On extension of Fuzzy connectives

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DEPTO. DE INFORM ÁTICA E MATEM ÁTICA APLICADA P ÓS-GRADUAÇ ÃO EM SISTEMAS E COMPUTAÇ ÃO

On Extension of Fuzzy Connectives

Eduardo Silva Palmeira

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Eduardo Silva Palmeira

On Extension of Fuzzy

Connectives

Thesis presented to the Graduate Program in Systems and Computation of Department of In-formatics and Applied Mathematics of Federal University of Rio Grande do Norte in a fulfill-ment of the Requirefulfill-ments for the degree of Phi-losophy Doctor .

Advisor:

Prof. Dr. Benjam´ın Ren´e Callejas Bedregal

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Acknowledgements

I have a lot of thanks to do, because a thesis is not built in days or weeks, it is the result of a lengthy effort and a lot of persistence and perseverance, where we deal every day with frustrations and achieve-ments in the struggle for the consolidation of ideas and conjectures that constitute this work today.

First of all I thank God who gave me the gift of life and that, in good and bad times, was always by my side watching and illuminating my path. For the infinite blessings He give me every day, praise and glory be given in His name.

I thank my parents Pedro and Telma, who directly or indirectly have always believed in me, giving me full support and that, even in the face of life’s difficulties, never failed to give me everything they could so that I tread my path to greatness in life. Also to my brothers Cristina and Danillo and my nephews Hendrick and Enzo for all the warmth.

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overpowering covered with the purest feeling of love, able to renew me and make me forget everything bad that might exist in the world. For my master, professor Benjam´ın Bedregal, for believing in me from the beginning and for putting all possible credibility in my work. Cer-tainly this work would not be the same, if it not had the orientation of a person so qualified and competent, able to demonstrate a humil-ity so great as to always treat their students with equal professional. Moreover, I would like to thank you for your friendship and partner-ship, and for the beers and casual conversations. I leave this course happy for having won much more than a research partner for doing a great friend.

Also, for GIARA research group from Pamplona-Spain, in the figure of the professor Humberto Bustince, thank you very much for giving me the opportunity to be researching with them for six months and the experience that they provided me with a personal and professional enrichment of great value.

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Let M be a sublattice of lattice L and K be a fuzzy operator (eg. a t-norm, a t-conorm, a fuzzy fuzzy negation, etc.) on M. So, how can we extend this operator from M to L preserving the most possible number of its properties?

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List of Figures

2.1 Hasse diagrams of lattices L and M . . . 13

2.2 The relaxed idea of sublattice . . . 16

2.3 Hasse diagrams of lattices M,L1,L2 and L3 . . . 18

2.4 Hasse diagrams of lattices M and L . . . 19

2.5 Hasse diagrams of lattices M and L . . . 26

4.1 Diagram of interval constructor . . . 81

4.2 Diagram of intuitive idea . . . 81

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List of Tables

2.1 Tables of retractions r1, r2 and r3. . . 18

2.2 Tables of pseudo-inverses s1, s2 and s3. . . 18

2.3 A functionI on L . . . 33

3.1 Implication onM . . . 59

3.2 Table of Abbreviations . . . 77

3.3 Properties preserved by TE _and _SE _{. . . .} ₇₇

3.4 Properties preserved by IE _{. . . .} ₇₈

3.5 Properties preserved by NE _{. . . .} ₇₈

4.2 Properties preserved by TE ⊙ and S⊙E . . . 104

4.3 Properties preserved by NE ⊙ . . . 104

5.1 Restricted equivalence function on latticeM . . . 108

5.2 Restricted equivalence function on latticeL . . . 113

5.3 The function IREF on lattice M . . . 115

6.2 Comparing results ofTE _and _TE ⊙ . . . 139

6.3 Comparing results ofSE _and _SE ⊙ . . . 139

6.4 Comparing results ofIE _and _IE ⊙ . . . 139

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Chapter 1 Introduction

This thesis is devoted to discuss about the problem of extending fuzzy connec-tives from sublattices to a greater one in such way to provide a suitable extension of them able to preserve the largest number of their properties.

Besides being a naturally interesting and challenging mathematical problem (Hestenes [1941]; Uspenskii [1966]; Whitney [1934]), the problem of extending functions or operators arises in many situations in various branches of computer science such as image processing, mathematical morphology, computational se-mantics, object-oriented programming, etc. For instance, there is a very known operator in object-oriented programming named inheritance and behavioral sub-typing that works as an extension operator through which one can extend a subclass to a superclass, leveraging their behaviors (methods) and variables (at-tributes). In mathematical morphology, erosion and dilation operators play a similar role switching images between lattices.

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1.1 On Extension Problem

The problem of extending functions is very know and studied by many re-searchers in many branches of knowledge. Particularly, in exact sciences, the problem of extending functions is widely studied in some areas such as mathe-matics, physics, computer science, etc (see Hans-Peter and Shapiro [1997]; Hori-uchi and Murakami [1993];Murota and Shioura[2000]). In a general way, we can describe this problem as follows: how to extend a given function from a subset to entire domain preserving its main properties, that is, if Lis an arbitrary set and supposing f is a function defined and possessing a property P on a nonempty subset M of L how can f be extended to L in order to preserve property P for the elements of L_\M? In other words, which is the best choice to definef(x) for elements x_∈L_\M?

The answer for this is: It depends! This is very simple if we want only to construct a new function that has L as its domain. In this case, it is enough to define f(x) = a for a suitable and fixed a belonging to L (i.e., define f as a constant function for the elements belonging to L_\M). However, this task becomes more complex if we want that extension of f preserve its properties. For instance, an important theorem of analysis states that a continuous function f defined on a bounded closed setM can be extended to the whole space preserving its continuity Apostol[1974]; Bartle and Sherbert [1982]; Lima [1982].

1.2 Fuzzy Logic

Emerging from the important work “Fuzzy Sets Theory” proposed by Zadeh

[1965], fuzzy logic is a generalization of classical logic which typically considers for membership degrees values in the unit interval [0,1] instead of only_{0,1_}, but in modern fuzzy logic, lattices are used to range these degrees Goguen [1967];

Gr¨atzer [2011];Trillas [1979].

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Pardo and de la Fuente[2010]; Takagi[2009]; Zadrozny and Kacprzyk [2009]. In literature, one may find a substantial amount of works that introduce a fuzzy version of classic concepts as well as some others that deal with the improvement of techniques and tools utilizing fuzzy mathematics B´ona[2006]; Gr¨atzer [2009];

H´ajek [1998]; Jacas and Recasens [1994]; Liu [2011]; Mitra and Pal [2005]; Siler and Buckley [2005];Takano [2002].

Talking about applications, fuzzy logic has been widely used to generalize and define important theories and tools in control theory, artificial intelligence, etc. According to Bojadziev and Bojadziev[1996]

“Fuzzy sets and fuzzy logic are powerful mathematical tools for mod-eling and controlling uncertain systems in industry, humanity, and nature; they are facilitators for approximate reasoning in decision making in the absence of complete and precise information. Their role is significant when applied to complex phenomena not easily de-scribed by traditional mathematics.”

As for every logic, an appropriate definition of connectives and rules is essen-tial. In fuzzy logic, conjunctions are interpreted by triangular norms (t-norms) and disjunctions are generalized by triangular conorms (t-conorms). Such as in classical case, a t-conorm S can be constructed from a given t-norm T, and a fuzzy negation N, by

N(S(x, y)) =T(N(x), N(y)) (1.1) In this case, t-conorm S is said to be N-dual of T (by dually principe).

Another concept which plays an important role in fuzzy logic are De Morgan triples (see Calvo [1992]; da Costa et al. [2011]; Garc´ıa and Valverde [1989];

Klement et al. [2000]), a fuzzy version of De Morgan’s law. More specifically, a triple _hT, S, N_i where T is a t-norm, S is a t-conorm and N is a negation, may be called a De Morgan triple if _hT, S, N_i satisfies both the equality (1.1) and

N(T(x, y)) = S(N(x), N(y)) (1.2)

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In particular, notice that t-norms, t-conorms, fuzzy negations and implications are functions as well as most fuzzy other operators so, the problem of extending functions can be considered for them. In this context, one of the main works is due toSaminger-Platz et al.[2008], in which the authors propose a way to extend a given t-norm T from a complete sublattice (in the sense of the usual concept of sublattice) M to a bounded lattice L. However, their extension method is very drastic due to the action of function x∗ _{= sup}

M{y | y 6L x, y ∈M ∪ {0L,1L}}

(since it works as a “collapsing function”, see (3.1)). This is motivated by the fact that they wish to propose the least possible extension of a t-norm T.

