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On the structure of generalized BL-algebras. (English) Zbl 1109.06011
A residuated lattice is an algebra $$L=(L,\wedge ,\vee ,\cdot ,e,\backslash ,/)$$ such that $$(L,\wedge ,\vee )$$ is a lattice, $$(L,\cdot ,1)$$ is a monoid and multiplication is both left and right residuated, with $$\backslash$$ and $$/$$ as residuals, i.e. $$a\cdot b\leq c\Leftrightarrow a\leq c/b\Leftrightarrow b\leq a\backslash c$$ for all $$a,b,c\in L.$$ A generalized BL-algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities $$x\wedge y=((x\wedge y)/y)y=y(y\backslash (x\wedge y)).$$ In the first part of this paper the authors shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. By observing that the idempotents in GBL-algebra form a subalgebra of elements that commute with all other elements, subsequently they construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. Also, they give equational bases for the varieties generated by such algebras; this construction provides a new way of order-embedding the lattice of varieties of $$l$$-groups into the lattice of varieties of integral GBL-algebras. We remark that the results of this paper also apply to pseudo BL-algebras.

##### MSC:
 06D35 MV-algebras 06F05 Ordered semigroups and monoids 03G25 Other algebras related to logic 08B15 Lattices of varieties
##### Keywords:
generalized BL-algebra; basic logic; residuated lattice
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