Fuzzy relation and fuzzy function over fuzzy sets: a retrospective

Abstract

In this paper, some earlier results dealing with fuzzy relations on fuzzy sets by one of the present authors are recapitulated to clarify the motivational background of the present study. Then concepts like fuzzy homorelation, correlation, cohomorelation, fuzzy function and fuzzy morphism are introduced and discussed. The presentation is uniform and integrated in the sense that every concept is developed on fuzzy set as base and is organized from the angle of foundational aspects of fuzzy set theory.

1 Introduction

This paper is a study in retrospect of the fundamental notions, the building blocks of fuzzy set theory, viz., fuzzy relation and fuzzy function. It seems that the first paper on fuzzy relation was written by Zadeh himself (Zadeh 1971a). Because of its wide applicability and also due to natural theoretical interest there had been numerous researches on fuzzy relation theory in the 1970s and 1980s. It is now almost impossible to present the complete list. However, outside the publications of Chakraborty and Ahsanullah (1992); Chakraborty and Banerjee (1991); Chakraborty and Das (1981, 1983, 1983a, b; 1987); Chakraborty et al. (1985) only a fragment is included in the reference (Bandler and Kohout 1986a, b; Bezdek and Harris 1978; Di Nola et al. 1989; Höhle and Stout 1991; Kaufmann 1975; Ovchinnikov and Riera 1982; Ovchinnikov 1981; Rosenfold 1975; Sanchez 1976; Sessa 1984; Yager 1981; Yeh and Bang 1975; Zadeh 1971a, b). Two features are noticeable in these works. Firstly, in most cases fuzzy relations are defined on crisp sets; and secondly, the notion of fuzzy function is not ‘properly’ fuzzy. There had been, of course, few researches that take care of these points but these are mainly category theoretic studies (Etyan 1981; Higgs 1973; Höhle and Stout 1991). Besides, the consequences that are drawn from the definitions are generally in the crispized direction such as the crisp partitioning of a set under fuzzy equivalence relation (Zadeh 1971a) and crisp ordering of a set under fuzzy orders (Zadeh 1971b), etc. These developments, though often quite useful, are not uniform and integrated. One of the present authors with his earlier collaborators published a number of research articles keeping an eye on the above points. But unfortunately these works did not draw much attention of the community (as will be evident from the bibliography of Klir and Yuan 1995). The notions of fuzzy relations and fuzzy function have undergone various generalizations in the past two decades. In these papers several proposals are made to address the second issue, namely, proper fuzzification of the notion of a function (Demirci and Recasens 2004; Gilles 1984; Ha j́ ek 1998; Klawonn 2000; Perfilieva 2011); but the first tradition, viz., defining fuzzy relation, function, etc., on crisp sets, by and large continues. This paper aims at the following:

• (i) to present the basic concepts of fuzzy set theory systematically as is done in a standard text book of classical informal set theory and

• (ii) to present the notions retaining the flavor of fuzziness to a great extent.

The first point deserves clarification. The development of intuitive set theory rests on the following initial steps: defining sets as collection of objects in a universe, defining the cartesian product of sets, defining relations as subsets of the cartesian product of sets, and finally defining functions as binary relations satisfying certain conditions. We shall follow this route, and in this sense we call the present approach ‘integrated’.

We shall, however, take the closed interval [0, 1] as the value set althrough and take $\wedge$ (min), $\vee$ (max) and $1 - x$ as the operations for conjunction (intersection), disjunction (union) and negation (complementation), respectively. In this sense, this approach is uniform. $\wedge$ and $\vee$ have some obvious advantages over other t-norms and s-norms, and at the same time some generalities are lost. But we want to explore here how much the basic notions of a particular fuzzy set theory can be built in an integrated way in the above-mentioned sense.

In Sect. 2, we shall recapitulate some of the earlier results forming the background of the current study. Section 3 deals with the notions of fuzzy homorelation, correlation and cohomorelation. In Sect. 4 fuzzy functions, morphisms are defined and investigated. Then there are some concluding remarks pointing on a few open issues.

2 Basic notions and earlier work

To make the paper self-contained, let us present some preliminary ideas in this section.

A fuzzy set, as defined in Zadeh (1965), is a function which assigns a value from [0, 1] to every element of a universal set, say $U$. The operations intersection ($\cap$), union ($\cup$) and complementation ($^{c}$) are defined as mentioned in the introduction.

The notion of cartesian product of ordinary sets has been generalized in fuzzy context by the following definition.

Definition 1

(Chakraborty and Das 1981) For two fuzzy sets $\tilde{A}$, $\tilde{B}$ on a universe $U$, the product is a fuzzy set on $U \times U$ defined as $\tilde{A} \times \tilde{B}$ $(x, y)$ = $\min \{\tilde{A}(x), \tilde{B}(y)\}$    for all $x, y \in U$.

In Chakraborty and Das (1981), a detailed study on properties of the product of fuzzy sets can be found.

Definition 2

(Klir and Yuan 1995) Let $\tilde{A}$ be a fuzzy set on the universal set $U$. A fuzzy set $\tilde{B}$ on $U$ is said to be a fuzzy subset of $\tilde{A}$, denoted by $\tilde{B} \subseteq \tilde{A}$ if for all $x \in U$, $\tilde{B}(x) \le \tilde{A}(x)$ holds.

In the study of fuzzy sets, $\alpha$-cut is an important notion which is an ordinary set defined as below.

Definition 3

(Klir and Yuan 1995) Let $\tilde{A}$ be a fuzzy set on $U$, i.e., $\tilde{A}$ maps every element of $U$ to the set [0, 1]. For any $\alpha \in [0, 1]$, the $\alpha$-cut of $\tilde{A}$ is defined as the set $\{x \in U : \tilde{A}(x) \ge \alpha \}$. The strict $\alpha$-cut of $\tilde{A}$ is the set $\{x \in U : \tilde{A}(x) > \alpha \}$.

The strict 0-cut of a fuzzy set $\tilde{A}$ is called the support of $\tilde{A}$. Throughout the paper, we shall denote the support of a fuzzy set $\tilde{A}$ by $A$.

In classical set theory, as mentioned in the introduction, a binary relation $R$ from a set $A$ to a set $B$ is defined as a subset of $A \times B$, i.e., $R \subseteq A \times B$. In fuzzy context, one kind of generalization of this definition is obtained by introducing a binary fuzzy relation $\tilde{R}$ as a fuzzy (sub)set of $A \times B$, i.e., $\tilde{R}$ maps every element of $A \times B$ to [0, 1]. This approach is taken in Zadeh (1965, 1971a), and in almost all the literature on fuzzy relation till date. The main departure in Chakraborty and Das (1983) took place with the introduction of fuzzy relation on fuzzy sets. We shall pursue this line and show by examples why and how this minor modification is significant.

Definition 4

(Chakraborty and Das 1983) For any finite $n$, a fuzzy $n$-ary relation $\tilde{R}$ on fuzzy sets taken in the order, say $\tilde{A}_{1}$, $\tilde{A}_{2}, \cdots \tilde{A}_{n}$ over a universe $U$, is a fuzzy subset of $\tilde{A}_{1} \times \tilde{A}_{2} \times \ldots \times \tilde{A}_{n}$. For $n = 2$, a binary fuzzy relation $\tilde{R}$ from a fuzzy set $\tilde{A}$ to a fuzzy set $\tilde{B}$ is a fuzzy subset of $\tilde{A}$ $\times$ $\tilde{B}$, i.e., for any $(x, y)$ from the universe of discourse $U \times U$,

$\tilde{R}$ $(x, y)$ $\le$ $(\tilde{A}$ $\times$ $\tilde{B})$ $(x, y)$ = $\min \{\tilde{A}(x), \tilde{B}(y)\}$.

The difference from the existing literature of fuzzy relation is quite clear here. Let us consider the following example.

Example 1

Let $U$ = $\{x_{1}, x_{2}, x_{3}, x_{4}\}$.

So $\tilde{R}$ is a fuzzy relation from $\tilde{A}$ to $\tilde{B}$ since for each $i, j$, $i, j \in \{1, 2, 3, 4\}$, $\tilde{R}(x_{i}, x_{j}) \le \tilde{A} \times \tilde{B}(x_{i}, x_{j})$.

Let us consider an $\tilde{R}^{\prime }$ such that $\tilde{R}^{\prime }(x_{1}, x_{1})$ = 1 and $\tilde{R}^{\prime }(x_{4}, x_{3})$ = .5, and rest of the entries get the same value as $\tilde{R}$. According to our definition, $\tilde{R}^{\prime }$ will not be considered as a fuzzy relation from $\tilde{A}$ to $\tilde{B}$. Both $\tilde{R}$ and $\tilde{R}^{\prime }$ are mappings from $U \times U$ to [0, 1], and hence from $A \times B$ to [0, 1], where $A$, $B$ are the supports of $\tilde{A}$ and $\tilde{B}$, respectively, but while the first one is a fuzzy relation from $\tilde{A}$ to $\tilde{B}$, in our sense, the second one is not. Besides, following the integrity principle, imposition of this restriction plays a role in the definition of various properties of relations, especially, the property of fuzzy reflexivity as shown below.

In Chakraborty and Das (1983a, b), various types of reflexivity are defined.

• (i) $\tilde{R}$ is reflexive if $\tilde{R}(x, x)$ = 1 for all $\tilde{A}(x) >$ 0, i.e., $x \in A$, where $A$ = $\{x \in U: \tilde{A}(x) > 0\}$. In this case $\tilde{A}$ turns out to be a crisp set since by definition $\tilde{R} \subseteq \tilde{A} \times \tilde{A}$. This is taken as the usual definition of reflexivity in the existing literature.

• (ii) $\tilde{R}$ is reflexive of order $\alpha > 0$ if $\tilde{R}(x, x)$ = $\alpha$ for all $x \in A$.

• (iii) $\tilde{R}$ is $\epsilon$-reflexive, $\epsilon > 0$, if $\tilde{R}(x, x)$ $\ge \epsilon$ for all $x \in A$.

