.
In this paper, we discuss the structure of some special classes of semirings . Here the properties of Viterbi semirings; PRD semirings satisfying the identity ab + a = a, for all a, b in S and semirings satisfying the identity a + ab + a = a, for all a, b in S are studied. We proved that (S, +, •) is a semiring satisfying the identity a + ab + a = a, in which if S contains the multiplicative identity which is also an additive identity, then S is a multiplicatively subidempotent semiring. Keywords 1.INTRODUCTIONIn recent years interest in the study of partially ordered and fully ordered semigroups, groups, semirings, semimodules, rings and fields has been increasing enormously. Much of the theory of rings applied to arbitrary semirings. Some mathematicians go so far as to say that semirings are really the more fundamental concept, and specializing to rings should be seen in the same light as specializing to say algebra over the complex numbers. In this paper, we investigate the additive and multiplicative properties of some special classes of Semirings. Definition 1.5An element `a' of a semiring S is multiplicatively subidempotent if and only if a + a 2 = a and S is multiplicatively subidempotent if and only if each of its elements is multiplicatively subidempotent. Definition 1.6A viterbi semiring is a semiring in which S is additively idempotent and multiplicatively subidempotent.i.e., a + a = a and a + a 2 = a, for all 'a' in S.The concept of viterbi Semiring is taken from the book of Jonathan S. Golan [3], entitled "Semirings and Affine Equations over Them: Theory and Applications".
In this paper, we study the properties of semiring with identity ab + a = a and also study the properties of ordered semiring satisfying the identity ab + a = a, for all a, b in S. This paper contains mainly two sections. In section 1, the structure of semirings satisfying the identity ab + a = a, for all a, b in S are studied. In section 2, we characterize totally ordered semirings satisfying the identity ab + a = a, for all a, b in S.
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