‎Triangle Algebras and Relative Co-annihilators

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.