• (iv) $\tilde{R}$ is weakly reflexive if $0 < \tilde{R}(x, x)$ $\ge \tilde{R}(x, y)$ for all $x \in A$.

• (v) $\tilde{R}$ is absolutely reflexive if $\tilde{R}(x, x)$ $> 0$ for all $x \in A$.

• (vi) $\tilde{R}$ is $\omega$-reflexive if $\tilde{R}(x, x)$ = $\tilde{A}(x)$ for all $x \in A$.

Ideas like weakly reflexive, $\epsilon$-reflexive, and $\omega$-reflexive are interesting abstractions of the notion of classical reflexivity. $\epsilon$-reflexivity fixes a non-zero threshold for the degree of relatedness of an element with itself. Whereas, weak reflexivity generalizes the idea that each pair $(x, x)$ is related, and the relatedness grade of $x$ with $x$ is greater or equal to the relatedness grade with any other $y$. This gives an emphasis on the significance of the relatedness of an element to itself over the other elements.

A deeper philosophical standpoint is reflected in the notion of $\omega$-reflexivity. It is accepted that $\tilde{A}(x)$ may be interpreted as the degree of existence (belongingness) of $x$ in the concept represented by the fuzzy set $\tilde{A}$. Now a philosophical position states that the extent to which an entity belongs to a concept is the extent of the entity’s self-identity relative to the concept. The self-identity is in turn reflected in the identity relation $I$ which is to be reflexive. That means, $I(x, x)$ = the extent of self-identity of $x$ with respect to the concept $\tilde{A}$ = the degree of existence of $x$ in $\tilde{A}$ = $\tilde{A}(x)$. This notion is captured in $\omega$-reflexivity. Even if somebody does not want to enter into philosophy, the idea of $\omega$-reflexivity is an extremely elegant mathematical notion which depends on defining a relation $\tilde{R}$ with respect to the underlying fuzzy set $\tilde{A}$. If by a reflexive relation one means $\tilde{R}(x, x)$ = 1 always (which is in vogue), and $\tilde{R}$ is to be considered as a subset of the product $\tilde{A} \times \tilde{A}$, $\tilde{A}$ has to be a crisp set. This destroys the beauty. For further clarification we present a few examples.

Example 2

Let us see different kinds of reflexivity over the fuzzy set $\tilde{A}$ considered in the Example 1. So, the cartesian product $\tilde{A} \times \tilde{A}$ will be as follows:

We give examples of fuzzy relations $\tilde{R}_{1}$ and $\tilde{R}_{2}$ on $\tilde{A}$ which are weakly reflexive (i.e., $0 < \tilde{R}(x, x) \ge \tilde{R}(x, y)$) and $\omega$-reflexive (i.e., $\tilde{R}(x, x)$ = $\tilde{A}(x)$), respectively.

It is to be observed that both $\tilde{R}_{1}$ and $\tilde{R}_{2}$ obey the restriction $\tilde{R}_{l}(x_{i}, x_{j}) \le \tilde{A} \times \tilde{A}(x_{i}, x_{j})$, $l = 1, 2$. Also if we consider $\epsilon$-reflexivity, $\tilde{R}_{1}$ is a .6-reflexive but not .7-reflexive.

Note 1

There is an interesting definition of reflexivity in Bodenhofer (2002), called $E$-reflexivity where $E$ is a fuzzy equivalence relation. This is defined as $E(x, y) \le \tilde{R}(x, y)$ which is interpreted crisply as if $x$ is equivalent to $y$, then $x$ is $\tilde{R}$-related to $y$. So, it follows that $E(x, x) \le \tilde{R}(x, x)$. But as in all the existing literature $E(x, x)$ is taken to be 1, $\tilde{R}(x, x)$ has to be 1. From the definitions of reflexivity given in (ii) to (vi), it is clear that this confinement is overcome in our context.

Other properties of a fuzzy binary relation are defined Chakraborty and Das (1983a, b) as follows:

• (vii) $\tilde{R}$ is antireflexive if for all $x \in A$, $\tilde{R}(x, x)$ = 0.

• (viii) $\tilde{R}$ is symmetric if $\tilde{R}(x, y)$ = $\tilde{R}(y, x)$ for all $x, y \in A$.

• (ix) $\tilde{R}$ is antisymmetric if for $x \ne y$, either $\tilde{R}(x, y)$ $\ne$ $\tilde{R}(y, x)$ or $\tilde{R}(x, y)$ = $\tilde{R}(y, x)$ = 0, for all $x, y \in A$.

• (x) $\tilde{R}$ is perfect antisymmetric if for $x \ne y$, $\tilde{R}(x, y)$ $> 0$ implies $\tilde{R}(y, x)$ $= 0$, for all $x, y \in A$.

It is to be noted here, that $\tilde{R}_{1}$, $\tilde{R}_{2}$ of example 2 are both antisymmetric relation; but while $\tilde{R}_{2}$ is a perfect antisymmetry $\tilde{R}_{1}$ is not.

Note 2

In a series of papers Bodenhofer (1999, 2002, 2008) has dealt with fuzzy ordering relation. He has proposed two generalizations of antisymmetry. One is $T$-antisymmetry, defined by $x \ne y$ implies $T(\tilde{R}(x, y), \tilde{R}(y, x))$ = 0, where $T$ is any t-norm. Bodenhofer in fact always assumes that the t-norm is left-continuous; but the definition also can be formulated for any t-norm. The other notion has been defined with respect to a fuzzy equivalence relation $E$ on a crisp set, a fuzzy relation $\tilde{R}$ on the same crisp set, and a t-norm $T$. The notion is called fuzzy antisymmetry or $T$-$E$-antisymmetry, and is defined by $T(\tilde{R}(x, y), \tilde{R}(y, x)) \le E(x, y)$. The difference between $T$-$E$ antisymmetry and our notion of fuzzy antisymmetry may be observed by taking $E$ as the crisp equality and $\tilde{R}$ as a fuzzy relation (this case must be within the intended range of the general definition). So, for $x \ne y$, $E(x, y)$ = 0 and hence according to Bodenhofer’s definition, $T(\tilde{R}(x, y), \tilde{R}(y, x))$ = 0. This implies, for many important t-norms such as minimum and product, at least one of the $\tilde{R}(x, y)$, $\tilde{R}(y, x)$ has to be 0. On the other hand, in the definition of antisymmetry given in (ix), for $x \ne y$, either $\tilde{R}(x, y)$ = $\tilde{R}(y, x)$ = 0 or $\tilde{R}(x, y) \ne \tilde{R}(y, x)$. Bodenhofer’s definition in the present context reduces to the definition of perfect antisymmetry, given in this paper. The same remark applies to his notion of $T$-antisymmetry.

Some other classifications of fuzzy relation according to Chakraborty and Das (1983a, b) are as follows:

• (xi) $\tilde{R}$ is transitive if $\tilde{R} \circ \tilde{R} \subseteq \tilde{R}$, where $\circ$ has its usual sup-min definition Klir and Yuan (1995).

• (xii) $\tilde{R}$ is a tolerance relation if it is reflexive and symmetric.

• (xiii) $\tilde{R}$ is a pre-order if it is reflexive and transitive.

• (xiv) $\tilde{R}$ is a partial order if it is reflexive, antisymmetric and transitive.

• (xv) $\tilde{R}$ is a strict order if it is antisymmetric and transitive.

• (xvi) $\tilde{R}$ is a non-strict order if it is reflexive, perfect antisymmetric and transitive.

• (xvii) $\tilde{R}$ is total order or linear order if for $x \ne y$, $\tilde{R}(x, y) \vee \tilde{R}(y, x) > 0$, for $x, y \in A$.

Depending on different definitions for reflexivity different types of equivalence relation (or Similitude relation), like weak similitude, absolute similitude, $\omega$-similitude, etc., and different types of tolerance relations are obtained. For example, an $\omega$-similitude is an $\omega$-reflexive, symmetric, and transitive relation. Let us consider the following example explaining different specifications of fuzzy relation mentioned above.

Example 3

Let $\tilde{A}$ be a fuzzy set over some universe $U$ such that $\tilde{A}(x_{1})$ = .7, $\tilde{A}(x_{2})$ = .8, $\tilde{A}(x_{3})$ = .9, and 0 is assigned to all the other elements of the universe. That is, $A$, the support of $\tilde{A}$, consists of $x_{1}, x_{2}, x_{3}$. Let $\tilde{R}$ be a binary fuzzy relation on $\tilde{A}$ such that

It is easy to check that $\tilde{R}$ is a $\omega$-reflexive, also weak reflexive, symmetric, transitive relation. So, $\tilde{R}$ is a $\omega$-similitude as well as weak similitude relation. It is also .7-similitude.

Some of the results from Chakraborty and Das (1983a, b); Chakraborty and Sarkar (1987); Chakraborty et al. (1985) are presented below.

Fuzzy equivalence

Theorem 1

If $\tilde{R}_{1}$, $\tilde{R}_{2}$ are $\omega$-similitudes on $\tilde{A}$ then $\tilde{R}_{1} \cup \tilde{R}_{2}$ is a $\omega$-similitude on $\tilde{A}$ iff $\tilde{R}_{1} \cup \tilde{R}_{2}$ = $\tilde{R}_{1} \circ \tilde{R}_{2}$.

Theorem 2

If $\tilde{R}_{1}$, $\tilde{R}_{2}$ are similitudes of one of the above-mentioned types on $\tilde{A}$, then $\tilde{R}_{1} \cap \tilde{R}_{2}$ is a similitude of the same type.

The importance of the classical idea of equivalence relation lies in the fact that an equivalence relation divides the universe of discourse into disjoint equivalent classes. Here, in the context of fuzzy relation over fuzzy sets, the point of departure is as follows.

