Elsevier

Fuzzy Sets and Systems

Volume 425, 30 November 2021, Pages 62-82
Fuzzy Sets and Systems

A survey of categorical properties of L-fuzzy relations

https://doi.org/10.1016/j.fss.2021.03.014Get rights and content

Abstract

An L-fuzzy relation is a relation valued on a complete lattice L with a monoidal structure. This paper reviews four categories of L-fuzzy relations each modelling an area where Fuzzy Set Theory can be applied. We review the notions of these multivalued binary relations and present some basic properties of the corresponding categories aiming at applications in areas such as Computing Science, Linear Logic and Quantum Mechanics. The emphasis is on the monoidal aspects of the categories. Monoidal categories are one of the most applied kinds of categories. A monoidal view of Fuzzy Relations may widen the spectrum of applications of Fuzzy Set Theory.

Introduction

“The significance of this work may lie more in its point of view than in any particular results.” Goguen [1]

Category theory is a conceptual framework with the power to describe general abstract structures in mathematics and how these structures interact or relate. In Category Theory, the structure of monoidal categories has numerous applications in several areas [2]. For example, in Logic, this rich structure can be used to define models for linear logic [3], [4], [5]. On the other hand, in Computer Science, symmetric monoidal categories and their graphical calculus (string diagrams [6]) are used to model diagrams in a network-style diagrammatic language [7], [8] and this graphical language has applications in many fields such as machine learning [9], game theory [10], programming language [11], concurrency [12], functional programming [13] and other applications [14], [15]. Besides, the dagger symmetric monoidal categories have applications in Quantum Mechanics, in the theory of open quantum systems, and string theory among others [16], [17], [18].

Fuzzy Set Theory deals with problems characterised by uncertainties and inaccuracies in data and information [19]. Fuzzy Set Theory has been successfully applied in areas such as computer science [20], logic [21], quantum mechanics [22], [23], quantum logic [24], medicine [25], control engineering, and artificial intelligence [19], [26], [27].

Although Category Theory and Fuzzy Set Theory are independently developed, many approaches and applications connect them [1], [28], [29]. A particularly interesting one for the present survey is their approach to what is a relation.

Binary relations are usually defined as subsets of the Cartesian product of two sets RA×B, or as binary valued (characteristic) function as R:A×BB, where B={0,1} [30]. Besides, sets have an associated membership or characteristic function χA:UB, where U is a universal set, and A the set in question.

Generalising the valuation set from B={0,1} to an arbitrary set K takes us to the realm of multivalued sets and multivalued relations. Common terminology is to call multivalued entities with K=B, crisp entities. In contrast, when the valuation set is the unit interval [0,1], the entities are called fuzzy entities.

By restricting K=L, where L is a complete Brouwerian lattice was proposed by Goguen [1]. Starting from Goguen's work, we consider valuations on monoidal lattices L to obtain additional structures on the corresponding categories.

Due to the recent success of categorical foundations in several areas [4], [31], [32], [33], another purpose of this survey is to provide a categorical description of the various types of approaches to multivalued relations. The point of views adopted here of using monoidal categorical model theory as foundations for Fuzzy Sets Theory allows for a cleaner organisation of the subject and hopefully will pave the way for co-operations of the fields involved. In particular, widening the gamut of applications of Fuzzy Set Theory.

Finally, it is possible to achieve a higher level of abstraction and generality using the Kleisli Category construction, a path we did not follow in the present work (see Section 7 for detailed comments). One of our aims is the popularisation of the relational categorical models for the fuzzy set theory researchers. We opted for a more concrete approach explicitly giving the constructions since one of the purposes of the present survey is to show to the fuzzy sets community the relationships between categories of fuzzy sets and (categorical) models of quantum mechanics, linear logic, and computer science. We strove to concretely and systematically enumerate the categories' properties and contrast the models.

There are different proposals of categorical models for Linear Logic with a fuzzy perspective [34]. Schalk and Paiva describe a method for constructing models of Linear Logic based on the category of sets and relations (Rel) [35].

