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

Abstract: In this paper, we introduce the notion of a pseudo-irreducible filter in a residuated lattice and compare this concept with related notions such as prime and maximal filters. Then, we recall the Boolean lifting property for filters and present useful characterizations for this property using pseudo-irreducible filters and the residuated lattice of fractions. Next, we study the Boolean lifting property of the radical of a filter. Furthermore, we introduce weak MTL-algebras and residuated lattices that have the transitional property of radicals decomposition (TPRD) as generalizations of several algebraic structures, including Boolean algebra, MV-algebra, BL-algebra, MTL-algebra, and Stonean residuated lattice. Moreover, by comparing weak MTL-algebras with other classes of residuated lattices, we address an open question concerning the Boolean lifting property of the radical of a residuated lattice. Finally, we give a topological answer to an open question about the Boolean lifting property of the radical of a residuated lattice. Several additional results are also obtained, further enriching the understanding of the Boolean lifting property in residuated lattices.