Definition 5

(Chakraborty and Das 1983a) If $\tilde{R}$ is a similitude on $\tilde{A}$ then for each $x \in A$, two types of function, viz., $f_{x}^{\tilde{R}}$ and $F_{x}^{\tilde{R}}$ are defined in the following way.

$$\begin{aligned} f_{x}^{\tilde{R}} (y)&= \tilde{R}(x, y),\quad \mathrm{for\, all}\, y \in U~\mathrm{and} \\ F_{x}^{\tilde{R}} (y)&= \tilde{A}(y) \quad \quad \quad \mathrm{if}\, \tilde{R}(x, y) > 0\\&= 0 \quad \quad \quad \quad \quad \mathrm{if}\, \tilde{R}(x, y) = 0. \end{aligned}$$

$\tilde{f}_{x}^{\tilde{R}}$ is actually Zadeh’s similarity class (Zadeh 1971a). $\tilde{f}_{x}^{\tilde{R}}$ and $\tilde{F}_{x}^{\tilde{R}}$ coincide with the idea of equivalence class in case of ordinary sets. Classically, we know that $y$ is in the equivalence class of $x$, say $[x]_{R}$ if and only if $Rxy$ holds. Reflection of this idea could be found in the definition of $f_{x}^{\tilde{R}}$. On the other hand, $F_{x}^{\tilde{R}}$ generalizes the idea $y$ is in $[x]_{R}$, the equivalence class of $x$, if and only if $y$ is in $A$ such that $Rxy$ holds. Later, in Chakraborty and Das (1983b), the definition of $F_{x}^{\tilde{R}}$ has been replaced by the following one, which generalizes that the equivalence class $[x]_{R}$ of $x$ is the same as the union of all equivalence classes $[y]_{R}$ such that $y$ is in $[x]_{R}$.

$$\begin{aligned} \mathcal {F}_{x}^{\tilde{R}} = \cup _{y : f_{x}^{\tilde{R}}(y) > 0} f_{y}^{\tilde{R}}. \end{aligned}$$

In the case of $f_{x}^{\tilde{R}}$, $f_{x}^{\tilde{R}}(y)$ $> 0$ does not imply $f_{x}^{\tilde{R}}$ = $f_{y}^{\tilde{R}}$, while $\mathcal {F}_{x}^{\tilde{R}}(y)$ $> 0$ implies $\mathcal {F}_{x}^{\tilde{R}}$ = $\mathcal {F}_{y}^{\tilde{R}}$ and either $\mathcal {F}_{x}^{\tilde{R}}$ and $\mathcal {F}_{y}^{\tilde{R}}$ are equal or disjoint. But $\cup \mathcal {F}_{x}^{\tilde{R}}$ does not always coincide with the fuzzy set $\tilde{A}$. In Chakraborty et al. (1985) a definition for partition in fuzzy context is given. In ordinary context, these versions for equivalence class do not differ, but that they make difference in fuzzy context is clear from the above definitions and results.

Definition 6

(Chakraborty et al. 1985) A set of fuzzy subsets $\{\tilde{P}_{i}\}_{i \in I}$ of the fuzzy set $\tilde{A}$ forms a partition of $\tilde{A}$ if $\tilde{P}_{i} \ne \phi$ for all $i \in I$, $\tilde{P}_{i} \cap \tilde{P}_{j}$ = $\phi$ for $i \ne j$ and $\cup _{i \in I} \tilde{P}_{i}$ = $\tilde{A}$.

This definition of partition is different from that introduced in Bezdek and Harris (1978); Ovchinnikov and Riera (1982). It is clear that a partition $\{\tilde{P}_{i}\}_{i \in I}$ of the fuzzy set $\tilde{A}$ gives rise to a partition $\{P_{i}\}_{i \in I}$ of $A$, the support of $\tilde{A}$, where $x \in P_{i}$ iff $\tilde{P}_{i}(x) > 0$. A similitude relation corresponding to a partition can be constructed as shown in Theorem 3.

Theorem 3

(Chakraborty et al. 1985) If $\{\tilde{P}_{i}\}_{i \in I}$ is a partition of $\tilde{A}$ then there exists a fuzzy relation $\tilde{R}$ which is $\omega$-similitude such that $\tilde{R}(x, y) > 0$ iff $x, y \in P_{i}$ for some $i$.

Theorem 4

(Chakraborty et al. 1985) Let $\lambda$ = $\{\tilde{P}_{i}\}_{i \in I}$ of $\tilde{A}$ be a partition of $\tilde{A}$ in the sense mentioned in Definition 6. A fuzzy relation $\tilde{R}_{\lambda }$ is defined on $\tilde{A}$ as $\tilde{R}_{\lambda }$ $(x, x^{\prime })$ = $\tilde{P}_{i_{x}}(x) \wedge \tilde{P}_{i_{x}}(x^{\prime }) \wedge \tilde{P}(x^{\prime })$ where $\tilde{P}_{i_{x}}$ is that element of $\lambda$ for which $\tilde{P}_{i_{x}}(x) > 0$ and $\tilde{P}$ is a fuzzy subset of $\tilde{A}$ such that $\tilde{P} \cap \tilde{P}_{i} \ne \phi$ for all $i \in I$. Then $\tilde{R}_{\lambda } \circ \tilde{R}_{\lambda }^{-1}$ is a weakly reflexive and symmetric relation.

This $\tilde{R}_{\lambda }$ $(x, x^{\prime })$ actually generalizes a concept, viz., standard for an element and read as ‘the degree to which $x^{\prime }$ is a standard for $x$’.

Fuzzy tolerance

In the context of fuzzy tolerance relation the notions of pre-class, tolerance class have been introduced in Chakraborty et al. (1985). Given a tolerance relation $\tilde{T}$ on $\tilde{M}$, a pre-class, as defined in Chakraborty et al. (1985), is a fuzzy subset $\tilde{A} \subseteq \tilde{M}$ such that $\tilde{A} \times \tilde{A} \subseteq \tilde{T}$. A tolerance class is a pre-class $\tilde{A}$ such that there is no pre-class $\tilde{B}$ with respect to the tolerance relation $\tilde{T}$ such that $\tilde{A} \subset \tilde{B}$. Given any tolerance relation $\tilde{T}$ on $\tilde{M}$ with finite support $M$ the following algorithm gives the construction of a tolerance class.

Step (i) Take any $x_{1} \in M$, and assign $\tilde{A}(x_{1}) = \tilde{T}(x_{1}, x_{1})$.

Step (ii) Take $x_{2}$ other than $x_{1}$ from $M$. Assign $\tilde{A}(x_{2})$ = $\tilde{T}(x_{1}, x_{2})$ if any one of the following holds.

• $\tilde{A}(x_{1}) > \tilde{T}(x_{2}, x_{2}) > \tilde{T}(x_{1}, x_{2})$

• $\tilde{A}(x_{1})$ = $\tilde{T}(x_{2}, x_{2}) > \tilde{T}(x_{1}, x_{2})$

• $\tilde{T}(x_{2}, x_{2}) > \tilde{A}(x_{1}) > \tilde{T}(x_{1}, x_{2})$

Assign $\tilde{A}(x_{2})$ = $\tilde{T}(x_{2}, x_{2})$, otherwise. The method in which the value is assigned to $x_{2}$ relative to $x_{1}$ ensures the assignment of maximum admissible value (m.a.v) to $x_{2}$ with respect to $x_{1}$.

Step (iii) Take $x_{3}$ other than $x_{1}, x_{2}$ and consider the m.a.v for $x_{3}$ with respect to $x_{1}$ and $x_{2}$. Assign $\tilde{A}(x_{3})$ = (m.a.v due to $x_{1}$) $\wedge$ (m.a.v due to $x_{2}$).

Step (iv) Repeat step (iii) for any element $x_{4}$ other than $x_{1}, x_{2}, x_{3}$, and continue till the set $M$ gets exhausted.

Example 4

Let $\tilde{A}$ be a fuzzy set such that $\tilde{A}(x_{1})$ = .7, $\tilde{A}(x_{2})$ = .3, $\tilde{A}(x_{3})$ = .9, and zero for all other elements of the universe. That is, $A$ = $\{x_{1}, x_{2}, x_{3}\}$. A fuzzy relation $\tilde{R}$ on $\tilde{A}$ is given below.

Clearly this is a $\omega$-reflexive, symmetric fuzzy relation. $\tilde{R}$ is not transitive because $\tilde{R} \circ \tilde{R}(x_{1}, x_{2})$ = .3 $\nleq \tilde{R}(x_{1}, x_{2})$. Now there are six possible arrangement of the elements of $A$. Among these arrangements, following the above algorithm for constructing tolerance classes the following different tolerance classes are obtained.

Arrangements of $A$	Tolerance classes
$\{x_{1}, x_{2}, x_{3}\}$ $\{x_{1}, x_{3}, x_{2}\}$	$\tilde{T}_{1}(x_{1})$ = .7, $\tilde{T}_{1}(x_{2})$ = .1, $\tilde{T}_{1}(x_{3})$ = .5
$\{x_{2}, x_{3}, x_{1}\}$ $\{x_{3}, x_{2}, x_{1}\}$	$\tilde{T}_{2}(x_{1})$ = .1, $\tilde{T}_{2}(x_{2})$ = .3, $\tilde{T}_{2}(x_{3})$ = .9
$\{x_{2}, x_{1}, x_{3}\}$	$\tilde{T}_{3}(x_{1})$ = .1, $\tilde{T}_{3}(x_{2})$ = .3, $\tilde{T}_{3}(x_{3})$ = .5
$\{x_{3}, x_{1}, x_{2}\}$	$\tilde{T}_{4}(x_{1})$ = .5, $\tilde{T}_{4}(x_{2})$ = .1, $\tilde{T}_{4}(x_{3})$ = .9

One can prove also that all the tolerance classes in $\tilde{M}$ relative to $\tilde{T}$ may be obtained by the above construction (Das et al. 1998). The tolerance classes constructed from each $x \in A$, considering $x$ as the first element of the sequence, all reduce to $f_{x}^{\tilde{R}}$ (see Definition 5) if and only if $\tilde{R}$ is an equivalence relation. Subsequently, fuzzy tolerance topology has been developed in Chakraborty and Ahsanullah (1992).

