Boolean Products of BL-Algebras

Boolean Lifting Properties for Bounded Distributive Lattices

“…In ring theory, the Lifting Idempotents Property (LIP) ( [21], [1], [6], [16]), which is the property that the idempotent elements can be lifted modulo every left (respectively right) ideal of a (unitary) ring, is closely related to clean rings and exchange rings (in the commutative case, rings with LIP coincide to clean rings and to exchange rings). It is well known that the idempotent elements of a commutative unitary ring R form a Boolean algebra, called the Boolean center of R. Properties similar to LIP have been studied in other algebras which have a Boolean center, namely residuated structures: MV-algebras ( [10]), BL-algebras ( [9], [17]) and residuated lattices ( [11], [18], [12], [7]); the property studied in these algebras refers to the lifting of the Boolean elements modulo filters, and was called the Boolean Lifting Property (BLP). In [7], we have introduced a Boolean Lifting Property (BLP) modulo filters for bounded distributive lattices, we have studied the BLP for residuated lattices from the algebraic and the topological point of view, and we have proved that the BLP is transferrable, through the reticulation functor, between the class of residuated lattices and the class of bounded distributive lattices.…”

Boolean lifting property for residuated lattices

“…It turns out that residuated lattices with BLP are exactly the quasi-local residuated lattices, which have been introduced in [19] as a generalization for quasi-local MV-algebras ( [9]) and quasi-local BL-algebras ( [7]). Residuated lattices with BLP also coincide with those residuated lattices whose lattice of filters is dually B-normal.…”

Free algebras in varieties of BL-algebras generated by a BLn-chain

“…Therefore each nontrivial BL-algebra can be represented as a weak Boolean product of directly indecomposable BL-algebras (see [5] and [23]). The terms of use, available at https://www.cambridge.org/core/terms.…”