Structures of Anti-Inverse Semirings
N. Sheela, A. Rajeswari · Annals of Pure and Applied Mathematics · 2018
In this paper we study the structures of an anti-inverse semiring.It has been proved that, for a semiring (S, +,•), if (S,•) is anti-inverse idempotent semigroup, we define a relation 'σ' on a semigroup S by a σ b implies ab n = b n+1 , ba n = a n+1 for any positive integer n and for any a, b in S then σ is congruence on S and also S is distributive.weproved that if (S,+,.)be an anti-inverse semiring then S is Quasiseparative, weakly separative, separative and (S,.) is normal.If (S ,.) be an anti-inverse Archimedean semigroup and if S is weakly separative then it is weakly reductive.If (S,+,.)be an anti-inverse semiring, we define a relation ρ on a semigroup S as a ρ b if and only if a 2 = ab =ba for all a, b in S then (S,+,., ρ) is a partially order semiring.We determine the additive and multiplicative structure of these anti-inverse semirings and the modern interest in semirings arises primarily from fields of applied mathematics such as optimization theory, the theory of discrete event dynamical systems, automata theory, as well as from the allied areas of theoretical computer science and theoretical physics.