In this paper, we study residuated lattices in order to give new characterizations for dense, regular and Boolean elements in residuated lattices and investigate special residuated lattices in order to obtain new characterizations for the directly indecomposable subvariety of Stonean residuated lattices. Free algebra in varieties of Stonean residuated lattices is constructed. We introduce in re...

Stonean residuated lattices are closely related to Stone algebras since the bounded lattice reduct of a distributive Stonean residuated lattice is a Stone algebra. In the present work we follow the ideas presented by Chen and Grätzer and try to apply them for the case of Stonean residuated lattices. Given a Stonean residuated lattice, we consider the triple formed by its Boolean skeleton, its a...

At present, the filter theory of $BL$textit{-}algebras has been widelystudied, and some important results have been published (see for examplecite{4}, cite{5}, cite{xi}, cite{6}, cite{7}). In other works such ascite{BP}, cite{vii}, cite{xiii}, cite{xvi} a study of a filter theory inthe more general setting of residuated lattices is done, generalizing thatfor $BL$textit{-}algebras. Note that fil...

Several algebraic structures (namely HW, BZMV , Stonean MV and MV∆ algebras) related to many valued logical systems are considered and their equivalence is proved. Four propositional calculi whose Lindenbaum– Tarski algebra corresponds to the four equivalent algebraic structures are axiomatized and their semantical completeness is given.

By a symmetric residuated lattice we understand an algebra A = (A,∨,∧, ∗,→,∼, 1, 0) such that (A,∨,∧, ∗,→, 1, 0) is a commutative integral bounded residuated lattice and the equations ∼∼ x = x and ∼ (x ∨ y) =∼ x∧ ∼ y are satisfied. The aim of the paper is to investigate properties of the unary operation ε defined by the prescription εx :=∼ x → 0. We give necessary and sufficient conditions for ...

Let K be a compact space, and denote by K̃ its hyperStonean envelope. We discuss the class of spaces K with the property that K̃ is homeomorphic to Ĩ, the hyper-Stonean envelope of the closed unit interval I. Certainly each uncountable, compact, metrizable space K belongs to this class. We describe several further classes of compact spaces K for which K̃ = Ĩ. In fact, K̃ = Ĩ if and only if the Bana...