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Filters and ideals in the generalization of pseudo-BL algebras. (English) Zbl 1436.06028

Summary: In this paper, we introduce the notion of quasi-pseudo-BL algebras as the generalization of pseudo-BL algebras and quasi-pseudo-MV algebras. First, we investigate the properties of quasi-pseudo-BL algebras and show the subdirect product composition of any quasi-pseudo-BL algebra. Especially, some properties of good quasi-pseudo-BL algebras are presented. Second, we discuss the filters of quasi-pseudo-BL algebras and prove that there exists a bijective correspondence between normal filters and filter congruences on a quasi-pseudo-BL algebra. The properties of some special filters are also discussed. Finally, we study the ideals of quasi-pseudo-BL algebras and investigate some connections between ideals and filters of a quasi-pseudo-BL algebra.

MSC:

06D35 MV-algebras
03G25 Other algebras related to logic
06B10 Lattice ideals, congruence relations
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