Some new characterization of ordered semigroups in terms of (λ;θ)-fuzzy bi-ideals

Abstract

In this paper, we demonstrate a new concept of fuzzy bi-ideals called a \(( \lambda ,\theta ) \)-fuzzy bi-ideals of an ordered semigroup. Fuzzy ideals of type \(( \lambda ,\theta) \) are the generalization of fuzzy bi-ideals and an \(( \in ,\in\vee q) \)-fuzzy bi-ideals of an ordered semigroup. We show that \( U( \mu ,t) ( \neq \emptyset ) \) is a bi-ideal if and only if the fuzzy subset \(\mu\) is a \(( \lambda ,\theta ) \)-fuzzy i-ideal of \(S\) for all \(t\in (\lambda ,\theta ]\). Similarly, \(A\) is a bi-ideal if and only if the characteristic function \(\mu _{A}\) of \(A\) is a \(( \lambda ,\theta ) \)-fuzzy bi-ideal of \(S\). With the help of some examples, we show that \(( \lambda ,\theta ) \)-fuzzy bi-ideals (\(( \lambda ,\theta ) \)-fuzzy subsemigroups) are neither fuzzy bi-ideals (fuzzy subsemigroups ) nor \(( \in ,\in \vee q) \)-fuzzy bi-ideals (\(( \in ,\in \vee q) \)-fuzzy subsemigroups) of an ordered semigroup \(S\). Finally, the characterization of completely ordered semigroups in terms of \(( \lambda ,\theta ) \)-fuzzy bi-ideals is given.