Transactions on Fuzzy Sets and Systems (May 2024)
Triangle Algebras and Relative Co-annihilators
Abstract
Triangle algebras are an important variety of residuated lattices enriched with two approximation operators as well as a third angular point (different from 0 and 1). They provide a well-defined mathematical framework for formalizing the use of closed intervals derived from a bounded lattice as truth values, with a set of structured axioms. This paper introduces the concept of relative co-annihilator of a subset within the framework of triangle algebras. As filters of triangle algebras, these relative co-annihilators are explored and some of their properties and characterizations are given. A meaningful contribution of this work lies in its proof that the relative co-annihilator of a subset $T$ with respect to another subset $Y$ in a triangle algebra $\mathcal{L}$ inherits specific filter's characteristics of $Y$. More precisely, if $Y$ is a Boolean filter of the second kind, then the co-annihilator of $T$ with respect to $Y$ is also a Boolean filter of the second kind. The same statement applies when we replace the Boolean filter of the second kind with an implicative filter, pseudo complementation filter, Boolean filter, prime filter, prime filter of the third kind, pseudo-prime filter, or involution filter, respectively. Finally, we establish some conditions under which the co-annihilator of $T$ relative to $Y$ is a prime filter of the second kind.
