Weakly Irreducible Filter in Strong Quasi-Ordered Residuated Systems

Abstract

In this article, the notion of weakly irreducible filters in strong quasi-ordered residuated systems is introduced and analyzed. It is shown that any weakly irreducible filter is a prime (and therefore, irreducible) filter. It is also proved that if the lattice F(A) of all filters in a strong quasi-ordered residuated system A is distributive, then any irreducible filter in A is weakly irreducible in A.