Annals of Communications in Mathematics

Abstract:In this paper, we introduce the notion of a fuzzy soft tri-ideal over \( \Gamma \)-semiring. We characterize the regular \( \Gamma \)-semiring in terms of fuzzy soft tri-ideals, and study some of the properties. \( M \) is a regular \( \Gamma \)-semiring, \( E \) be a parameters set and \( A \subseteq E \). If \( (\mu, A) \) is a fuzzy soft left tri-ideal over \( M \), then \( (\mu, A) \) is a fuzzy soft right ideal over \( M \).

AbstractIn this paper, we introduce the notion of interval valued fuzzy prime ideals of Γ−semirings.We study, some properties of prime ideals of a Γ−semirng in terms of interval valued fuzzy ideals.

AbstractIn this section, we introduce the notions of a left and a right translational invariant fuzzy subsets of a Γ−semigroup M, as well as the concept of a unit with respect to a fuzzy subset, and study their properties. We also prove that if μ is a translational invariant fuzzy subset of a commutative Γ−semigroup with unity, then the principal ideal generated by an element and μ that contains a unity element is a prime ideal of the Γ−semigroup.

AbstractIn this paper, we introduce the notion of a tri-quasi ideal and a fuzzy tri-quasi ideal as a further generalization of ideals, left ideals, right ideals, bi-ideals, quasi ideals, and interior ideals. We characterize the regular semigroup in terms of tri-quasi ideals, fuzzy tri-quasi ideals and study some of their properties. This generalization enables mathematicians to explore new relationships and enhancing the understanding of these structures. We establish that, a semigroup is a regular semigroup if and only if B ∩ I ∩ L ⊆ BIL, for any tri-quasi ideal B, ideal I and left ideal L of a semigroup, and for a semigroup, if μ is a fuzzy left tri-ideal of a semigroup then μ is a fuzzy tri-quasi ideal.

Abstract:This paper studies \( B \)-algebras through the Boolean-centre-valued operators \( x^\nabla = 1 \to x \) and \( x^\Delta = 0 \leftarrow x \). The operator \( {}^\Delta \) defines a centre-valued negation satisfying \( (x \vee y)^\Delta = x^\Delta \wedge y^\Delta \) for all \( x,y \in A \), together with the inequality \( x^\Delta \vee y^\Delta \leq (x \wedge y)^\Delta \). Equality in the latter holds on the Boolean centre \( B(A) \); an explicit example confirms that restricting to \( B(A) \) is necessary. The Boolean centre, equipped with the restricted operations, satisfies the equations of one standard axiomatisation of Nelson algebras [14]. An involutive distributive lattice \( \mathcal{N}(A) = A \times A \) is constructed together with an injective map \( \iota(x) = (x,x^\Delta) \). The embedding preserves joins globally and preserves meets on \( B(A) \). Prime-spectrum and prime-filter representation results are obtained for \( B \)-algebras. Under an injectivity condition on the pseudo-supplement map, an ultrafilter representation follows. The bi-implication \( x \leftrightarrow y = (x \Rightarrow y) \wedge (y \Rightarrow x) \) characterises equality and satisfies a transitivity inequality.