In this sense, seeking to have a more general way to extend fuzzy connectives we would like to propose a method which could be more flexible than one provided by Saminger-Platz et al. [2008]. To do so, we consider a generalized version of the concept of sublattice in which M is not necessarily a subset ofL, but can be seen as a copy embedded in L.

Specifically speaking, we propose the following concept of sublattice: let M and L be two bounded lattices. It is said that M is a (r, s)-sublattice of L if there is a retraction r : L _−→ M with pseudo inverse s : M _−→ L such that r_◦s=idM; i.e. M is a (r, s)-sublattice of Lif M is a retract ofL(see Definition

2.10). A natural example of sublattice in this sense can be given by taking a bounded lattice K asM and the interval lattice K constructed from it as L (see Example 4.1).

Based on notion of (r, s)-sublattice, our early research has led us to provide a way to extend t-norms, t-conorms and fuzzy negations (named extension method via retractions, see Palmeira and Bedregal[2012]), in which we try to achieve two goals: (1) generalize the extension of t-norms presented in Saminger-Platz et al.

[2008]; (2) in order to preserve the largest possible number of properties of these fuzzy connectives which are invariant by retractions. However, despite the first goal has been achieved, the extension proposed in Palmeira and Bedregal [2012] does not completely fulfill (2) since some important properties related to t-norms and t-conorms are not preserved such as continuity, for example. For extension of fuzzy negation this method does not preserve involution property and hence it does not preserve strong negations (see Remark 3.2).

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via retractions proposed inPalmeira and Bedregal[2012];Palmeira et al.[2009] we have turned our attention to investigate extensions that are more effective in pre-serving properties of fuzzy connectives even this new extensions do not fulfill goal (1). The inspiration for this comes from the following important fact: the interval constructor (seeBedregal and Takahashi[2006];Bedregal et al.[2006b]) provides a natural way to identify a lattice F with F _{via retractions. So, we developed} inPalmeira et al. [2012b] a new method to extend t-norms (t-conorms and fuzzy negations) using a special mapping named e-operator as in Definition 4.1 (this method is named extension method via e-operator). To verify that this method is more efficient than extension method via retractions in preserving properties we investigated in Palmeira et al. [2012b] same issues as in Palmeira and Bedregal

[2012] and results were quite satisfactory. Every problem presented by extension method via retractions were overcome. Another advantage of extension method via e-operator becomes evident when we are extending t-norms (and t-conorms) since some constraints in hypotheses of Theorem 3.1 are not necessary anymore, namely M could not be a lower (r, s)-sublattice of L (see Theorem 4.1).

1.3 Restricted Equivalence Functions

Basically our studies are theoretical, however determine a wide range of pos-sible applications. One of our main interest is applying our extension methods for operators used in several branches of knowledge that lead with images. Our motivation for choosing images comes from the fact that many works discuss about images defined on lattices nowadays, and it is natural to generalize con-cepts related to image processing for lattices in order to obtain a much more general framework than [0,1]. In particular bounded lattices are of interest since intensities in images can be considered as taking values in such lattices.

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• To provide a formalization of the concept of restricted equivalence functions on bounded lattices;

• To present a characterization theorem of these functions via implications; • Apply our two extension methods for extending restricted equivalence

func-tions and test which one has a better behavior.

Results of this research period were submitted for specialized journals on fuzzy sets (see, Palmeira et al. [submitted 2013a,s]) where we made a formalization of definition of restricted equivalence functions on bounded lattices and its exten-sions.

1.4 Objectives

The main objectives of this thesis is developing a consistent formalization of methods for extending lattice-valued fuzzy connectives and other fuzzy operators which preserve the largest possible number of their properties. Moreover, we discuss about the efficiency of these methods for extending restricted equivalence functions.

Hence, seeking to achieve expected results we turn our attention to investigate the following specific goals:

• Propose a suitable definition of the generalized notion of sublattices ((r, s )-sublattices) relaxing some classical constraints of this concept using retrac-tions;

• Define a way to extend fuzzy connectives (t-norms, t-conorms, fuzzy nega-tions and implicanega-tions) from a (r, s)-sublattice to a lattice which is able to preserves their properties. In this case, we develop two different methods: (1) extension method via retraction seeking to generalize the method pro-posed by Saminger-Platz et al.[2008] and (2) a method efficient in preserv-ing properties of extended connectives (extension method via e-operator); • Apply both extension method for other fuzzy operators namely t-subnorms,

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• Establish the relation between extension and conjugate of a fuzzy connective and other operators;

• Define restricted equivalence functions on bounded lattices and study some related properties and operators. Also, apply our extension methods for extending it;

• Develop a study about restricted dissimilarity functions and normal Ee,N

-functions defined on bounded lattices and its extensions.

1.5 Structure of Thesis

This thesis is composed of six chapters which are divided in sections in which we discuss about, lattices, extension, fuzzy connectives, restricted equivalence functions and other operators. We begin describing the research problem of this thesis and making a discussion about state of art on our research area in Chapter 1 (Introduction). Other chapters are organized as follows:

• In Chapter 2we make a specific formalization of main concepts used along this work such as lattices, automorphisms, retractions, sublattices, continu-ity and fuzzy connectives which constitute our theoretical bases;

• Chapter 3 is devoted to present our extension method via retractions. We have chosen t-norms, t-conorms, fuzzy negations, fuzzy implications, De Morgan triples and t-subnorms for applying this method and to test it efficiency in preserving properties. For a better organization, we divided this chapter in six section in order to study these issues separately. We also discuss about the relation between extension and conjugate in Section 3.5;

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• Restricted equivalence functions are addressed in Chapter 5. In section 5.1, we present the definition of L-REF and then we characterize them by implications in Section 5.2. Section 5.3 and 5.4 are dedicated to introduce the notion of restricted dissimilarity functions and normal Ee,N-functions

on bounded lattices and to prove some results. Section 5.5 discusses about restricted equivalence functions on L([0,1]) and presents a generalization of the REF characterization theorem given in Bustince et al. [2006]. In Sections 5.6 and 5.7 we apply our extension method for L-REF;

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Chapter 2 Preliminaries

This chapter is devoted to present and discuss about main concepts and results we are leading in this work which constitute the framework of our studies.

We start in Section 2.1 making a formalization of lattices and its morphisms. In which follows we introduce the key-concept of (r, s)-sublattices, a generalized way to define the notion of sublattice, some useful examples and related defini-tions. Continuity for lattices is considered at the end of this section.

In Section 2.2 we turn our attention to recall the usual notions of lattice-valued fuzzy connectives such as t-norms, t-conorms, fuzzy negations and fuzzy implications. Also in this framework we discuss about t-subnorms and De Morgan triples.

It is important to point out here that most of definitions and results present is this chapter are from the literature, but some of them were proposed by us. So, to highlight this fact definitions and results that came from literature we cite its respective authors.

2.1 Lattice Theory: Definitions and

Construc-tions

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L represents a bounded lattice.

Some elementary concepts will not be shown here, however for a further read-ing about such concepts we recommend Bedregal et al. [2007]; Birkhoff [1973];

Burris and Sankappanavar [2005];Chen and Pham [2001];de Cooman and Kerre

[1994]; Klement and Mesiar [2005]; Klement et al. [2000]; Klir and Yuan [1995];

Lowen [1996].

2.1.1 Bounded Lattices and Homomorphisms

It is very known that the concept of lattices has two approaches, namely: an algebraic and an order-theoretical approaches.

Definition 2.1 Birkhoff [1973] Let L be a nonempty set and 6L be a partial

order on it. We define _hL,6Li as an ord-lattice if for all a, b∈ L the set {a, b}

has a supremum and an infimum. If there are two elements 0L and 1L in L such

that 0L 6L x (bottom) and x 6L 1L (top) for each x∈ L, then hL,6L,0L,1Li is

called a bounded lattice.

Definition 2.2 Birkhoff [1973] Let L be a nonempty set. If _∧L and ∨L are two

binary operations on L, then _hL,_∧L,∨Li is an alg-lattice provided that for each

x, y, z _∈L, the following properties stand:

1. x_∧Ly=y∧Lx and x∨Ly =y∨Lx (commutativity);

2. (x_∧Ly)∧Lz=x∧L(y∧Lz)and(x∨Ly)∨Lz =x∨L(y∧Lz)(associativity);

3. x_∧L(x∨Ly) =x and x∨L(x∧Ly) = x (absorption law).