Dedekind categories is one of the algebraic formalisation studied in [36] to develop the fuzzy relational calculus. Besides, the theory of Goguen categories (based on Dedekind categories) constitutes a good framework to reason about L-fuzzy controllers and other applications of fuzziness in Computer Science [37], [38].

Throughout the literature there are numerous proposals supporting that Quantum Mechanics can be modelled using categories [16], [39]. Specifically, the dagger symmetric monoidal categories (†-SMC) and compact (closed) categories with basis structures which can describe many characteristics of Quantum Mechanics [17].

Marsden and Genovese in [40] define the category Rel(Q) of crisp sets and relations valued in a quantale Q. They incorporate an algebraic signature structure, an algebraic Q-relation (Σ,E), to capture convexity, one of their goals. They also show that these constructions give concrete descriptions of the mathematical objects of interest, suitable for use in applications. They apply their technique to the construction of new and existing models for natural language processing and applications in cognition [40].

Special attention is then given to categories that have multivalued relations as morphisms. Starting with the structure of the category Rel and its numerous applications [33], [41], and relating monoidal categories with notions of fuzzy relations, we hope to widen the scope of applications of both fields and strengthen their relationship.

We want to emphasise our approach to investigate the monoidal properties of the categories involved aiming at applications. In [42] the authors defined the category with fuzzy sets and relations - Rel[0,1], investigated basic categorical properties and enriched this category with the monoidal structure coming from t-norms, but here we investigate the monoidal properties of the more general case of relations valued in a general complete Brouwerian lattice L. In their work, B. Coecke in [43] and M. Barr in [34] respectively, study the monoidal aspects of the categories Rel (crisp sets and crisp relations) and RelL(L) (L-fuzzy sets and L-fuzzy relations), but here we add the study of the †-SMC structure in the category RelL(L). Finally, M. Winter and E. Jackson do not investigate the monoidal properties of the category LRel (crisp sets and L-fuzzy relations) in [44], which we do here.

One purpose of this review is to collect in one place, in a systematical way, the properties of the following monoidal categories with multivalued relations as morphisms:

  • 1.

    Rel: crisp sets and crisp relations [43, p. 223].

  • 2.

    Rel(L): L-fuzzy sets and crisp relations [42].

  • 3.

    LRel: crisp sets and L-fuzzy relations [44].

  • 4.

    RelL(L): L-fuzzy sets and L-fuzzy relations1 [34].

In each case we investigate the properties the category has and when it is possible to obtain a (monoidal) categorical model [45].

In Section 3, we provide a brief review of the monoidal categorical properties and structures in the category Rel.

In Section 4, we approach the category of L-fuzzy sets and crisp relation Rel(L) and, in addition, to provide some categorical constructions (product and coproduct), we study the monoidal structure in this category and show that the category Rel(L) is a symmetric monoidal category and that there is an involutive functor (†) only for a particular subcategory of Rel(L). Moreover, we describe the structures of internal comonoid and monoid.

The authors in [36], [46] have shown some basic lattice theoretical properties within the category of sets and fuzzy relations. The generalisation to the category of sets and L-fuzzy relation is seen in [38] as a particular case of the Dedekind and the Goguen categories. In Section 5, we exhibit the product and the coproduct in the category LRel and demonstrated that this category is dagger symmetric monoidal. Furthermore, we also consider the structures of internal comonoid and monoid.

M. Barr [34] defines the category of L-fuzzy sets and L-fuzzy relations and shows that this category is a model for Linear Logic. In Section 6, based on M. Barr's construction, we describe the product and coproduct in category RelL(L) and we exhibit that the category RelL(L) is symmetric monoidal. In addition to this, we also present the structures of internal comonoid and monoid for this category.

In a nutshell, we propose a systematic review of categories with sets and relations in the L-fuzzy environment and we investigate some basic categorical properties such as product, coproduct, tensor, negation, dagger structure, internal monoid and comonoid.