Fuzzy order

The notion of ordering plays an important role in the study of fuzzy relation over fuzzy sets. It is already discussed in Notes 1, 2 that the notion of reflexivity and antisymmetry according to Bodenhofer (2002) are quite different from our notions of reflexivity and antisymmetry. So, it is quite expected that the notion of ordering in our context is different from his notion of ordering.

The following two propositions relate to the synthesis of perfect antisymmetric relations.

Proposition 1

(Chakraborty and Sarkar 1987) If $\tilde{R}_{1}$, $\tilde{R}_{2}$ are two perfect antisymmetric relations, then $\tilde{R}_{1} \cup \tilde{R}_{2}$ is also a perfect antisymmetric relation iff $\tilde{R}_{i}$ $(x, y) > 0$ implies $\tilde{R}_{j}$ $(y, x)$ = $0$, $i, j$ = $1, 2$.

Proposition 2

(Chakraborty and Sarkar 1987) Let $\tilde{R}_{1}$, $\tilde{R}_{2}$ be two perfect antisymmetric relations. Then $\tilde{R}_{1} \cup \tilde{R}_{2}$ is symmetric iff for all $x, y \in A$, $\tilde{R}_{1}$ $(x, y)$ = $\tilde{R}_{2}$ $(y, x)$.

One of the significances of perfect antisymmetric relation lies in the idea of analyzing a fuzzy relation into two perfect antisymmetric components. In this regard, max–min resolution and almost cyclic resolution are of importance. The following few definitions will give an idea of the above-mentioned notions.

Definition 7

(Chakraborty and Sarkar 1987) Let $\tilde{R}$ be a fuzzy relation on $\tilde{A}$. Define $\tilde{R}_\mathrm{max}$, $\tilde{R}_\mathrm{min}$ on $\tilde{A}$ in the following way.

$\tilde{R}_\mathrm{max}(x, x)$ = $\tilde{R}_\mathrm{min}(x, x)$ = $\tilde{R}(x, x)$,   for all $x \in A$

For $x \ne x^{\prime }$ of $A$,   if $\tilde{R}(x, x^{\prime }) \ge \tilde{R}(x^{\prime }, x)$, then

$\tilde{R}_\mathrm{max}(x, x^{\prime })$ = $\tilde{R}(x, x^{\prime })$, $\tilde{R}_\mathrm{max}(x^{\prime }, x)$ = 0,

$\tilde{R}_\mathrm{min}(x, x^{\prime })$ = 0, and $\tilde{R}_\mathrm{min}(x^{\prime }, x)$ = $\tilde{R}(x^{\prime }, x)$.

It follows immediately that $\tilde{R}_\mathrm{max}$, $\tilde{R}_\mathrm{min}$ are perfect antisymmetric, $\tilde{R}_\mathrm{max} \cup \tilde{R}_\mathrm{min}$ = $\tilde{R}$, and $\tilde{R}_\mathrm{max} \cap \tilde{R}_\mathrm{min}$ is a diagonal relation, where non-diagonal entries are always zero. Also, if $\tilde{R}$ is reflexive, then $\tilde{R}_\mathrm{max}, \tilde{R}_\mathrm{min}$ are so. If $\tilde{R}$ is a transitive relation, the components preserve the same in some cases.

Theorem 5

(Chakraborty and Sarkar 1987) Let $\tilde{R}$ be an antisymmetric and transitive relation on $\tilde{A}$ and $\tilde{R}_{1}$, $\tilde{R}_{2}$ be the components of $\tilde{R}$ obtained by max–min resolution. Then $\tilde{R}_{1}$, $\tilde{R}_{2}$ are also transitive.

In case of almost cyclic resolution, preservation of transitivity into the components of $\tilde{R}$ comes directly from the construction of its components. In order to define almost cyclic resolution let us define some intermediate concepts below.

Definition 8

(Chakraborty and Sarkar 1987)

• (i) For a set $\{a, b, c\}$ of three elements each of the following sets of ordered pairs is called an almost cyclic arrangement. $\{ab, bc, ac\}$, $\{ba, cb, ca\}$, $\{ba, ac, bc\}$, $\{ab, ca, cb\}$, $\{ac, cb, ab\}$, $\{ca, bc, ba\}$.

• (ii) If the above sets can be paired off in such a way that elements of one set are just in reverse order of that of the others, then sets belonging to such pairs are called complementary arrangements.

• (iii) Two almost cyclic arrangements $\mathcal {P}$ and $\mathcal {Q}$ of $\{a, b, c\}$ and $\{a^{\prime }, b^{\prime }, c^{\prime }\}$, respectively, are said to be in harmony if and only if there do not exist elements $x$ and $y$ such that $xy \in \mathcal {P}$, and $yx \in \mathcal {Q}$, and vice versa.

• (iv) Let $S$ be any finite set. It is possible to assign almost cyclic arrangements to all 3-element subsets of $S$ so that any two such arrangements are in harmony. (That this would be possible is intuitively clear, but a formal proof could be given by induction.) A totality of all such arrangements on all 3-elements subsets of $S$ constitute an almost cyclic harmonic arrangements of the set $S$. It is clear that the complementary arrangements of two almost cyclic arrangements in harmony of 3-elements sets are also in harmony. From this it follows that the complementary arrangements of an almost cyclic harmonic arrangements of a finite set is also almost cyclic harmonic arrangements.

• (v) Let $\tilde{R}$ be a fuzzy relation on $\tilde{A}$ such that $A$ is finite. Let $\mathcal {A}$, and $\mathcal {A}^{\prime }$ be an almost cyclic harmonic arrangements and its complementary arrangement of $A$, respectively. We define relations $\tilde{R}_{1}$ and $\tilde{R}_{2}$ as follows: $\tilde{R}_{1}(x, x^{\prime })$ = $\tilde{R}(x, x^{\prime })$ if $(x, x^{\prime }) \in \mathcal {A}$,                 = 0,    if $(x, x^{\prime }) \in \mathcal {A}^{\prime }$ $\tilde{R}_{2}(x, x^{\prime }) = \tilde{R}(x, x^{\prime })$   if $(x, x^{\prime }) \in \mathcal {A}^{\prime }$,                 $= 0,$   if $(x, x^{\prime }) \in \mathcal {A}$., and $\tilde{R}_{1}(x, x)$ = $\tilde{R}_{2}(x, x)$ = $\tilde{R}(x, x)$ for all $x \in A$. This resolution of $\tilde{R}$ into $\tilde{R}_{1}$, $\tilde{R}_{2}$ is called almost cyclic resolution.

It is easy to observe that $\tilde{R}_{1}$, $\tilde{R}_{2}$ are perfect antisymmetric relations, and $\tilde{R}_{1} \cup \tilde{R}_{2}$ = $\tilde{R}$, $\tilde{R}_{1} \cap \tilde{R}_{2}$ is a diagonal relation, reflexivity of $\tilde{R}$ implies reflexivity (of the same type) of its components and transitivity of $\tilde{R}$ is preserved into its components straight away from the construction of the components.

The importance of these two methods lies in the fact that with the help of such resolutions Szpilrajn–Marczewski-like extension theorems may be proved for $\tilde{R}$. Using almost cyclic resolution the linearization of $\tilde{R}$, in the sense mentioned in the following theorem, can be obtained.

Theorem 6

(Chakraborty and Sarkar 1987) Let $\tilde{R}$ be a partial order relation on $\tilde{A}$ where $A$ is finite. Then there exists a linear order extension of $\tilde{R}$ in $\tilde{A}$.

In case of max–min resolution, the above claim of linearization may not work. However, following Zadeh’s method of extension (Zadeh 1971a), a fuzzy relation $\tilde{R}^{\prime }$ can be so chosen that $\tilde{R} \cup \tilde{R}^{\prime }$ becomes a linear extension of $\tilde{R}$ provided $\tilde{R}_{2} \cup \tilde{R}^{\prime }$ is transitive, where $\tilde{R}_{2}$ is obtained by max–min resolution of $\tilde{R}$ (see Definition 7). In this case also the support of the base fuzzy set of $\tilde{R}$ has been considered finite. In Chakraborty and Sarkar (1987) a detailed study on pre-order, partial order, linear order, strict as well as non-strict order has been made. A scheme for linearizing a partial order with an infinite base set is also given in Chakraborty et al. (1985).

A few constants, viz., order of antisymmetry, measure of antisymmetry are introduced in Chakraborty and Sarkar (1987). These constants have interesting and practical interpretation in the field of networking of any kind of flow.

It will be observed that defining all the notions on a fuzzy set, apart from the foundational obligation, generalizes many of the existing notions non-trivially and produces interesting results. Continuing the same direction as above Chakraborty and Ahsanullah studied fuzzy topology on fuzzy sets (Chakraborty and Ahsanullah 1992), and Chakraborty and Banerjee developed fuzzy algebraic structures on fuzzy sets in categorical framework (Chakraborty and Banerjee 1991).

3 Homorelation, correlation, cohomorelation

In mathematics relation, preserving mappings from one relational structure to another is a very important notion. These are called homomorphisms. Our intension is to define similar concepts in fuzzy context. We want to propose concepts of fuzzy relation-preserving fuzzy functions from one fuzzy relational structure to another where a fuzzy relational structure is a pair $(\tilde{A}, \rho )$ consisting of $\tilde{A}$, a fuzzy set and $\rho$, a fuzzy relation on $\tilde{A}$. The actual notions will be presented in Definition 15. But as a step we first define more generalized notions of homorelation, correlation, cohomorelation, etc.

