‎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.