Generalized co-annihilators in residuated lattices

Abstract

The aim of this paper is to extend the notion of the generalized co-annihilator of residuated lattices. We introduce the concepts of a generalized co-annihilator of a given subset of a residuated lattice $\mathfrak{A}$. We prove that generalized co-annihilators relative to a filter F of $\mathfrak{A}$ are again filters and moreover pseudocomplements in the lattice $\textbf{Fi}(\mathfrak{A})\textbf{[}F,A\textbf{]}$ of all filters of $\mathfrak{A}$ containing $F$. Also, for a given filter F of $\mathfrak{A}$ we prove that the set $Co-An^F(\mathfrak{A})$ of all generalized co-annihilators relative to F forms a complete Boolean lattice.