3.1 Homorelation, correlation, cohomorelation

Definition 9

A fuzzy relation $\tilde{R}$ from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$ is called

• (i) a homorelation iff $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) \ge \rho (x, x^{\prime })$ implies $\tau (y, y^{\prime }) \ge \rho (x, x^{\prime })$,

• (ii) a correlation iff $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) \ge \rho (x, x^{\prime })$ implies $\tau (y, y^{\prime }) \le \rho (x, x^{\prime })$ and

• (iii) a cohomorelation iff $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) \ge \rho (x, x^{\prime })$ implies $\tau (y, y^{\prime }) = \rho (x, x^{\prime })$, for all $x, x^{\prime } \in A$ and $y, y^{\prime } \in B$.

Proposition 3

If $\tilde{R}$ is a homorelation (correlation / cohomorelation) from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$ and $\tilde{R}$ $^{\prime }$ $\subseteq \tilde{R}$ then $\tilde{R}$ $^{\prime }$ is also a homorelation (correlation / cohomorelation).

Corollary 1

For any fuzzy relation $\tilde{R}$ $^{\prime }$ from $\tilde{A}$ to $\tilde{B}$, $\tilde{R}$ $^{\prime }$ $\cap$ $\tilde{R}$ is a homorelation (correlation / cohomorelation) if $\tilde{R}$ is so.

Theorem 7

Let $\tilde{R}_{1}$ be a correlation from $(\tilde{A}, \rho )$ to $(\tilde{B}, \sigma )$ and $\tilde{R}_{2}$ be a correlation from $(\tilde{B}, \sigma )$ to $(\tilde{C}, \tau )$. Then $\tilde{R}_{1} \circ \tilde{R}_{2}$ is a correlation from $(\tilde{A}, \rho )$ to $(\tilde{C}, \tau )$, provided in the definition of $\tilde{R}_{1} \circ \tilde{R}_{2}$ the supremum is attained.

The following observations indicate that the notions of homorelation and correlation are very close to the prevalent notion of fuzzy functions that have come up in past two decades (Demirci and Recasens 2004; Gilles 1984; Klawonn 2000; Perfilieva 2011).

In the existing literature of fuzzy function, a fuzzy function, usually defined from $(X, \rho )$ to $(Y, \tau )$ where $X, Y$ are ordinary sets, and $\rho$, $\tau$ are fuzzy equivalence relations, is a fuzzy relation $\tilde{R}$ over $X \times Y$ such that the following conditions hold.

• (I) $\tilde{R}(x, y) \wedge \rho (x, x^{\prime }) \le \tilde{R}(x^{\prime }, y)$,

• (II) $\tilde{R}(x, y) \wedge \tau (y, y^{\prime }) \le \tilde{R}(x, y^{\prime })$ and

• (III) $\tilde{R}(x, y) \wedge \tilde{R}(x, y^{\prime }) \le \tau (y, y^{\prime })$.

Generally, instead of $\wedge$, a t-norm is taken. In Perfilieva (2011), Perfilieva defined a notion of perfect function $\tilde{R}$ which satisfies (I), (II), (III) (using t-norm for $\wedge$) and the condition that for all $x \in X$, there exists $y \in Y$ such that $\tilde{R}(x, y)$ = 1. Also, according to Perfilieva (2011) $\tilde{R}$ is said to be (strong) surjective if it also satisfies that for all $y \in Y$, there exists $x \in X$ such that $\tilde{R}(x, y)$ = 1. According to Klawonn Klawonn (2000) $\tilde{R}$ is called a fuzzy function if it satisfies (I), (II), (III) (using t-norm for $\wedge$), along with $\sup _{x \in X}\tilde{R}(x, y)$ = 1 for all $y \in Y$. In Demirci and Recasens (2004), they defined several notions like partial function, fuzzy function, strong fuzzy function, etc. According to Demirci and Recasens (2004), a fuzzy relation $\tilde{R}$ from $(X, \rho )$ to $(Y, \tau )$ is called a strong fuzzy function if the following conditions hold.

(f1) For each $x$, there is $y$ such that $\tilde{R}(x, y)$ = 1, and

(f2) $T(\tilde{R}(x, y), \tilde{R}(x^{\prime }, y^{\prime }), \rho (x, x^{\prime })) \le \tau (y, y^{\prime })$, where $T$ is a t-norm. $\tilde{R}$ is called a perfect fuzzy function (Demirci and Recasens 2004) if $\tilde{R}$ satisfies (I), (II), (III) (taking a t-norm $T$ for $\wedge$), and (f1).

Observation 1 If $\tilde{R}$ is a strong fuzzy function in the sense of Demirci et al. with minimum as t-norm, then it can be shown that $\tilde{R}$ turns out to be a homorelation restricted to the context of ordinary sets.

Observation 2

• (i) Let us assume that $\rho (x, x^{\prime }) \le \tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime })$, the left hand side of the condition of homorelation/correlati-on holds for $x, x^{\prime }, y$. Then $\rho (x, x^{\prime }) \le \tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y)$ implies $\rho (x, x^{\prime }) \le \tilde{R}(x, y), \tilde{R}(x^{\prime }, y)$. So, $\rho (x, x^{\prime }) \wedge \tilde{R}(x, y)$ = $\rho (x, x^{\prime }) \le \tilde{R}(x^{\prime }, y)$. (Condition I)

• (ii) Let $\tilde{R}$ from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$ be a correlation, and the left hand side of the condition of correlation hold for $x, y, y^{\prime }$, i.e., $\rho (x, x) \le \tilde{R}(x, y) \wedge \tilde{R}(x, y^{\prime })$. So, $\tau (y, y^{\prime }) \le \rho (x, x) \le \tilde{R}(x, y) \wedge \tilde{R}(x, y^{\prime })$. Hence $\tau (y, y^{\prime }) \le \tilde{R}(x, y), \tilde{R}(x, y^{\prime })$. Therefore, $\tilde{R}(x, y) \wedge \tau (y, y^{\prime })$ = $\tau (y, y^{\prime }) \le \tilde{R}(x, y^{\prime })$. (Condition II)

• (iii) Let $\tilde{R}$ be a homorelation, and $\rho$ be a $\omega$-reflexive relation. Let us assume the left hand side of the condition of homorelation hold for $x, y, y^{\prime }$. That is, $\rho (x, x) \le \tilde{R}(x, y) \wedge \tilde{R}(x, y^{\prime })$. Then $\rho (x, x) \le \tau (y, y^{\prime })$. Also, $\tilde{R}(x, y), \tilde{R}(x, y^{\prime }) \le \tilde{A}(x)$ = $\rho (x, x)$. So, $\rho (x, x)$ = $\tilde{R}(x, y) \wedge \tilde{R}(x, y^{\prime })$ and hence we have $\tilde{R}(x, y) \wedge \tilde{R}(x, y^{\prime }) \le \tau (y, y^{\prime })$. (Condition III)

In Sect. 4 we shall discuss fuzzy homomorphism, comorphism, etc. Before that, let us present some results on the $\gamma$-cuts of the above notions.

3.2 $\gamma$-homorelation, $\gamma$-correlation, $\gamma$-cohomorelation

Definition 10

A fuzzy relation $\tilde{R}$ from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$, where $\tilde{R} \subseteq \tilde{A} \times \tilde{B}$, is said to be a

• (i) $\gamma$-homorelation iff $\tilde{R}(x, y)$ $\wedge$ $\tilde{R}(x^{\prime }, y^{\prime }) > \gamma$ implies $\tau (y, y^{\prime }) \ge \rho (x, x^{\prime })$,

• (ii) $\gamma$-correlation iff $\tilde{R}(x, y)$ $\wedge$ $\tilde{R}(x^{\prime }, y^{\prime }) > \gamma$ implies $\tau (y, y^{\prime }) \le \rho (x, x^{\prime })$ and

• (iii) $\gamma$-cohomorelation iff $\tilde{R}(x, y)$ $\wedge$ $\tilde{R}(x^{\prime }, y^{\prime }) > \gamma$ implies $\tau (y, y^{\prime }) = \rho (x, x^{\prime })$, for any $x, x^{\prime } \in A$, $y, y^{\prime } \in B$ and $\gamma \in [0, 1)$.

Restricting the above definitions in two-valued context we can see, for $\gamma$ = 0, the respective definitions of homorelation, correlation, and cohomorelation specify the different relational-structure preserving conditions between the structures $(\tilde{A}, \rho )$ and $(\tilde{B}, \tau )$.

Proposition 4

If $\tilde{R}$ is a $\gamma$-homorelation ($\gamma$-correlation/ $\gamma$-cohomorelation) from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$ then $\tilde{R}$ is a $\gamma ^{\prime }$-homorelation ($\gamma ^{\prime }$-correlation / $\gamma ^{\prime }$-cohomorelation) from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$ for $\gamma ^{\prime } > \gamma$.

Proposition 5

If $\tilde{R}$ is a $\gamma _{1}$-homorelation and $\gamma _{2}$-correlation then $\tilde{R}$ is a $\gamma _{1} \vee \gamma _{2}$-cohomorelation.

Proposition 6

If $\tilde{R}$ is a $\gamma$-homorelation ($\gamma$-correlation / $\gamma$-cohomorelation) from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$ and $\tilde{R}$ $^{\prime } \subseteq \tilde{R}$ then $\tilde{R}$ $^{\prime }$ is a $\gamma$-homorelation ($\gamma$-correlation / $\gamma$-cohomorelation) from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$.

Theorem 8

Let $\tilde{R}_{1}$ and $\tilde{R}_{2}$ be, respectively, $\gamma _{1}$-homorelation and $\gamma _{2}$-homorelation (correlation/cohomorelation) from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$. Then

• (i) $\tilde{R}_{1} \cap \tilde{R}_{2}$ is a $\gamma _{1} \wedge \gamma _{2}$-homorelation (correlation/cohomorelation) and

• (ii) $\tilde{R}_{1} \cup \tilde{R}_{2}$ is a $\gamma$-homorelation (correlation/cohomorelation) for $\gamma \ge \gamma _{1} \vee \gamma _{2}$.