If in Lthere are elements0Land1Lsuch that, for allx∈L, x∨L0L =x(bottom)

and x_∧L1L =x (top), then hL,∧L,∨L,0L,1Li is a bounded lattice .

A very interesting fact is that Definitions 2.1 and 2.2 are equivalent. Indeed, it is known from the literature that given an alg-lattice L, the relation

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defines a partial order on L and hence L can be seen as an ord-lattice and vice-versa. This allows us to use both definitions indiscriminately. We consider in this whole work the algebraic notion of this concept (Definition 2.2) taking into account that from this structure we can always define a partial order on Lwhat is very important for us to compare elements. The reason for choosing it rises up from the lattice homomorphism as we discuss in rest of this subsection.

Remark 2.1 From now on when we say that Lis a bounded lattice it means that L has a structure as in Definition 2.2. Otherwise, an appropriate distinction will be made.

Definition 2.3 Birkhoff [1973] A lattice L is called a complete lattice if every subset of it has a supremum and an infimum element. Notice that every complete lattice is bounded.

Example 2.1 The set [0,1] endowed with the operations defined by x _∧ y = min_{x, y_} and x_∨y= max_{x, y_} for all x, y _∈[0,1] is a complete bounded lattice in the sense of Definitions 2.2 and 2.3 which has 0 as the bottom and 1 as the top element.

Example 2.2 For allx, y _∈[0,1]it is possible to define the interval setL([0,1]) = {[x, y] ; 06 x6y61_} . This set equipped with the operations

[x, y]_∧L[w, z] = [x∧w, y∧z] and [x, y]∨L[w, z] = [x∨w, y∨z].

with a_∧b = min(a, b) and a_∨b = max(a, b), is a complete bounded lattice (in the sense of Definition 2.2) which has [0,0] and [1,1] as a bottom and a top respectively. It is easy to see that this lattice is also obtained by considered in L([0,1]) the partial order [a, b]6₂ [c, d] if and only if a6c and b6d.

Remark 2.2 When 6_L is a partial order on L and there are two elements x and y belonging to L such that neither x 6L y nor y 6L x, these elements are

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Definition 2.4 Davey and Priestley[2002] Let(L,6_L,0L,1L)and(M,6M,0M,1M)

be bounded lattices. A mappingf :L_−→M is said to be a lattice ord-homomorphism if, for all x, y _∈L, it follows that

1. If x6_Ly then f(x)6_M f(y); 2. f(0L) = 0M and f(1L) = 1M.

Definition 2.5 Davey and Priestley [2002] Let (L,_∧L,∨L,0L,1L) and (M,∧M,

∨M,0M,1M) be bounded lattices. A mapping f : L −→ M is said to be a lattice

alg-homomorphism if, for all x, y _∈L, we have 1. f(x_∧Ly) = f(x)∧M f(y);

2. f(x_∨Ly) = f(x)∨M f(y);

3. f(0L) = 0M and f(1L) = 1M.

Definition 2.6 Hungerford [2000] A given lattice homomorphism f on L is called:

1. A monomorphism if it is injective;

2. An epimorphism if f is surjective;

3. An isomorphism when f is bijective. An automorphism is an isomorphism from a lattice to itself.

Proposition 2.1 Every alg-homomorphism is an ord-homomorphism.

Proof: _Let _f _: _L _−→ _M _{be an alg-homomorphism. Since} _x 6_L y if only if x_∧Ly =x, therefore f(x) = f(x∧Ly) = f(x)∧M f(y) and hencef(x)6M f(y).

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However, in general, the reciprocal of Proposition 2.1 does not hold. If f : L _−→ M is an ord-homomorphism, since x _∧L y 6L x and x∧L y 6L y, so

f(x_∧Ly)6M f(x) andf(x∧Ly)6M f(y). Thus, f(x∧Ly)6M f(x)∧Mf(y) =

inf_{f(x), f(y)_}, however it is possible for f(x_∧Ly) 6= inf{f(x), f(y)} to occur.

For example, consider the lattices Land M, as depicted in Hasse diagram shown in Figure 2.1. Nevertheless, the map f : L _−→ M defined by f(0L) = 0M,

f(1L) = 1M, f(x) =u and f(y) =v, preserves infimum and supremum elements

and, hence, is an ord-homomorphism, though it is not an alg-homomorphism as ∧ operation is not preserved.

L ◦ ◦ ◦ ◦ 1L x y 0L ❅ ❅ ❅ ❅ ❅_❅ M ◦ ◦ ◦ ◦ ◦ 1M u v w 0M ❅ ❅ ❅ ❅ ❅_❅

Figure 2.1: Hasse diagrams of lattices L and M

In other words the Proposition2.1says that every alg-homomorphism is order-preserving. Due to this fact we have chosen to use the algebraic approach to lattice homomorphisms (to see the other way to define lattice homomorphisms we recommend Birkhoff [1973]; Davey and Priestley [2002]). From now on, alg-homomorphisms will be called just alg-homomorphisms for simplicity.

2.1.2 Automorphisms and Conjugates

Proposition 2.2 Let L be a bounded lattice. Then a function f :L_−→L is an automorphism if and only if

1. f is bijective and

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Proof: _If_f_{is an automorphism, from Proposition}_2.1_{we just need to prove that} x6L ywhenf(x)6Lf(y). Suppose thatf(x)6Lf(y), thenf(x)∧Lf(y) = f(x)

and so, as f is a lattice homomorphism, f(x _∧Ly) = f(x). Thus, since f is

bijective, x_∧Ly=x and therefore x6Ly.

Conversely, suppose that either 1. or 2. is not satisfied. If 1. is violated, i.e., if f is not bijective, then it is not an automorphism by definition. If 2. is violated, then we have two possible cases:

(i) either there are x, y _∈ L such that x 6_L y but not f(x) 6_L f(y). However, then f(x)_∧Lf(y)<Lf(x) = f(x∧Ly), and thusf is not a homomorphism, and

hence neither an automorphism;

(ii) or there are x, y _∈Lsuch that f(x)6_Lf(y) but notx6_L y. Putz =x_∧Ly.

Then z <L x, and f(x∧Ly) =f(z)6=f(x) =f(x)∧Lf(y) due to the fact that

f is bijective; again this mean that f is not a homomorphism, and hence not an automorphism.

Remark 2.3 We denote the set of all automorphisms on a bounded lattice L by Aut(L) . This set endowed with the composition operation is a group that has as neutral element the identity function idL. In algebra, an important tool is

the action of the groups on sets Burris and Sankappanavar [2005]; Hungerford

[2000]. In our case, the action of automorphism group transforms lattice functions in other lattice functions.

Definition 2.7 Bustince et al. [2003] Let L be a bounded lattice and Ln ₌ _L

× · · · ×L be a cartesian product of L n-times. Given a function f : Ln _→ _{L, the}

action of an automorphism ρ over f results in the function fρ _:_Ln _→ _L _defined

as

fρ₍_x

1, . . . , xn) =ρ−1(f(ρ(x1), . . . , ρ(xn))) (2.2)

In this case, fρ _{is called a conjugate of} _f_.

Notice that if f :Ln _→_L _{is a conjugate of} _g _: _Ln _→ _L _and _g _{is a conjugate}

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composition) and iff is a conjugate ofg, theng is also a conjugate off since the inverse of an automorphism is also an automorphism. Thus, an automorphism action on set ofn-ary functions onL(LLn

) determines an equivalence relation on LLn

.

Let f : Ln _→ _L _{be a conjugate of} _g _: _Ln _→ _L_. _If _f₍_x

1, . . . , xn) 6L

g(x1, . . . , xn) for every (x1, . . . , xn)∈Ln then we denote it by f 6g.

2.1.3 Retractions and Sublattices

Recall that the classical notion of sublattice is given as following.

Definition 2.8 Birkhoff [1973] An ordinary sublattice of a lattice L is a subset M of L such that x, y _∈M imply x_∧Ly∈M and x∨Ly∈M.

Nevertheless, we would like to work in a more flexible framework of sublattice where M does not need to be a subset of L. Focusing on this idea, we have the interest to define a generalized notion of sublattice using retractions.

