Co-annihilators and fuzzy co-annihilators in EQ-algebras

Abstract

In this paper, we introduce the co-annihilator ^⊥A of a set A and the co-annihilator of (F : a) of a relative to a prefilter (filter) F in an EQ-algebra ɛ. We investigate related properties of them, and obtain that the lattice of all prefilters $\mathcal{PF}\left(\varepsilon \right)$ forms a pseudo-complemented lattice and the collection of all co-annihilators $\mathcal{COA}\left(\varepsilon \right)$ forms a Boolean algebra in a separated EQ-algebra ɛ. Moreover, we introduce the notion of Δ-co-annihilators in an EQ_Δ-algebra ɛ_Δ and conclude that the collection of all Δ-prefilters ${\mathcal{PF}}_{\Delta }\left({\varepsilon }_{\Delta }\right)$ in an ℓEQ_Δ-algebra ɛ_Δ constitutes a relative pseudo-complemented lattice. Finally, we introduce and investigate two types of fuzzy co-annihilators Ann _R  (μ) and Ann (μ, ν) in ɛ. We come to a conclusion that the set of all fuzzy filters $\mathcal{FF}\left(\varepsilon \right)$ in a residuated ℓEQ-algebra ɛ forms a relative pseudo-complemented lattice whence → is the Gödel residuated implication in St ^r  (μ, ν).