Theorem 9

Let $\tilde{R}_{1}$ be a $\gamma _{1}$-homorelation (correlation/cohomorelation) from $(\tilde{A}, \rho )$ to $(\tilde{B}, \sigma )$ and $\tilde{R}_{2}$ be a $\gamma _{2}$-homorelation (correlation/cohomorelation) from $(\tilde{B}, \sigma )$ to $(\tilde{C}, \tau )$. Then $\tilde{R}_{1} \circ \tilde{R}_{2}$ is a $\gamma _{1} \vee \gamma _{2}$-homorelation (correlation/cohomorelation) from $(\tilde{A}, \rho )$ to $(\tilde{C}, \tau )$.

Proof

Let us prove $\tilde{R}_{1} \circ \tilde{R}_{2}$ is a $\gamma _{1} \vee \gamma _{2}$-homorelation from $(\tilde{A}, \rho )$ to $(\tilde{C}, \tau )$. If not, then there exist $x, x^{\prime } \in A$ and $z, z^{\prime } \in C$ such that $\tilde{R}_{1}\circ \tilde{R}_{2}$ $(x, z) \wedge \tilde{R}_{1}\circ \tilde{R}_{2}$ $(x^{\prime }, z^{\prime })$ $>$ $\gamma _{1} \vee \gamma _{2}$ but $\tau (z, z^{\prime })$ $<$ $\rho (x, x^{\prime })$. That is, $\sup _{y \in B}$ $[\tilde{R}_{1}(x, y) \wedge \tilde{R}_{2}(y, z)]$ $\wedge$ $\sup _{y \in B}$ $[\tilde{R}_{1}(x^{\prime }, y) \wedge \tilde{R}_{2}(y, z^{\prime })]$ $>$ $\gamma _{1} \vee \gamma _{2}$. Then for $x, x^{\prime } \in A$, $z, z^{\prime } \in C$ there exist some $y, y^{\prime } \in B$ such that $\tilde{R}_{1}(x, y) \wedge \tilde{R}_{2}(y, z)$ $>$ $\gamma _{1} \vee \gamma _{2}$ and $\tilde{R}_{1}(x^{\prime }, y^{\prime }) \wedge \tilde{R}_{2}(y^{\prime }, z^{\prime })$ $>$ $\gamma _{1} \vee \gamma _{2}$. Hence, $\tilde{R}_{1}(x, y)$, $\tilde{R}_{2}(y, z)$ $> \gamma _{1}, \gamma _{2}$ and $\tilde{R}_{1}(x^{\prime }, y^{\prime })$, $\tilde{R}_{2}(y^{\prime }, z^{\prime })$ $> \gamma _{1}, \gamma _{2}$. Therefore, $\tilde{R}_{1}(x, y) \wedge \tilde{R}_{1}(x^{\prime }, y^{\prime }) > \gamma _{1}$ implies $\sigma (y, y^{\prime }) \ge \rho (x, x^{\prime })$ and $\tilde{R}_{2}(y, z) \wedge \tilde{R}_{2}(y^{\prime }, z^{\prime }) > \gamma _{2}$ implies $\tau (z, z^{\prime }) \ge \sigma (y, y^{\prime })$. Hence $\tau (z, z^{\prime }) \ge \rho (x, x^{\prime })$. This contradicts the assumption. Hence $\tilde{R}_{1} \circ \tilde{R}_{2}$ is a $\gamma _{1} \vee \gamma _{2}$-homorelation. $\square$

Definition 11

$\tilde{R}$ is a fuzzy relation from $\tilde{A}$ to $\tilde{B}$. $\tilde{R}^{-1}$ is a fuzzy relation from $\tilde{B}$ to $\tilde{A}$ where $\tilde{R}^{-1}$ is defined as $\tilde{R}$ $(x, y)$ = $\tilde{R}^{-1}$ $(y, x)$. An element $x \in A$ is $\gamma$-compatible with another element $x^{\prime } \in A$ iff there exist sequences $\{x_{1}, x_{2}, \ldots , x_{n}\}$ and $\{y_{1}, y_{2}, \ldots , y_{n+1}\}$ on $A$ and $B$, respectively, such that $\tilde{R}(x, y_{1})$ $\wedge$ $\tilde{R}^{-1}(y_{1}, x_{1})$ $\wedge$ $\tilde{R}(x_{1}, y_{2})$ $\wedge$...$\wedge$ $\tilde{R}(x_{n}, y_{n+1})$ $\wedge$ $\tilde{R}^{-1}(y_{n+1}, x^{\prime })$ $> \gamma$.

Lemma 1

If $x \in A$ is $\gamma$-compatible with $x^{\prime } \in A$ then $x^{\prime }$ is $\gamma$-compatible with $x$.

Lemma 2

If $x_{i} \in A$ is $\gamma$-compatible with $x_{j} \in A$ and $x_{j}$ is $\gamma$-compatible with $x_{k} \in A$, then $x_{i}$ is $\gamma$-compatible with $x_{k}$.

Definition 12

For a fuzzy relation $\tilde{R}$ from $\tilde{A}$ to $\tilde{B}$, a relation $C_{A}^{\gamma }$ is defined on $A \times A$ in the following way.

$(x_{i}, x_{k})$ $C_{A}^{\gamma }$ $(x_{j}, x_{l})$ iff $x_{i}$ is $\gamma$-compatible with $x_{j}$ and $x_{k}$ is $\gamma$-compatible with $x_{l}$. The pairs $(x_{i}, x_{k})$ and $(x_{j}, x_{l})$ are called $\gamma$-compatible ordered pairs.

Theorem 10

If $\tilde{R}$ is a fuzzy relation from $\tilde{A}$ to $\tilde{B}$ such that for all $x \in A$ there is $y \in B$ satisfying $\tilde{R}(x, y) > \gamma$ then $C_{A}^{\gamma }$ on $A \times A$ is an equivalence relation.

Proof

That $C_{A}^{\gamma }$ is a symmetric and transitive relation is quite clear from Lemmas 1 and 2 Let us prove that $C_{A}^{\gamma }$ is reflexive. Let $(x_{i}, x_{j}) \in A \times A$. As for $x_{i} \in A$ there is $y_{i} \in B$ such that $\tilde{R}(x_{i}, y_{i}) > \gamma$ we have $\tilde{R}(x_{i}, y_{i}) \wedge \tilde{R}^{-1}(y_{i}, x_{i})> \gamma$. (since $\tilde{R}(x, y)$ = $\tilde{R}^{-1}(y, x)$). Hence $x_{i}$ is $\gamma$-compatible with $x_{i}$. And hence $(x_{i}, x_{j}) C_{A}^{\gamma } (x_{i}, x_{j})$. $\square$

Definition 13

Let $\tilde{R}$ be a fuzzy relation from $\tilde{A}$ to $\tilde{B}$. Then $R^{\gamma }$ is a relation from $A \times A$ to $B \times B$, defined by $(x , x^{\prime })$ $R^{\gamma }$ $(y , y^{\prime })$ iff $\tilde{R}(x, y)$ $\wedge$ $\tilde{R}(x^{\prime }, y^{\prime }) > \gamma$, for all $x, x^{\prime } \in A$ and $y, y^{\prime } \in B$.

Theorem 11

Let $\tilde{R}$ be a fuzzy relation from $\tilde{A}$ to $\tilde{B}$ such that (i) for all $x \in A$, there is $y \in B$ satisfying $\tilde{R}(x, y) > \gamma$ and (ii) for all $y \in B$, there is $x \in A$ satisfying $\tilde{R}(x, y) > \gamma$. Then $\gamma$-compatible ordered pairs of $A \times A$ are $R^{\gamma }$ related only to $\gamma$-compatible ordered pairs of $B \times B$.

Definition 14

Let $\tilde{R}$ be a fuzzy relation from $\tilde{A}$ to $\tilde{B}$, $\rho$ be a fuzzy relation on $\tilde{A}$ and $\gamma$ be any element from $[0, 1)$. Then $\rho ^{\gamma }$ on $\tilde{B}$ is defined as follows. $\rho ^{\gamma }(y, y^{\prime })$ = $\sup _{x, x^{\prime } \in A}$ $\{\rho (x, x^{\prime }) :\tilde{R}(x, y)\wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma \}$, for any $y, y^{\prime } \in B$.

Lemma 3

$\tilde{R}$ is a $\gamma$-homorelation from $(\tilde{A}, \rho )$ to $(\tilde{B}, \rho ^{\gamma })$.

Lemma 4

$\tilde{R}$ is a $\gamma$-homorelation from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$ if and only if $\rho ^{\gamma } \subseteq \tau$.

Theorem 12

Let $\tilde{R}$ be a fuzzy relation from $\tilde{A}$ to $\tilde{B}$ and $\rho$ be a fuzzy relation on $\tilde{A}$ where $A$ and $B$ are finite. Then for each $\gamma \in [0, 1)$ there exist relations $\rho _{m}$ and $\rho _{m}^{\gamma }$ on $\tilde{A}$ and $\tilde{B}$, respectively, such that

• (i) $\tilde{R}$ is a $\gamma$-cohomorelation from $(\tilde{A}, \rho _{m})$ to $(\tilde{B}, \rho _{m}^{\gamma })$,

• (ii) $\rho \subseteq \rho _{m}$ and

• (iii) if $\tilde{R}$ is a $\gamma$-cohomorelation from $(\tilde{A}, \rho _{m})$ to $(\tilde{B}, \tau )$ then $\rho _{m}^{\gamma } \subseteq \tau$.

Proof

Let us first construct fuzzy relations $\rho _{0}$, $\rho _{1}$, $\rho _{2} \ldots$ on $\tilde{A}$ and $\rho _{0}^{\gamma }$, $\rho _{1}^{\gamma }$, $\rho _{2}^{\gamma }$ ...on $\tilde{B}$ in the following way.