Definition 2.9 Burris and Sankappanavar[2005] A homomorphismrof a lattice L onto a lattice M is said to be a retraction if there exists a homomorphism s of M into L which satisfies r_◦s=idM. A lattice M is called a retract of a lattice

L if there is a retraction r, of L onto M.

Remark 2.4 In the literature, homomorphism s presented in Definition 2.9 has no specific name. But here, this function play an important hole in our studies and by this reason we give to it a special name, viz. a pseudo-inverse of retraction r.

Notice that, if a lattice M is a retract of latticeL, then we have an identifica-tion ofM with a subsetK =s(M) ofL in which it is carried on some properties of M to K, including its lattice structure via retraction r (see Figure 2.2). In this case,K works as a algebraic copy ofM (i.e. they are isomorphic) embedded into Lsince r is an isomorphism when restricted to K.

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Figure 2.2: The relaxed idea of sublattice

isomorphisms). In other words,M is a(r, s)-sublattice ofLif there is a retraction r of L onto M with pseudo-inverse s:M _→L.

Proposition 2.3 The following statements hold:

1. M is a (r, s)-sublattice of L if and only if for all x, y _∈ M we have that r(s(x)_∧Ls(y))∈M and r(s(x)∨Ls(y))∈M;

2. If M is an ordinary sublattice of L (in the sense of Definition 2.8) then M is a (r, s)-sublattice of L where s is the identity function and r(x) = sup_{z _∈L _| z 6x_};

3. If M is a(r, s)-sublattice of L then s(M) equipped with restriction of oper-ation of L is an ordinary sublattice of L.

Proof: _Straight.

Based on the above proposition we can list at least four main advantages in working with (r, s)-sublattice notion instead of the classical one:

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• Algebraic invariance: Since M is isomorphic with a subset s(M) of L every property invariant under homomorphisms that holds in M works on s(M) as well;

• Flexible identification: In Definition 2.10, a pseudo-inverse s of a re-traction r can not be unique. This means that if there exist more than one pseudo-inverse for the same retraction it is possible to identifyM with a subset of L in different ways1 _{what give us the possibility to chose the} best one for our proposes. But it must be clear that when we say that M is a (r, s)-sublattice of L we are considering the existence of at least one pseudo-inverses and fixing it. However, no matter which pseudo-inverse is taken, every result presented here remains working;

• Subclasses of sublattices: Considering (r, s)-sublattice notion allows us to define some subclasses of this concept (see Definition2.11) what it is not possible for ordinary sublattices.

Remark 2.5 Throughout this thesis, it is used the concept of (r, s)-sublattice as in Definition 2.10 instead of retract. Whenever the usual definition of sublattice is used and it is not clear from the context, this sublattice will be called ordinary sublattice.

Definition 2.11 Every retraction r : L _−→ M (with pseudo-inverse s) which satisfiess_◦r6idL2 (idL6s◦r) is called a lower (an upper) retraction . In this

case, M is a lower (an upper) retract of L.

Example 2.3 One can easily see in Figure 2.3 thatM is a lower retract (but it is not an upper retract) of L1, M is an upper retract (but is not a lower retract) of L2 and M is a retract (but is neither an upper nor a lower retract) of L3. In fact, the unique possible retractions are ri : Li −→ M with i ∈ {1,2,3} defined

as in the Table 2.1. Their pseudo-inverses si : M → Li with i ∈ {1,2,3} are

respectively given as in the Table 2.2.

1_{For each pair (}_{r, s}_{) we have a different identification between}_M _{and its image by}_s_. 2_If_f _and_g _{are functions on a lattice}_L_{it is said that}_f ₆_g _{if and only if}_f₍_x₎₆

Lg(x) for

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M ◦ ◦ ◦ ◦ d b c a ❅ ❅ ❅ ❅ ❅_❅ L1 ◦ ◦ ◦ ◦ ◦ 5 3 4 2 1 ❅ ❅ ❅ ❅ ❅_❅ L2 ◦ ◦ ◦ ◦ ◦ 6 5 3 4 2 ❅ ❅ ❅ ❅ ❅_❅ L3 ◦ ◦ ◦ ◦ ◦ ◦ 6 5 3 4 2 1 ❅ ❅ ❅ ❅ ❅_❅

Figure 2.3: Hasse diagrams of lattices M,L1, L2 and L3 x r1

1 a 2 a 3 b 4 c 5 d

x r2 2 a 3 b 4 c 5 d 6 d

x r3 1 a 2 a 3 b 4 c 5 d 6 d Table 2.1: Tables of retractions r1,r2 and r3.

Example 2.4 Let M and L be bounded lattices as shown in Figure 2.4. A map-ping r :L _−→M given by r(x) = sup_{z _∈ M _| s(z)6L x} is a lower retraction

whose pseudo-inverse is a mapping s:M _−→Ldefined by s(1M) = 1L, s(a) =v,

s(b) = x, s(c) = y, s(d) = z and s(0M) = 0L. Therefore, it follows that M is a

(r, s)-sublattice of L in the sense of Definition 2.10.

Remark 2.6 Note that, given a lower retraction it is possible sometimes to define an upper retraction with the same pseudo-inverse. For instance, let L and M be

x s1 s2 s3

a 1 2 1

b 3 3 3

c 4 4 4

d 5 6 6

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M ◦ ◦ ◦ ◦ ◦ ◦ 1 a b c d 0 ❅ ❅ ❅ ❅ ❅_❅ L ◦ ◦ 1 t u ◦ ◦ ◦ ◦ ◦ ◦ v x y z 0 ❅ ❅ ❅ ❅ ❅_❅ ❅ ❅ ❅

Figure 2.4: Hasse diagrams of lattices M and L

lattices as shown in Figure 2.4. If r is a lower retraction with pseudo-inverse s as defined in the Example 2.4, then the function r′ _{given by} _r′₍_x_{) = inf}_{_z _∈

M _| s(z)>_L x_} is an upper retraction since idL6 s◦r′. It is easy to check that

s is also a pseudo-inverse of r′_.

It is worth noting that ifM is a (r, s)-sublattice ofLthen there is a retraction r from L onto M, but it is not required to r to be a lower or an upper retrac-tion. Nevertheless, as shown in the above remark, there may be more than one retraction from L onto M with the same pseudo-inverse. This is a very useful particularity of Definition 2.10 and we would like to highlight it in a definition.

Definition 2.12 Let M be a (r1, s)-sublattice of L. We say that

1. M is a lower (r1, s)-sublattice of L if r1 is a lower retraction. Notation: M < L with respect to (r1, s);

2. M is an upper (r1, s)-sublattice of L whenever r1 is an upper retraction. Notation: M > L with respect to (r1, s);

3. Ifr1 is a lower retraction and there is an upper retractionr2 :L−→M such that its pseudo-inverse is also s, then M is called a full(r1, r2, s)-sublattice of L. Notation: M EL with respect to (r1, r2, s).

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An immediate consequence of the definition of lower (upper) retraction is that, if M EL then it follows that s_◦r1 6idL6s◦r2.

Proposition 2.4 LetK, M andLbe bounded lattices. IfKEMELthenKEL. Proof: _{We shall prove that there exists a lower and an upper retraction} _r _and r′ _from _L _onto _K_{, respectively, both with a pseudo-inverse} _s _{such that} _s_◦_r ₆

idL 6s◦r′.

Supposing K EM with respect to (r1, r2, s1) and M E L with respect to (r3, r4, s2), it follows that

r1◦s1 =r2◦s1 =idK and s1◦r1 6idM 6s1◦r2 and

r3◦s2 =r4◦s2 =idM and s2◦r3 6idL 6s2◦r4

Thus, letting r =r1◦r3,r′ =r2◦r4 and s=s2◦s1 then, for all x∈K, we have

r_◦s(x) = r1(r3(s2(s1(x)))) =r1(s1(x)) =x=idK(x)

r′_◦s(x) = r2(r4(s2(s1(x)))) =r2(s1(x)) =x=idK(x)

and hence

s_◦r=s2◦(s1 ◦r1)◦r3 =s2◦r3 6idL 6s2◦r4 6s2◦(s1◦r2)◦r4 =s◦r′ Since r, r′ _and _s _{are homomorphisms (composition of homomorphisms is also a}

homomorphism) then KEL with respect to (r, r′_{, s}_).

