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...

This contribution focuses on inverse fuzzy transforms (shortly inverse F-transforms) over residuated lattices introduced by I. Perfilieva and their approximation properties. We will try to reduce some requirements used in the original work to prove Approximation theorem. Moreover, we show in which sense F-transforms are the best approximations. Keywords— Fuzzy transform, Approximation, Extensio...

The paper presents results on approximation in residuated lattices given that closeness is assessed by means of biresiduum. We describe central points and optimal central points of subsets of residuated lattices and provide properties of these points. In addition, we present algorithms for two problems regarding optimal approximation.

In this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice. We give certain characterizations of fuzzy semipartitions and fuzzy partitions over a complete residuated lattice, as well as over a linearly ordered complete Heyting algebra. In the latter case, for a fuzzy equivalence relation over a linearly ordered comple...

We show that the size reduction problem for fuzzy automata is related to the problem of solving a particular system of fuzzy relation equations. This system consists of infinitely many equations, and finding its general solution is a very difficult task, so we first consider one of its special cases, a finite system whose solutions, called right invariant fuzzy equivalences, are common generali...

Formal concept analysis is a particular method of analysis of relational data. In addition to tools for data analysis, formal concept analysis provides elaborated mathematical foundations for relational data. In the course of the last decade, several attempts appeared to extend formal concept analysis to data with graded (fuzzy) attributes. Among these attempts, an approach based on residuated ...

We extend Cayley’s and Holland’s representation theorems to idempotent semirings and residuated lattices, and provide both functional and relational versions. Our analysis allows for extensions of the results to situations where conditions are imposed on the order relation of the representing structures. Moreover, we give a new proof of the finite embeddability property for the variety of integ...