We will see that the four categories here addressed to have a symmetrical monoidal structure (the structure necessary for several categorical models).

Section snippets

Lattices

Consider a poset (P,), i.e. a set P partially ordered by ≤. Given elements a,bP, the least upper bound ab of the set {a,b} on P, is called the join of a and b. Similarly, the greatest lower bound ab of the set {a,b} is called the meet of a and b. Note that, for an arbitrary poset P, ab and ab may not exist.

Definition 2.1

A lattice L is a poset such that for all a,bL there exists the join ab and the meet ab.

For our purposes the most important class of lattices is the following:

Definition 2.2

Let L be a lattice and a,

The category of crisp sets and crisp relations

In this section, we cover the properties of the category Rel [43], category of crisp sets and crisp relations (Crisp-Crisp).

Definition 3.1

The category Rel is defined as follows:

  • 1.

    The objects are crisp sets: A,B,C...;

  • 2.

    The morphisms are crisp relations R:AB, i.e. R:A×BB;

  • 3.

    For R:AB and S:BC the composite is defined by:SR:={(a,c)A×C|bB;(a,b)R,(b,c)S};

  • 4.

    The identity is defined by: IdA:={(a,a)A×A|aA}.

The category of L-fuzzy sets and crisp relations

Goguen in [1] explores the categorical properties of the L-fuzzy sets and functions. In [42] the authors defined the category with fuzzy sets and relations.

In this section, we present the category Rel(L) of L-fuzzy sets and crisp relation (LFuzzy - Crisp).

We describe the product and coproduct based on the construction in Rel. In addition, we study the monoidal structure in Rel(L) and we exhibit that Rel(L) is a symmetric monoidal category.

We will see that there is an involutive functor (†) only

The category of crisp sets and L-fuzzy relations

The authors in [36], [46] have developed monoidal categorical properties with the category of sets and fuzzy relations.

The generalisation for the category LRel, of sets and L-fuzzy relation, is seen in [38] as a particular case of the Dedekind and Goguen categories.

In this section, we exhibit the product and the coproduct in the category LRel and that this category is dagger symmetric monoidal. Moreover, we exhibit the structures of internal comonoid and monoid.

As mentioned earlier, the

The category of L-fuzzy sets and L-fuzzy relations

M. Barr showed that the category of L-fuzzy sets and L-fuzzy relations where L is a complete ⁎-autonomous lattice gives a ⁎-autonomous category and shows that this category is a model for Linear Logic [34].

In this section, we systematize properties and structures of category RelL(L) of L-fuzzy sets and L-fuzzy relations (LFuzzy-LFuzzy).

Some properties and definitions in this section were based on M. Barr's construction [34]. In particular, we describe the product and coproduct in category RelL(L

Kleisli category

We have described separate and concretely some basic properties of four monoidal categories with multivalued relations as morphisms. As pointed out by one of the anonymous referees, the four categories are particular cases of a more general and abstract concept: the Kleisli category of a monad in a given category [50].

Definition 7.1

The Kleisli category of a monad (M:CC,μ,η), denoted CM, is a category where:

  • 1.

    An object of CM is an object C of the category C;

  • 2.

    A morphism f:CCMC of CM is a morphism f:CCMC of C;

Conclusion

We saw that Rel(L) and RelL(L) are symmetric monoidal categories but have no dagger structure. In contrast, the categories Rel and LRel are enriched with sufficient structure to express many features of Quantum Mechanics [39], [43]. In addition, categories Rel and RelL(L) are models for Linear Logic. Besides, the four categories investigated here have the necessary structure to model many fundamental aspects of Computer Science as in [7], [8], [14], [15].

Therefore, this survey presented a broad

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We sincerely thank the anonymous referees whose comments and suggestions helped improve the quality of this manuscript.

This work was partially supported by the Brazilian National Council for Scientific and Technological Development (CNPq) grant number 421849/2016-9.

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