Proposition 2.5 Let M⋖L with respect to (r, s). For every nonempty set A_⊆ M it holds thats(supA) = sups(A). In other words, a pseudo-inverse of a lower retract preserves supremum element.

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On one hand, for all t _∈ s(A) there is a k _∈ A such that t = s(k). Since k 6M a for each k ∈ A, then t = s(k)6L s(a) by monotonicity of s, i.e. s(a) is

an upper bound of s(A) and hence a′ ₆

Ls(a) sincea′ = sups(A).

On the other hand, if we take an arbitrary element k _∈ A it follows that s(k) 6L a′ that implies k = r(s(k)) 6M r(a′) for all k ∈ A which means that

r(a′_{) is an upper bound of} _A_{. Thus by properties of supremum we have that}

a6_M r(a′_{) and hence} _s₍_a₎₆

Ls(r(a′))6L a′ since r is a lower retraction.

Analogously, we can prove the following.

Proposition 2.6 Let M⋗_L _{with respect to}₍_{r, s}₎_{. For every nonempty set}_A _⊆ M it holds that s(infA) = infs(A), i.e. a pseudo-inverse of an upper retract preserves infimum element.

Remark 2.8 It is important to point out that in Proposition 2.5 (Proposition 2.6) the hypothesis that r should be a lower retraction (an upper retraction) is just used to prove inequality s(supA) 6L sups(A) (infs(A) 6L s(infA)). It

means that sups(A) 6L s(supA) (s(infA) 6L infs(A)) always holds no matter

which kind of retraction r is (i.e., lower, upper or neither).

2.1.4 Pseudo-quasi-metrics and Continuity

Intuitively, a distance among two values is a non-negative real value and so a distance function on a set X is a function d : X _×X _−→ R+_{, where} _R+ _{is the} set of non-negative real numbers. In mathematics, several conditions have been considered for distance functions, but the more used ones are those for metric distances. However, in order to consider a reasonable general notion of distance, we will consider the notion of pseudo-quasi-metric distance.

Definition 2.13 Kim [1968] Let X be a set. A functiondX :X×X −→R+, is

a pseudo-quasi-metrics 1 _{if for each} _{x, y} _∈_X _{it satisfies the following conditions:}

1_{In other papers, such as in}_Kasahara_[₁₉₆₈_{], pseudo-quasi-metrics are called pre-metrics .}

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1. dX(x, x) = 0 and

2. dX(x, z)6dX(x, y) +dX(y, z)

In mathematics, an important requirement for functions is its continuity, which intuitively means that small changes in the input result in small changes in the output of function. Therefore, it is necessary to consider some way of measuring changes, which can be made by using some distance notion.

Definition 2.14 Smyth [1992] Let dX and dY pseudo-quasi-metrics on sets X

and Y, respectively. A function f :X _−→Y is continuous w.r.t. dX and dY, or

just (dX, dY)-continuous , if for each x ∈ X and ǫ > 0 there exists δ > 0 such

that for any y_∈X, if dX(x, y)6δ then dY(f(x), f(y))6ǫ.

Proposition 2.7 Let X be a set and d be a pseudo-quasi-metric on X. Then d2

X :X×X −→R+ defined by dX2 ((x1, x2),(y1, y2)) =

p

dX(x1, y1)2+dX(x2, y2)2 is also a pseudo-quasi-metric.

Proof: _{Straightforward.}

Proposition 2.8 Let dX and dY be pseudo-quasi-metrics on the sets X and Y,

respectively. Iff :X _−→Y is(dX, dY)-continuous thenf2 :X −→Y ×Y defined

by f2₍_x_{) = (}_f₍_x₎_{, f}₍_x₎₎ _is ₍_d

X, d2Y)-continuous.

Proposition 2.9 LetdX,dY anddZ be pseudo-quasi-metrics onX, Y andZ,

re-spectively. If g :X _−→Y and f :Y _−→Z are(dX, dY) and (dY, dZ)-continuous

then f _◦g is (dX, dZ)-continuous.

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2.2 Fuzzy Connectives

Typically fuzzy logic considers for membership degrees values the unit interval [0,1]. This logic is very rich due to its connectives (t-norm, t-conorm, fuzzy negation and implication) can be defined in various different ways and have many different properties since the unique consensus about how to define them is that truth functions of these connectives have to behave classically on extremal values 0 and 1. For instance, there are several paper in the literature presenting different definitions for fuzzy implications and variances of it.

Modern fuzzy logic consider lattices to range its degrees of truth for having a much more general framework. In this section, we discuss about these fuzzy connectives on bounded lattices taking into account definitions given inBaczy´nski and Jayaram [2008]; Bedregal et al. [2006b, 2013]. Moreover, we discuss about De Morgan triples and t-subnorms.

2.2.1 T-norms and T-conorms

It presented here a short formalization for the notion of t-norm and t-conorm on bounded lattices. Moreover, some results are demonstrated as well.

Definition 2.15 Bedregal et al. [2006b] Let L be a bounded lattice. A binary operation T :L_×L_−→L is a t-norm if, for all x, y, z _∈L, it satisfies:

(T1) T(x, y) =T(y, x) (commutativity);

(T2) T(x, T(y, z)) =T(T(x, y), z) (associativity);

(T3) If x6_L y then T(x, z)6_L T(y, z), _∀ z _∈L (monotonicity); (T4) T(x,1L) =x (boundary condition).

Definition 2.16 Klement and Mesiar [2005] Let L be a bounded lattice and dL

a pseudo-quasi-metric on L. A t-norm T on L is called

1. Archimedean if for any x, y _∈ L_\{0L,1L}, there exists n ∈ N such that

Tn₍_x₎₆

Ly, where for any m∈N

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2. dL-nilpotent if it is (d2L, dL)-continuous and each x∈L\{0L,1L} is a

nilpo-tent element of T, i.e. there exists n _∈N such that Tn₍_x_{) = 0} L;

3. idempotent if for each x_∈L, T(x, x) = x; and

4. positive if T(x, y) = 0L if and only if either x= 0L or y= 0L;

5. An element a _∈ L is called a zero divisor if there exists a b _∈ L such that T(a, b) = 0L.

Example 2.5 Let L be a bounded lattice. Thus, the function T : L_×L _→ L defined by T(x, y) = x_∧Ly is a t-norm that generalize classical fuzzy t-norm of

minimum, i.e. TM : [0,1]×[0,1]→ [0,1] such that TM(x, y) = min{x, y} for all

x, y _∈[0,1].

Dually, it is possible to define the concept of t-conorms.

Definition 2.17 Bedregal et al. [2006b] Let L be a bounded lattice. A binary operation S :L_×L_−→L is said be a t-conorm if, for all x, y, z _∈L, we have:

(S1) S(x, y) = S(y, x) (commutativity);

(S2) S(x, S(y, z)) =S(S(x, y), z) (associativity);

(S3) If x6y then S(x, z)6S(y, z), _∀ z _∈L (monotonicity); (S4) S(x,0L) = x (boundary condition).

Remark 2.9 Notice that T(x, y) 6L x (or T(x, y) 6L y) and x 6L S(x, y) (or

y 6L S(x, y)) for all x, y ∈L. In fact,T(x, y)6Lx∧y6Lx andx6Lx∨Ly 6L

S(x, y).

Example 2.6 Given an arbitrary bounded lattice L, the function S given by S(x, y) = x_∨L y for all x, y ∈ L is a t-conorm on L that generalize the

clas-sical fuzzy t-conorm of maximum, i.e. SM(x, y) = max{x, y} for all x, y ∈[0,1].

Proposition 2.10 Letρ be an automorphism onL. A t-conorm S :L_×L_−→L satisfies

S(x, y) = 1L if and only if x= 1L or y= 1L (2.4)

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Proof: _{We have that} _Sρ₍_{x, y}_{) = 1}_L _{if and only if} _ρ−1₍_S₍_ρ₍_x₎_{, ρ}₍_y_{))) = 1}_L _if and only if S(ρ(x), ρ(y)) = 1L if and only if ρ(x) = 1L or ρ(y) = 1L (by (2.4)) if

and only if x= 1L or y= 1L.

Similarly, it can be proved the following.

Proposition 2.11 Let ρ be an automorphism on L. A t-norm T :L_×L_−→ L is positive if and only if Tρ _{satisfies also it.}

2.2.2 Fuzzy Negations

A natural extension of notion of fuzzy negations can be done by considering arbitrary bounded lattices as possible sets of truth values.

