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

Jipsen, P.; Montagna, F.

Algebra Univers. 60, No. 4, 381-404 (2009).

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.

Reviewer: Dana Piciu (Craiova)

MSC:

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

Keywords:

generalized BL-algebras; residuated lattices; basic logic; generalized MV-algebras; lattice-ordered groups; hoops