$\rho _{0}$ = $\rho$,

$\rho _{0}^{\gamma }$ $(y,y^{\prime })$ = $\sup _{x, x^{\prime } \in A}$ $\{\rho _{0}(x,x^{\prime }): \tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma \}$ for any $y, y^{\prime } \in B$,

$\rho _{1}$ $(x, x^{\prime })$ = $\sup _{y, y^{\prime } \in B}$ $\{\rho _{0}^{\gamma }(y, y^{\prime }): \tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma \}$ for any $x, x^{\prime } \in A$,

$\rho _{i}$ $(x, x^{\prime })$ = $\sup _{y, y^{\prime } \in B}\{\rho _{i-1}^{\gamma }(y, y^{\prime }):\tilde{R}(x, y)\wedge \tilde{R}(x^{\prime }, y^{\prime })> \gamma \}$ for any $x, x^{\prime } \in A$,

$\rho _{i}^{\gamma }$ $(y, y^{\prime })$ = $\sup _{x, x^{\prime } \in A}$ $\{\rho _{i}(x, x^{\prime }): \tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma \}$ for any $y, y^{\prime } \in B$,

From Lemma 3, it is clear that $\tilde{R}$ is a $\gamma$-homorelation from $(\tilde{A}, \rho _{i})$ to $(\tilde{B}, \rho _{i}^{\gamma })$. Let us first prove that $\tilde{R}$ is a $\gamma$-correlation from $(\tilde{A}, \rho _{i})$ to $(\tilde{B}, \rho _{i-1}^{\gamma })$. By definition of $\rho _{i}$ and $\rho _{i-1}^{\gamma }$, we have for any $x, x^{\prime } \in A$, $\rho _{i}$ $(x, x^{\prime })$ $\ge$ $\rho _{i-1}^{\gamma }$ $(y, y^{\prime })$ if for $y, y^{\prime } \in B$, $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma$.

Hence $\tilde{R}$ is a $\gamma$-correlation from $(\tilde{A}, \rho _{i})$ to $(\tilde{B}, \rho _{i-1}^{\gamma })$.

Now for $x, x^{\prime } \in A$ and $y, y^{\prime } \in B$, satisfying

$\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma$ the following holds for any $i$.

$\rho _{i}^{\gamma }$ $(y, y^{\prime })$ $\ge$ $\rho _{i}$ $(x, x^{\prime })$ $\ge$ $\rho _{i-1}^{\gamma }$ $(y, y^{\prime })$ $\ge \cdots \ge$ $\rho _{1}^{\gamma }$ $(y, y^{\prime })$ $\ge$ $\rho _{1}$ $(x, x^{\prime })$ $\ge$ $\rho _{0}^{\gamma }$ $(y, y^{\prime })$ $\ge$

$\rho _{0}(x, x^{\prime }) = \rho (x, x^{\prime }) \ldots (1)$

Now as $A$ and $B$ are finite and for any $i$,

$\rho _{i}$ $(x, x^{\prime })$ $\le \tilde{A}(x) \wedge \tilde{A}(x^{\prime })$ and $\rho _{i}^{\gamma }$ $(y, y^{\prime })$ $\le \tilde{B}(y) \wedge \tilde{B}(y^{\prime })$, after a finite number, say $m$, $\rho _{m}$ $(x, x^{\prime })$ = $\rho _{m+1}$ $(x, x^{\prime })$. Hence $\tilde{R}$ is a $\gamma$-homorelation from $(\tilde{A}, \rho _{m})$ to $(\tilde{B}, \rho _{m}^{\gamma })$ as well as $\tilde{R}$ is a $\gamma$-correlation from $(\tilde{A}, \rho _{m})$ to $(\tilde{B}, \rho _{m}^{\gamma })$. That is, $\tilde{R}$ is a $\gamma$-cohomorelation from $(\tilde{A}, \rho _{m})$ to $(\tilde{B}, \rho _{m}^{\gamma })$. $\rho \subseteq \rho _{m}$ is clear from (1) and

by Lemma 4, if $\tilde{R}$ is a $\gamma$-homorelation from $(\tilde{A}, \rho _{m})$ to $(\tilde{B}, \tau )$ then $\rho _{m}^{\gamma } \subseteq \tau$. $\square$

Theorem 13

• (i) If for all $\gamma$, $\tilde{R}$ is a $\gamma$-homorelation from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$, then $\tilde{R}$ is a homorelation from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$.

• (ii) Let $\tilde{R}$ be a homorelation from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$, and $\gamma$ = $\sup _{x,x^{\prime },y,y^{\prime }}\{\rho (x, x^{\prime }):\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) \ge \rho (x, x^{\prime })\}$ $\ne$ 1. Then $\tilde{R}$ is a $\gamma$-homorelation from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$.

Proof

• (i) Let for any $\gamma \in [0, 1)$, $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma$ implies $\tau (y, y^{\prime }) \ge \rho (x, x^{\prime })$ hold for any $x, x^{\prime } \in A$, $y, y^{\prime } \in B$. Let $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) \ge \rho (x, x^{\prime })$ hold for any arbitrarily fixed $x, x^{\prime } \in A$, $y, y^{\prime } \in B$. Now two cases arise: $\rho (x, x^{\prime })$ = 0 and $\rho (x, x^{\prime }) \ne$ 0. If $\rho (x, x^{\prime })$ = 0, the result holds by default. If $\rho (x, x^{\prime }) \ne$ 0, let us choose an element $\gamma$ such that 0 $\le \gamma < \rho (x, x^{\prime })$. Hence $\gamma <$ 1 as $\rho (x, x^{\prime }) \le$ 1, and $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma$. Then as $\tilde{R}$ is a $\gamma$-homorelation $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma$ implies $\tau (y, y^{\prime }) \ge \rho (x, x^{\prime })$.

• (ii) Let $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) \ge \rho (x, x^{\prime })$ imply $\tau (y, y^{\prime }) \ge \rho (x, x^{\prime })$, for any $x, x^{\prime } \in A$, $y, y^{\prime } \in B$, and $\gamma$ = $\sup _{x, x^{\prime }, y, y^{\prime }}\{\rho (x, x^{\prime }): \tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) \ge \rho (x, x^{\prime })\} \ne$ 1. For any $x, x^{\prime } \in A$, $y, y^{\prime } \in B$, $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) > \gamma$ implies $\tilde{R}(x, y) \wedge \tilde{R}(x^{\prime }, y^{\prime }) \ge \rho (x, x^{\prime })$, and hence $\tau (y, y^{\prime }) \ge \rho (x, x^{\prime })$. That is $\tilde{R}$ is a $\gamma$-homorelation.$\square$

Corollary 2

Let $\tilde{R}$ be a homorelation from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$, where $\tilde{A}$ is a non-normal fuzzy set with finite support $A$. Then there is some $\gamma$ such that $\tilde{R}$ is a $\gamma$-homorelation from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$.

4 Fuzzy function, homomorphism, comorphism, cohomomorphism

The definition of fuzzy function by conditions (I), (II), and (III) only, seems to be too general and difficult to work with. By imposing further conditions more handy definitions are obtained in Demirci and Recasens (2004) and Perfilieva (2011) (see Sect. 3). The direction is towards crispization to some extent. As an instance, one can consider Demirci and Recasens’s (2004) perfect fuzzy function, strong fuzzy function, and Perfilieva’s (2011) perfect and surjective fuzzy function. Klawonn (2000), however, refrained from such attempts to some extent; he imposed $\sup _{x \in X}\tilde{R}(x, y)$ = 1 instead of demanding existence of one such $x$ giving $\tilde{R}(x, y)$ = 1.

It should be recalled that in the year 1973 Higgs wrote a celebrated but unpublished article (Higgs 1973) in which the morphisms of the category satisfy conditions (I), (II), (III), and (IV), which is $\sup _{y \in Y}\tilde{R}(x, y)$ = $\tilde{A}(x)$. The meaning of (IV) in the classical context is equivalent to the requirement of a function that every element of the domain has an image. This point has been further clarified in the Note 3 after definition 15.

In Banerjee and Chakraborty (2003) various categories of various types of fuzzy sets with fuzzy identities have been studied that include Higgs’s category also. In all these categories identity morphisms are $\omega$-similitudes, and morphisms are fuzzy relations satisfying (I), (II), (III), (IV), and their various generalizations in terms of t-norms. Category theoretically these are elegant constructions but not so workable as fuzzy sets. Hence, in compliance with our target, as stated in the introduction, we propose below a set of notions which are all fuzzy relations from one fuzzy set to another with various levels of functionality conditions. In our definitions we do not depend on fuzzy equivalences in the domain and range sets. These will occur when we enter into the notions of homomorphism/comorphism.

Definition 15

(Chakraborty et al. 1985) A fuzzy relation $\tilde{F}$ from $\tilde{A}$ to $\tilde{B}$, denoted by $\tilde{F}: \tilde{A} \rightarrow \tilde{B}$, is said to be a fuzzy function if for any $x \in A$, $\tilde{A}(x)$ = $\sup _{y \in B}\tilde{F}(x, y)$.

Note 3

Classically, a function, say $F$ from a set $X$ to $Y$ is a rule such that for all $x \in X$ there is a unique $y \in Y$ satisfying the relation $F(x, y)$. Here, in this context let us analyze the definition for function. According to the definition, a fuzzy function $\tilde{F}$ from $\tilde{A}$ to $\tilde{B}$ is a fuzzy relation from $\tilde{A}$ to $\tilde{B}$, i.e., $\tilde{F}(x, y)$ $\le$ $\tilde{A}(x)$ $\wedge$ $\tilde{B}(y)$, for all $x \in A$, $y \in B$. That is, for any $x \in A$, $\tilde{F}(x, y)$ $\le$ $\tilde{A}(x)$, for all $y \in B$. Hence

$\sup _{y \in B}\tilde{F}(x, y) \le \tilde{A}(x) \ldots (1)$

On the other hand, generalizing the classical version for function that is, for each $x \in X$, there is $y \in Y$ such that $F(x, y)$ holds, in fuzzy context the following inequality is obtained. For each

$x \in A, \tilde{A}(x) \le \sup _{y \in B}\tilde{F}(x, y). (2)$

Combining (1) and (2) the definition for function is obtained in the fuzzy context. The domain of the function is assumed to be the set $\tilde{A}$ and the range of $\tilde{F}$, denoted by $Ran(\tilde{F})$ is defined as $Ran(\tilde{F})(y)$ = $\sup _{x \in A}$ $\tilde{F}(x, y)$, for each $y \in B$.

