Sugeno integrals and their particular cases such as weighted minimum and maximum have been used in multiple-criteria aggregation when the evaluation scale is qualitative. This paper proposes two new variants of weighted minimum and maximum, where the criteria weights have a role of tolerance threshold. These variants require the use of a residuated structure, equipped with an involutive negatio...

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 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...

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...

In this talk we are going to explore an interesting connection between the famous Burnside problem for groups, regular languages, and residuated lattices. Let K be a finitely axiomatized class of residuated lattices. Recall that the usual way of proving decidability of universal theory for K is to establish that K has the finite embeddability property (FEP) [2, 3]. It turns out that the method ...