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

Sarmin, Nor Haniza and Muhammad Khan, Faiz and Asghar Khan, Asghar Khan (2012) Some new characterization of ordered semigroups in terms of (λ;θ)-fuzzy bi-ideals. International Journal of Algebra and Statistics, 1 (1). pp. 22-32. ISSN 2314-4556

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Official URL: https://doi.org/10.20454/ijas.2012.376

## 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.

Item Type: Article semigroups in terms, characteristic function Q Science Science 33051 Fazli Masari 20 May 2014 10:08 28 Jan 2019 11:50

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