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On ideals in De Morgan residuated lattices. (English) Zbl 1449.03052

Summary: In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal lattice, we pay attention to prime, maximal, \(\odot\)-prime ideals and to ideals that are meet-irreducible or meet-prime in the lattice of all ideals. We introduce the concept of an annihilator of a given subset of a De Morgan residuated lattice and we prove that annihilators are a particular kind of ideals. Also, regular annihilator and relative annihilator ideals are considered.

MSC:

03G25 Other algebras related to logic
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
06B10 Lattice ideals, congruence relations
06B20 Varieties of lattices
06D35 MV-algebras
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