Definition 2.18 Bedregal et al.[2013] A mappingN :L_→Lis a fuzzy negation on L if the following properties are satisfied for each x, y _∈L:

(N1) N(0L) = 1L and N(1L) = 0L and

(N2) If x6_Ly then N(y)6_L N(x).

Moreover, a fuzzy negation N is strong if it also satisfies the involution property, i.e.

(N3) N(N(x)) =x for each x_∈L. N is strict if satisfies the property: (N4) N(x)<LN(y) whenever y <L x.

and it is called frontier if it satisfies property:

(N5) N(x)_{∈ {}0L,1L} if and only if x= 0L or x= 1L.

Example 2.7 Bedregal et al. [2013] IfL is an arbitrary bounded lattice, then the functions N⊥, N⊤:L→L defined by

N⊥(x) =

(

1L, if x= 0L;

0L, otherwise.

and

N⊤(x) =

(

0L, if x= 1L;

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for each x_∈L are fuzzy negations on L.

Remark 2.10 An element e _∈ L is an equilibrium point of a fuzzy negation N on L if N(e) = e. But differently from usual case and interval-valued case (see

Bedregal [2010]), strong fuzzy negations on lattices can not have an equilibrium point.

Example 2.8 Let L and M be bounded lattices as shown in the Figure 2.5. The function N1 : M → M defined by N1(0M) = 1M, N1(x) = y, N1(y) = x and

N1(1M) = 0M is a strongM-negation. Nevertheless, N1 has no equilibrium point. Now, consider a function N2 :L→L given by N2(0L) = 1L, N2(a) = e, N2(e) =

a, N2(1L) = 0L and N2(u) =u for each u∈ {b, c, d}. In this case, N2 is a strong fuzzy negation with three equilibrium points, namely b, c and d.

M ◦ ◦ ◦ ◦ 1M x y 0M ❅ ❅ ❅ ❅ ❅_❅ L ◦ ◦ ◦ ◦ ◦ ◦ ◦ 1L e

b c d

a 0L ❅ ❅ ❅ ❅ ❅_❅

Figure 2.5: Hasse diagrams of lattices M and L

Proposition 2.12 Let N be a strong fuzzy negation on L. Then 1. N is strict;

2. If N(x)6L N(y) then y6Lx; and

3. N is bijective.

Proof: _If _{y <}_L _x _{then by (N2),} _N₍_x₎6_L N(y). Supposing N(x) = N(y) then N(N(x)) = N(N(y)) and so x = y, which is a contradiction with the premise. Therefore, N(x)<LN(y).

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Since N is strict then N is trivially injective. Moreover, for any y _∈ L we haveN(N(y)) =y and hence N also is surjective. Thus, N is bijective.

From this proposition it follows that for each strong fuzzy negation we have that x_ky if and only if N(x)_kN(y). Notice that although every lattice admits a negation N, it is not true that every lattice admits an involutive negation.

Example 2.9 LetLbe any lattice such that there exists x0 ∈Lwithx0 6= 0L,1L.

Then the mapping:

N(x) =

  

 

0L if x= 1L;

1L if x= 0L;

x0 otherwise.

is a frontier negation. Notice that this example proves that for every lattice L with at least three elements it is possible to define a frontier negation.

Example 2.10 Consider latticeL0 obtained from latticeLin Figure2.5by omit-ting the pointa. Then, it does not exist a strong negation for this lattice. Indeed, if N is such negation, then we should have that0L< N(e)< N(b), N(c), N(d)<1L

which is not possible due to the injectivity of a strong negation. It is important to point out that the fact we can not define a strong negation over a lattices in general, as shown in this example, is not due to the partial order of the lattice. It is easy to see that it is not possible to define a strong negation on the linear ordered lattice L=_{0,1_{} ∪}[2,3]as well, considering the usual linear order of real numbers 6 on L.

Proposition 2.13 Let N : L _→ L be a function, ρ be an automorphism on L and i _{∈ {}1,2,3,4,5_}. N satisfies (Ni) if and only if Nρ _{satisfy (Ni). Moreover,}

e is an equilibrium point of N if and only if ρ−1₍_e₎ _{is an equilibrium point of} _Nρ_.

Proof: _Suppose _N _{satisfies (}_{N i}_{) with} _i_{∈ {}₁_,₂_,₃_,₄_}_{, then} (N1) Nρ₍₀

L) = ρ−1(N(ρ(0L))) = ρ−1(N(0L)) = ρ−1(1L) = 1L. Analogously it

can be proved that Nρ₍₁

L) = 0L;

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isotony of ρ−1_, _ρ−1₍_N₍_ρ₍_y₎₎₆

Lρ−1(N(ρ(x)));

(N3) Nρ₍_Nρ₍_x_{)) =}_ρ−1₍_N₍_ρ₍_ρ−1₍_N₍_ρ₍_x_{)))))) =} _ρ−1₍_N₍_N₍_ρ₍_x_{)))) =} _ρ−1₍_ρ₍_x_{)) =} x;

(N4) Straight; (N5) If Nρ₍_x₎_{∈ {}₀

L,1L}then, by Equation (2.2) and considering thatρ−1(0L) =

0Landρ−1(1L) = 1Lwe haveN(ρ(x))∈ {0L,1L}. Thus, by (N5),ρ(x)∈ {0L,1L}

which implies that x= 0L orx= 1L;

IfN(e) = ethenN(ρ(ρ−1₍_e_{))) =}_e_{and hence} _Nρ₍_ρ−1₍_e_{)) =}_ρ−1₍_N₍_ρ₍_ρ−1₍_e_{)))) =} ρ−1₍_e_).

Reciprocal is straightforward from the previous item and the fact that for any function f :L_→L, (fρ₎ρ−1

=f.

Corollary 2.1 Let N : L _→ L be a function and ρ be an automorphism on L. N is a (strong, frontier) fuzzy negation if and only if Nρ _{is a (strong, frontier)}

fuzzy negation.

Proof: _{Straightforward from Proposition} _2.13.

Klement et al. [2000]; Klir and Yuan [1995] observed that it is possible to obtain, in a canonical way, a fuzzy negation NT from a t-normT. This negation

is called natural negation of T or negation induced by T. In the most general case, where we consider a t-norm on a bounded latticeL, it is not always possible to obtain a natural negation, because the construction of NT is based in the

supremum of, possibly, infinite sets (this concept was generalized for lattices by

Bedregal et al. [2012a]).

Proposition 2.14 Let L be a complete lattice and T be a t-norm on L. Then the function NT :L→L defined by

NT(x) = sup{z ∈L | T(x, z) = 0L} (2.5)

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Proof: _{According to Definition} _2.18_{we shall prove that} _N_T _{satisfies (}_N_{1) and} (N2). Hence

(N1) NT(1L) = sup{z ∈ L | T(1L, z) = 0L} = sup{0L} = 0L and NT(0L) =

sup_{z _∈L _| T(0L, z) = 1L}= supL= 1L;

(N2) Ifx6_L ythen for anyz _∈L,T(x, z)6_LT(y, z) and therefore, ifT(y, z) = 0L then T(x, z) = 0L. So,{z ∈L |T(y, z) = 0L} ⊆ {z ∈L | T(x, z) = 0L}.

Hence,NT(y) = sup{z ∈L|T(y, z) = 0L}6L sup{z ∈L|T(x, z) = 0L}=

NT(x).

Theorem 2.1 Let T be a t-norm on L. If T is positive then NT =N⊥.

Proof: _If _x₆_{= 0}_L _and _z _∈_L _{then, by (T4),}_T₍_{x, z}_{) = 0}_L _{if and only if} _z _{= 0}_L_. So, by Equation (2.5),NT(x) = sup{0L}= 0L. Therefore, NT =N⊥.

Theorem 2.2 Let T be a t-norm on L. If NT is a frontier negation then each

x_∈L_\{0L} is a zero divisor of T.

Proof: _If _x ₆_{= 1}_L_{, then, as} _N_T _{is frontier,} _N_T₍_x₎ ₆_{= 0}_L _{and so sup}_{_z _∈ L _| T(x, z) = 0L} 6= 0L, that is, {z ∈ L | T(x, z) = 0L} 6= {0L}. Thus, since

T(x,0L) = 0L,{0L} ⊂ {z ∈L|T(x, z) = 0L}. Therefore, there existsz ∈L\{0L}

such that T(x, z) = 0L. Hence,x is a zero divisor of T.

