On the structure of generalized BL-algebras

Abstract

A generalized BL-algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities x ∧ y = ((x ∧ y)/y)y = y(y\(x ∧ y)). It is shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. We also observe that the idempotents in a GBL-algebra form a subalgebra of elements that commute with all other elements.Subsequently we construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. We also give equational bases for the varieties generated by such algebras. The construction provides a new way of order-embedding the lattice of -group varieties into the lattice of varieties of integral GBLalgebras.The results of this paper also apply to pseudo-BL algebras.

How Do $$\ell $$-Groups and Po-Groups Appear in Algebraic and Quantum Structures?

“…It was an open problem whether every pseudo BL-algebra is good, see Di Nola et al (2002b, Problem 3.21). This was answered in the negative in Dvurečenskij et al (2010), where a special type of pseudo BL-algebras generated by the -group of integers, Z, defined in Jipsen and Montagna (2006) was used, which we now call a kite. This construction was generalized in Dvurečenskij and Kowalski (2014) for arbitrary -groups.…”

“…Di Nola et al (2002b, Problem 3.21)). In Dvurečenskij et al (2010) it was solved in the negative showing that an algebra from Jipsen and Montagna (2006), which was using the -group of integers Z, and which we now call a kite, provides such a counterexample. This idea was generalized in Dvurečenskij and Kowalski (2014), where a construction of general kites using an arbitrary -group was studied and the basic properties of such pseudo BL-algebras were established.…”

On decomposition of pseudo BL-algebras

“…The following important lemma was proved in [JiMo,Lem 2] for generalized BL-algebras; we present it in the form suitable for pseudo BL-algebras.…”

“…Let G be an -group written multiplicatively and with neutral element e. We denote by (G − ) ∂ the dual poset of the lattice reduct of [JiMo,Lem 8] …”

Soft Constraints Processing over Divisible Residuated Lattices

“…We shall see that, as far as we are concerned with the implementation of local consistency techniques, for instance the k-hyperarc consistency enforcing algorithm presented in Section 4, prelinearity turns out to be redundant. Since prelinearity is exactly the property that specializes BL-algebras inside the class of DRLs [16], we are led to the latter as a defensible level of generality for our unifying evaluation framework. On the logical side, the variety of DRLs forms the algebraic semantics of an intersecting common fragment of basic logic and intuitionistic logic, called generalized basic logic [3].…”