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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
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