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The Blok-Ferreirim theorem for normal GBL-algebras and its application. (English) Zbl 1192.06011

Generalized BL-algebras (GBL-algebras for short) are divisible residuated lattices.
In this paper, the authors investigate normal GBL-algebras, that is, integral GBL-algebras in which every filter is normal. For these structures they prove an analogue of Blok and Ferreirim’s ordinal sum decomposition theorem. As applications they prove that the variety of commutative and integral GBL-algebras has the finite embeddability property, so the universal theory of commutative GBL-algebras is decidable and \(n\)-potent GBL-algebras are commutative.
Also, they present a representation theorem for finite GBL-algebras as poset sums of GMV-algebras.

MSC:

06F05 Ordered semigroups and monoids
03G25 Other algebras related to logic
06D35 MV-algebras
06F15 Ordered groups
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