‎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 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.