Theorem 2.3 Let T be a t-norm on L and ρ be an automorphism on L. Then N_Tρ =NTρ.

Proof: _Let _x_∈_L_{, then}

N_Tρ(x) = ρ−1₍_N

T(ρ(x)))

= ρ−1_(sup_{_z_∈_L _| _T₍_ρ₍_x₎_{, z}₎₌₀

L})

= ρ−1_(sup_{_z_∈_L _| _Tρ₍_{x, ρ}−1₍_z₎₎₌₀

L})

= sup_{ρ−1₍_z₎_∈_L _|_Tρ₍_{x, ρ}−1₍_z₎₎₌₀

L} (∗)

= sup_{z_∈L _| Tρ₍_{x, z}_{) = 0} L}

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(_∗) Notice that every monotone homomorphism preserves supremum element.

It is also possible to define this kind of negation from a given t-conorm S on a complete bounded lattice L, as one can see in the following proposition.

Proposition 2.15 Let L be a complete lattice and S be a t-conorm on L. Then the function NS :L→L defined by

NS(x) = inf{z ∈L | S(x, z) = 1L} (2.6)

is an L-negation called natural negation of S .

Proof:

(N1)

NS(1L) = inf{z ∈ L | S(1L, z) = 1L} = infL = 0L and NS(0L) = inf{z ∈

L _| S(0L, z) = 1L}= inf{1L}= 1L.

(N2)

If x 6_L y then for any z _∈ L we have that S(x, z) 6_L S(y, z) and hence if S(x, z) = 1L then S(y, z) = 1L. Thus, NS(y) = inf{z ∈ L | S(y, z) =

1L}6L inf{z ∈L |S(x, z) = 1L}=NS(x).

Proposition 2.16 Bedregal et al. [2013] Let S be a t-conorm on a complete bounded lattice L. If S is positive then NS =N⊤L

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2.2.3 Fuzzy Implications

In the literature several notions for fuzzy implications have been considered (see for example Baczy´nski [2004]; Bustince et al. [2003]; Fodor and Roubens

[1994];Mas et al.[2007];Yager[1983]). But, the unique consensus is that a fuzzy implication should have the same behavior than classical implication when the crisp case is considered Fodor and Roubens [1994], i.e. for values 0 and 1. Here we consider the notion given by Bedregal et al. [2013].

Definition 2.19 Bedregal et al. [2013] Let L be a bounded lattice. A function I :L2 _→_L _{is called a fuzzy implication if it satisfies the following properties:} (FPA) I(y, z)6_L I(x, z) whenever x6_Ly (First place antitonicity);

(SPI) I(x, y)6LI(x, z) whenever y6Lz (Second place isotonicity);

(CC1) I(0L,0L) = 1L (Corner condition 1);

(CC2) I(1L,1L) = 1L(Corner condition 2);

(CC3) I(1L,0L) = 0L (Corner condition 3)

The set of implications on L will be denoted by I_L_.

Example 2.11 Let L be a bounded lattice. Thus, functions I⊥, I⊤ :L×L → L

given by

I⊥(x, y) =

(

1L, if x= 0L or y= 1L;

0L, otherwise.

and

I⊤(x, y) =

(

0L, if x= 1L and y= 0L;

1L, otherwise.

for all x, y _∈L are fuzzy implications.

For all x, y, z _∈L, we define the following properties of I:

(WFPA) if x 6L y, x ¨ z and y ¨ z, then I(y, z) 6L I(x, z) (weak first place

antitonicity);

(WSPI) if y 6_L z, x ¨y and x ¨ z, then I(x, y) 6L I(x, z) (weak second place

isotonicity);

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(LB) I(0L, y) = 1L (left boundary condition);

(CC4) I(0L,1L) = 1L(corner condition 4);

(NP) I(1L, y) = y for each y∈L (left neutrality principle);

(EP) I(x, I(y, z)) =I(y, I(x, z)) for allx, y, z _∈L (exchange principle); (IP) I(x, x) = 1L for each x∈L(identity principle);

(OP) I(x, y) = 1L if and only if x6Ly (ordering property);

(LOP) if x6_L y then I(x, y) = 1L (left ordering property);

(WLOP) if x6_L y orx_kL y then I(x, y) = 1L (weak left ordering property);

(IBL) I(x, I(x, y)) = I(x, y) for all x, y, z _∈L (iterative Boolean law);

(CP) I(x, y) = I(N(y), N(x)) being N a strong L-negation (contrapositivity property);

(L-CP) I(N(x), y) =I(N(y), x) (left contraposition law); (R-CP) I(x, N(y)) =I(y, N(x)) (right contraposition law); (P) I(x, y) = 0L if and only if x= 1L and y= 0L (positivety) .

It is easy to verify that (SPI) and (CC1) imply (LB). Moreover, (FPA) and (CC2) imply (LB) and consequently (CC4).

Remark 2.11 Notice that if L is a totally ordered set then (WFPA), (WSPI) and (WLOP) are equivalent to (FPA), (SPI) and (LOP), respectively.

Example 2.12 Let L be a bounded lattice (see Figure 2.5) and N2 the strong fuzzy negation on L in Example 2.8. The function I :L2 _→_L _{given by}

I(x, y) =

     

    

1L, if x6L y;

N2(x), if y= 0L and x6= 0L;

y, if x= 1L;

e, otherwise.

(2.7)

satisfies the properties (F P A), (OP), (CP)(with respect to N2) and(P). It is a easy to see that it holds by Table 2.3.

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I 0L a b c d e 1L

0L 1L 1L 1L 1L 1L 1L 1L

a e 1L 1L 1L 1L 1L 1L

b b e 1L e e 1L 1L

c c e e 1L e 1L 1L

d d e e e 1L 1L 1L

e a e e e e 1L 1L

1L 0L a b c d e 1L

Table 2.3: A function I on L

Proof: _If _y 6_L z then N(z) 6_L N(y) and so by (FPA), I(N(y), N(x)) 6_L

I(N(z), N(x)) for each x _∈ L. Therefore, by (CP), I(N(N(x)), N(N(y))) 6_L

I(N(N(x)), N(N(z))). Hence, I(x, y)6_L I(x, z) whereas N is strong.

Proposition 2.18 Letρ be an automorphism onL, I :L_×L_→Lbe a function and P _{∈ {}(F P A),(SP I),(CC1),(CC2),(CC4),(LB),(RB)_}. I satisfies P if and only if Iρ _{also satisfies} _P_.

Proof: _{See Proposition 10 in} _{Bedregal et al.} _[₂₀₁₃_].

A special type of fuzzy implication that we would like to study is the (S, N )-implication, that is, an implication defined from a t-conorm S and a fuzzy nega-tion N.

Definition 2.20 Baczy´nski and Jayaram [2008] Let S be a t-conorm on L and N be a fuzzy negation on L. The function IS,N :L×L−→L given by

IS,N(x, y) = S(N(x), y) (2.8)

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Proposition 2.19 Bedregal et al. [2013] Let I :L_×L_→L be a function and ρ an automorphism on L. Thus, I is an (S, N)-implication on L generated from S and N if and only if Iρ _{is an} ₍_{S, N}₎_{-implication on}_L _{generated from} _Sρ _and_Nρ_.

In other words, (IS,N)ρ =ISρ_,Nρ.

Another special type of implication we would like to consider in this work is R-implication. Taking into account that there exists an isomorphism between classical two-valued logic and classical set theory, it is possible to see that if K and G are subsets of a set X then the identity

Kc

∪G= (K_\G)c ₌[

{P _⊆X _| K_∩P _⊆G_}

holds, where Kc _{is the complement of set}_K _(see_Baczy´_{nski and Jayaram}_[₂₀₀₈_]).

The R-implications (residual implications) are generalizations of the this iden-tity in fuzzy logic.

Definition 2.21 Baczy´nski and Jayaram [2008] Let L be a complete bounded lattice. A function I : L_×L _→ L is called an R-implication if there exists a t-norm T such that for all x, y _∈L we have

I(x, y) = sup_{t_∈L _| T(x, t)6_L y_} (2.9) We denote this implication generated from a t-norm T byIT .

To finish this section we present the notion of negations defined from implica-tions. There exists a natural way to define this particular class of fuzzy negations on [0,1] based on the fact that a propositional formula p is logically equivalent (in classic logic) to p _→⊥ where _⊥ denotes the absurd (see Lemma 1.4.14, pg 18 in Baczy´nski [2004]). Bedregal et al. [2013] have generalized this concept for bounded lattices as in the following proposition.

