On pseudo-irreducibility and Boolean lifting property of filters in residuated lattices

This paper advances residuated lattice theory by introducing pseudo-irreducible filters and establishing their fundamental connections to the Boolean lifting property (BLP). Also, key structural properties of these filters are established, and new characterizations of the BLP using pseudo-irreducible filters and the residuated lattice of fractions are derived. Further, we investigate the BLP of the radical of a filter by introducing weak MTL-algebras and the transitional property of radical decomposition (TPRD) as a unifying framework that generalizes Boolean algebras, MV-algebras, BL-algebras, MTL-algebras, and Stonean residuated lattices. By addressing an open question in the literature concerning the BLP of the radical of a residuated lattice, we provide algebraic and topological solutions grounded in the TPRD and the space of maximal filters. Complementary results deepening the understanding of BLP in residuated lattices are also established.

کلیدواژه ها:

Pseudo-irreducible filter ، Boolean lifting property ، residuated lattice ، Weak MTL-algebra ، Transitional property of radicals decomposition (TPRD)