Study of Pseudo BL–Algebras in View of Left Boolean Lifting Property

In this paper, we define left Boolean lifting property (right Boolean lifting property) LBLP (RBLP) for pseudo BL–algebra which is the property that all Boolean elements can be lifted modulo every left filter (right filter) and next, we study pseudo BL-algebra with LBLP (RBLP). We show that Quasi local, local and hyper Archimedean pseudo BL–algebra that have LBLP (RBLP) has an interesting behavior in direct products. LBLP (RBLP) provides an important representation theorem for semi local and maximal pseudo BL–algebra

Functorial Properties of the Reticulation of a Universal Algebra

The reticulation of an algebra A is a bounded distributive lattice whose prime spectrum of ideals (or filters), endowed with the Stone topology, is homeomorphic to the prime spectrum of congruences of A, with its own Stone topology. The reticulation allows algebraic and topological properties to be transferred between the algebra A and this bounded distributive lattice, a transfer which is facilitated if we can define a reticulation functor from a variety containing A to the variety of (bounded) distributive lattices. In this paper, we continue the study of the reticulation of a universal algebra initiated in [27], where we have used the notion of prime congruence introduced through the term condition commutator, for the purpose of creating a common setting for the study of the reticulation, applicable both to classical algebraic structures and to the algebras of logics. We characterize morphisms which admit an image through th

Boolean Lifting Properties for Bounded Distributive Lattices

In this paper, we introduce the lifting properties for the Boolean elements of bounded distributive lattices with respect to the congruences, filters and ideals, we establish how they relate to each other and to significant algebraic properties, and we determine important classes of bounded distributive lattices which satisfy these lifting properties