Proposition 2.20 Let L be a bounded lattice. If a function I : L_× L _→ L satisfies (FPA), (CC1) and (CC3) then the function NI :L→L defined for each

x_∈L by

NI(x) =I(x,0L) (2.10)

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Next lemma provides necessary conditions for NI to be a fuzzy negation and

also establishes some properties.

Lemma 2.2 Bedregal et al. [2013] Let L be a bounded lattice. If a function I :L_×L_→L satisfies (EP) and (OP) then

1. NI is a fuzzy negation;

2. x6LNI(NI(x)) for each x∈L;

3. NI ◦NI ◦NI =NI.

2.2.4 T-subnorms

As a generalization of the concept of t-norms, it was presented by Klement et al. [2000] the notion of triangular subnorms as follows.

Definition 2.22 A function F : [0,1]2 _−→ _[0_,_1] _{that satisfies, for all} _{x, y, z} _∈ [0,1], the properties

1. F(x, y) =F(y, x)

2. F(x, F(y, z)) =F(F(x, y), z)

3. F(x, z)6F(y, z) whenever x6y 4. F(x, y)6min(x, y)

is called a t-subnorm.

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Proposition 2.21 If F is a t-subnorm then the function T : [0,1]2 _−→ _[0_,_1]_, given by

T(x, y) =

(

F(x, y), if(x, y)_∈[0,1[2 min(x, y), otherwise is a t-norm.

Proof: _See _{Klement et al.} _[₂₀₀₀_].

Naturally, the concept of t-subnorm may be generalized for bounded lattices.

Definition 2.23 LetL be a bounded lattice. A functionF :L_×L_−→Lis called a t-subnorm on L if it satisfies the following properties:

1. F(x, y) =F(y, x)

2. F(x, F(y, z)) = F(F(x, y), z)

3. F(x, z)6_L F(y, z) whenever x6_Ly 4. F(x, y)6_Lx_∧Ly

Proposition 2.22 If F is a t-subnorm on bounded lattice L, then T defined by

T(x, y) =

(

F(x, y), if(x, y)_∈(L_\{1L})2

x_∧Ly, otherwise

is a t-norm on L.

Proof: _Since _F _{is a t-subnorm and} _x_∧_L_y _{is a t-norm, then}_T _{is commutative,} associative and monotone. Thus, we shall only prove thatT satisfies the boundary condition, i.e., T(x,1L) =x for each x∈L.

But, for each x _∈ L we have x = x_∧L 1L = T(x,1L) since x 6 1L for all

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Note that it is possible to define a t-subconormRdually from the definition of t-subnorm just by replacing the propertyF(x, y)6_L x_∧Lyforx∨Ly6LR(x, y).

Precisely, we have

Definition 2.24 LetLbe a bounded lattice. A function R:L_×L_−→L is called a t-subconorm on L if it satisfies the following properties

1. R(x, y) = R(y, x)

2. R(x, R(y, z)) = R(R(x, y), z)

3. R(x, z)6_LR(y, z) whenever x6_Ly 4. x_∨Ly6L R(x, y)

Of course, a dual proof of Proposition2.22can be given to prove the following proposition.

Proposition 2.23 IfR is a t-subconorm on bounded lattice L, thenS defined by

S(x, y) =

(

R(x, y), if(x, y)_∈(L_\{0L})2

x_∨Ly, otherwise

is a t-conorm on L.

2.2.5 De Morgan Triples

De Morgan’s laws represent, among other things, a way to relate disjunctive and conjunctive operators via negations. Both in set theory and formal logics these laws are very important tools to simplify equations, formal proofs and other tasks. Specifically, these laws are given by

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or

¬(α_∧β)_{≡ ¬}α_{∨ ¬}β and _¬(α_∨β)_{≡ ¬}α_{∧ ¬}β (2.12) In fuzzy logic, De Morgan’s laws can be naturally generalized using norms, t-conorms and negations as operators. Nevertheless, due to the several possibilities that exist to define these operators in fuzzy logic, De Morgan’s laws can be generalized in different ways as seen inCalvo[1992];da Costa et al.[2011];Garc´ıa and Valverde[1989];Koles´aros´a and Mesiar[2010];Palmeira and Bedregal[2012]. Here, we choose a version that we believe is the most general one because it does not impose any constraints for operators involved and it is more fateful with Equations (2.11) and (2.12).

Definition 2.25 Calvo [1992] Let T be a t-norm, S a t-conorm and N a fuzzy negation, all defined on the same bounded lattice L. We say that _hT, S, N_i is a De Morgan triple if, for all x, y _∈L, we have

1. N(T(x, y)) =S(N(x), N(y));

2. N(S(x, y)) =T(N(x), N(y)).

Remark 2.12 Naturally, every time we are talking about De Morgan triple_hT, S, N_i its operators are considered defined on the same lattice. In order to highlight the lattice concerned, we shall simply say that _hT, S, N_i is a De Morgan triple on L when T, S and N are a t-norm, a t-conorm and a fuzzy negation all defined on L.

It is important to point out that there are some fuzzy negations which are not involutive and hence for some t-norms, t-conorms and fuzzy negations only one of the items of Definition 2.25 holds true, as can be seen in the example below.

Example 2.13 Let L be the bounded lattice shown in Figure 2.4. It is easy to check that the function N : L _−→ L, given by setting N(0L) = 1L, N(z) = v,

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negation. Thus, considering the t-norm T and the t-conorm S defined by T 0 z x y v u t 1

0 0 0 0 0 0 0 0 0 z 0 z z z z z z z x 0 z x z x z x x y 0 z z y y y y y v 0 z x y v u v v u 0 z z y u y y u t 0 z x y v y v t 1 0 z x y v u t 1

S 0 z x y v u t 1 0 0 z x y v u t 1 z z z x y v 1 1 1 x x x x v v 1 1 1 y y y v y v 1 1 1 v v v v v v 1 1 1 u u 1 1 1 1 1 1 1 t t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

then it follows that

N◦T 0 z x y v u t 1

0 1 1 1 1 1 1 1 1

z 1 v v v v v v v

x 1 v x v x v x x

y 1 v v y y y y y

v 1 v x y z y z z

u 1 v v y y y y y

t 1 v x y z y z z

1 1 v x y z y z 0

S◦(N×N) 0_L z x y v u t 1_L

0 1 1 1 1 1 1 1 1

z 1 v v v v v v v

x 1 v x v x v x x

y 1 v v y y y y y

v 1 v x y z y z z

u 1 v v y y y y y

t 1 v x y z y z z

1 1 v x y z y z 0

Therefore N(T(a, b)) = S(N(a), N(b)) for all a, b _∈ L. However, on the other hand, we see that N _◦S ₆=T _◦(N _×N) as evidenced on the following tables:

N◦S 0 z x y v u t 1

0 1 v x y z y z 0

z v v x y z 0 0 0

x x x x z z 0 0 0

y y y z y z 0 0 0

v z z z z z 0 0 0

u y 0 0 0 0 0 0 0

t z 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0

T◦(N×N) 0 z x y v u t 1

0 1 v x y z y z 0

z v v x y z y z 0

x x x x z z z z 0

y y y z y z y z 0

v z z z z z z z 0

u y y z z z y z 0

t z z z z z z z 0

On extension of Fuzzy connectives

On Extension of Fuzzy Connectives

Eduardo Silva Palmeira

Eduardo Silva Palmeira

On Extension of Fuzzy

Connectives

Advisor:

Prof. Dr. Benjam´ın Ren´e Callejas Bedregal

Acknowledgements

List of Figures

List of Tables

Contents

Chapter 1

Introduction

1.1

On Extension Problem

1.2

Fuzzy Logic

1.3

Restricted Equivalence Functions

1.4

Objectives

1.5

Structure of Thesis

Chapter 2

Preliminaries

2.1

Lattice Theory: Definitions and

Construc-tions

2.1.1

Bounded Lattices and Homomorphisms

2.1.2

Automorphisms and Conjugates

2.1.3

Retractions and Sublattices

2.1.4

Pseudo-quasi-metrics and Continuity

2.2

Fuzzy Connectives

2.2.1

T-norms and T-conorms

2.2.2

Fuzzy Negations

2.2.3

Fuzzy Implications

2.2.4

T-subnorms

2.2.5

De Morgan Triples