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...

We solve several open problems on the cardinality of atoms in the subvariety lattice of residuated lattices and FL-algebras [4, Problems 17–19, pp. 437]. Namely, we prove that the subvariety lattice of residuated lattices contains continuum many 4-potent commutative representable atoms. Analogous results apply also to atoms in the subvariety lattice of FLi-algebras and FLo-algebras. On the othe...

The existence of lateral completions of `-groups is an old problem that was first solved, for conditionally complete vector lattices, by Nakano [5]. The existence and uniqueness of lateral completions of representable `-groups was first obtained as a consequence of the orthocompletions of Bernau [1], and later the proofs were simplified by Conrad [3], who also proved the existence and uniquenes...

In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Fi...

In 1998, Hájek established a representation theorem of BL-algebrs as all subdirect products of linear BL-algebrs. We establish a similar result for the much wider class of prelinear residuated algebras, in which neither the lattice structure nor the divisibility of the monoid operation is assumed. We show, in the case of prelinear residuated lattices, that this order embedding becomes a lattice...

Cancellative residuated lattices are a natural generalization of lattice-ordered groups (`-groups). Although cancellative monoids are defined by quasi-equations, the class CanRL of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of CanRL that cover the trivial variety, namely the varieties generated by the integers and the negative intege...

We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lat...

This work is towards the study of the relationship between fuzzy preordered sets and Alexandrov (left/right) fuzzy topologies based on generalized residuated lattices here the fuzzy sets are equipped with generalized residuated lattice in which the commutative property doesn't hold. Further, the obtained results are used in the study of fuzzy automata theory.

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follows, including various types of completeness theorems of substructural logics.