A residuated lattice is an ordered algebraic structure L = 〈L,∧,∨, · , e, \ , / 〉 such that 〈L,∧,∨〉 is a lattice, 〈L, ·, e〉 is a monoid, and \ and / are binary operations for which the equivalences a · b ≤ c ⇐⇒ a ≤ c/b ⇐⇒ b ≤ a\c hold for all a, b, c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as “dividing” on the right...

In this article we prove a set of preservation properties of the reticulation functor for residuated lattices (for instance preservation of subalgebras, finite direct products, inductive limits, Boolean powers) and we transfer certain properties between bounded distributive lattices and residuated lattices through the reticulation, focusing on Stone, strongly Stone and m-Stone algebras. 2000 Ma...

We investigate a relationship between extensionality of fuzzy relations and their Lipschitz continuity on generalized metric spaces. The duality of these notions is shown, and moreover, two particular applications of the extensionality property in the field of approximation are given.

We show how the firing rule of Petri nets relies on a residuation operation for the commutative monoid of natural numbers. On that basis we introduce closed monoidal structures which are residuated monoids. We identify a class of closed monoidal structures (associated with a family of idempotent group dioids) for which one can mimic the token game of Petri nets to define the behaviour of these ...

In this paper, we investigate functorial relations between Alexandrov fuzzy topologies and upper approximation operators in complete residuated lattices. We present some examples. AMS Subject Classification: 03E72, 03G10, 06A15, 06F07, 54A40

Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conu...

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...

We are concerned with the subvariety of commutative, bounded, and integral residuated lattices, satisfying divisibility and prelinearity, namely, BL-algebras. We give an explicit combinatorial description of the category that is dual to finite BL-algebras. Building on this, we obtain detailed structural information on the locally finite subvarieties of BL-algebras that are analogous to Grigolia...