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In this article, a new generalisation of fuzzy bi-ideals and intuitionsitic fuzzy bi-ideals of a semigroup considered so called picture fuzzy bi-ideals no a semigroup. It is well known that the intra-regular semigroups play an essential role in studying the structure, especially the decomposition, of semigroups. The purpose of this paper is to deal with the algebraic structure of semigroups by applying picture fuzzy set theory. As an application of our results we get characterisations of intra-regular regular semigroups in terms of picture fuzzy bi-ideals. We prove that a semigroup is both regular and intra-regular if and only if every picture fuzzy bi-ideal onis idempotent.

Characterisations of semigroups by the properties of their picture fuzzy bi-ideals