Proposition 7

(Chakraborty et al. 1985) $Dom(\tilde{F})(x)$ = $\sup _{y \in B}$ $\tilde{F}(x, y)$ for each $x \in A$.

Proposition 8

(Chakraborty et al. 1985) $Ran(\tilde{F})$ $\subseteq$ $\tilde{B}$.

It can be noticed that this definition of function does not require the condition of existence of some $y$ in the range set for each $x$ in the domain set. This point has been accommodated in the notion of semi-proper function. In proper function uniqueness of such an $y$ has been incorporated. The definitions are as follows.

Definition 16

(Chakraborty et al. 1985) A fuzzy function $\tilde{F}$ from $\tilde{A}$ to $\tilde{B}$ is said to be a semi-proper function if for each $x \in A$, there exist some $y \in B$ such that $\tilde{A}(x)$ = $\sup _{y \in B}$ $\tilde{F}(x, y)$ = $\tilde{F}(x, y)$, i.e., the supremum is attained at some $y \in B$.

Note 4

(i) Since here $y$ may not be unique, a fuzzy function is the counterpart of a multiple-valued function.

(ii) The departure from Demirci (Demirci and Recasens 2004) and Perfilieva (Perfilieva 2011) is to be marked. While for their function $\tilde{F}(x, y)$ = 1, for ours it is $\tilde{F}(x, y)$ = $\tilde{A}(x)$.

Definition 17

(Chakraborty et al. 1985) A fuzzy function $\tilde{F}$ from $\tilde{A}$ to $\tilde{B}$ is said to be a proper function if for each $x \in A$, there exists a unique $y \in B$ such that $\tilde{A}(x)$ = $\sup _{y \in B}$ $\tilde{F}(x, y)$ = $\tilde{F}(x, y)$, i.e., the supremum is attained at a unique $y \in B$.

Note 5

The notion of a proper fuzzy function may appear to be an ordinary function in fuzzy envelope. But one should notice that more restricted proper fuzzy functions are morphisms in Etyan’s (1981) category of fuzzy sets, and the category set of ordinary sets forms a proper subcategory Banerjee and Chakraborty (2003) of Etyan’s category.

Proposition 9

(Chakraborty et al. 1985) Let $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ and $\tilde{G}$ : $\tilde{B} \rightarrow \tilde{C}$ be two proper functions then $\tilde{G} \circ \tilde{F}$ : $\tilde{A} \rightarrow \tilde{C}$ is a proper function.

Definition 18

(Chakraborty et al. 1985) A proper function $\tilde{I}$ : $\tilde{A} \rightarrow \tilde{A}$ is called a near identity iff for all $x \in A$, $\sup _{x^{\prime } \in A}$ $\tilde{I}(x, x^{\prime })$ = $\tilde{I}(x, x)$ = $\tilde{A}(x)$.

Note 6

It is shown (Chakraborty et al. 1985) that for a function $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ and near identities $\tilde{I}_{A}$, $\tilde{I}_{B}$ on $\tilde{A}$ and $\tilde{B}$, respectively, $\tilde{F} \circ \tilde{I}_{A}$ $\ne$ $\tilde{F}$ and $\tilde{I}_{B} \circ \tilde{F}$ $\ne$ $\tilde{F}$.

Definition 19

A near identity $\tilde{I}_{A}$ : $\tilde{A} \rightarrow \tilde{A}$ is called an identity iff for all $x_{1}, x_{2} \in A$,

$$\begin{aligned} \begin{array}{ll} \tilde{I}_{A}(x_{1}, x_{2}) = \tilde{A}(x_{1})&{}\quad \mathrm{if} x_{1} = x_{2}.\\ \quad \quad \quad \quad \quad \quad \!=0,&{}\quad \mathrm{otherwise.} \end{array} \end{aligned}$$

Proposition 10

Let $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ a function and $\tilde{I}_{A}$, $\tilde{I}_{B}$ be identities on $\tilde{A}$ and $\tilde{B}$, respectively. Then $\tilde{F} \circ \tilde{I}_{A}$ = $\tilde{F}$ and $\tilde{I}_{B} \circ \tilde{F}$ = $\tilde{F}$.

Proof

Let $x \in A$ and $y \in B$. Then

$$\begin{aligned}&\tilde{F} \circ \tilde{I}_{A}(x, y) = \sup _{x^{\prime } \in A}[\tilde{I}_{A}(x, x^{\prime }) \wedge \tilde{F}(x^{\prime }, y)]\\&\quad \! = \tilde{A}(x)\!\wedge \! \tilde{F}(x, y) \!=\! \tilde{F}(x, y) \quad \!\! [Since \tilde{A}(x) \!=\! \sup _{y \in B} \tilde{F}(x, y)] \end{aligned}$$

Similarly, the other part can be proved. $\square$

Definition 20

Let $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ be a function and $\tilde{I}_{A}$, $\tilde{I}_{B}$ be identities on $\tilde{A}$ and $\tilde{B}$, respectively. $\tilde{F}$ is said to be invertible if there exists a function $\tilde{G}$ : $\tilde{B} \rightarrow \tilde{A}$ such that $\tilde{G} \circ \tilde{F}$ = $\tilde{I}_{A}$ and $\tilde{F} \circ \tilde{G}$ = $\tilde{I}_{B}$.

Theorem 14

Let $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ be invertible and $\tilde{G}$ : $\tilde{B} \rightarrow \tilde{A}$ be the function such that $\tilde{G} \circ \tilde{F}$ = $\tilde{I}_{A}$ and $\tilde{F} \circ \tilde{G}$ = $\tilde{I}_{B}$. Then (i) $\tilde{F}$ and $\tilde{G}$ are proper functions and (ii) $\tilde{G}$ is invertible.

Proof

(i) $\tilde{G} \circ \tilde{F}$ $(x, x) = \tilde{I}_{A}$ $(x, x) = \tilde{A}(x)$ and

$\tilde{G} \circ \tilde{F}$ $(x, x^{\prime }) = \tilde{I}_{A}$ $(x, x^{\prime }) = 0$ for $x \ne x^{\prime }$.

Now for $x, x^{\prime } \in A$, $\tilde{G} \circ \tilde{F}$ $(x, x^{\prime })$

= $\sup _{y \in B}$ $[\tilde{F}(x, y) \wedge \tilde{G}(y, x^{\prime })]$ = $\tilde{A}(x)$, $x$ = $x^{\prime }$

= 0, otherwise.

Hence for $x \ne x^{\prime }$, $\tilde{F}(x, y)$ = 0 or $\tilde{G}(y, x^{\prime })$ = 0 hold for any $y \in B$.    ...(1)

Similarly, for any $x \in A$, if $y \ne y^{\prime }$, then $\tilde{G}(y, x)$ = 0 or $\tilde{F}(x, y^{\prime })$ = 0.    ...(2)

But as for $x \in A$, $\tilde{A}(x)$ = $\sup _{y \in B}$ $\tilde{F}(x, y) >$ 0, there is some $y \in B$ such that $\tilde{F}(x, y) >$ 0, and for $y \in B$, as $\tilde{B}(y)$ = $\sup _{x \in A}$ $\tilde{G}(y, x) >$ 0, there is some $x \in A$ such that $\tilde{G}(y, x) >$ 0. Now using (1) and (2) we have -

for any $x \in A$, there is a unique $y \in B$ such that $\tilde{F}(x, y) >$ 0 and $\tilde{G}(y, x) >$ 0.    ...(3)

Hence $\tilde{G} \circ \tilde{F}$ $(x, x)$ = $\tilde{A}(x)$ = $\sup _{y \in B}$ $[\tilde{F}(x, y) \wedge \tilde{G}(y, x)]$, and (3) imply that for each $x \in A$, there is a unique $y \in B$ such that $\tilde{G} \circ \tilde{F}$ $(x, x)$ = $\tilde{A}(x)$ = $\tilde{F}(x, y) \wedge \tilde{G}(y, x)$.

That is, for $x \in A$, there is a unique $y \in B$, such that $\tilde{A}(x)$ $\le$ $\tilde{F}(x, y)$. Also, $\tilde{F}(x, y)$ $\le$ $\tilde{A}(x)$.

That is, for each $x \in A$, there is a unique $y \in B$, such that $\tilde{A}(x)$ = $\tilde{F}(x, y)$. Hence $\tilde{F}$ is a proper function.

Similarly, it can be proved that $\tilde{G}$ is a proper function. $\square$

Definition 21

Let the homorelation (correlation/cohomorelation) $\tilde{F}$ from $(\tilde{A}, \rho )$ to $(\tilde{B}, \tau )$ be in particular a function from $\tilde{A}$ to $\tilde{B}$. Then it is called a homomorphism (comorphism / cohomomorphism).

Definition 22

(i) For a function $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ if for all $y \in B$ there exists $x \in A$ such that $\tilde{F}(x, y)$ = $\tilde{A}(x)$ then $\tilde{F}$ is said to have property $(\alpha )$.

(ii) If $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ satisfies property $(\alpha )$ and for $y_{1} \ne y_{2}$ of $B$, the correspondings $x$’s are different, then $\tilde{F}$ is said to have property $(\alpha ^{\prime })$.

Definition 23

(i) A function $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ is said to be injective if $\tilde{F}$ is a proper function such that

$\sup _{y \in B}$ $\tilde{F}(x_{1}, y)$ = $\tilde{F}(x_{1}, y_{1})$

= $\tilde{F}(x_{2}, y_{1})$ = $\sup _{y \in B}$ $\tilde{F}(x_{2}, y)$

implies $x_{1}$ = $x_{2}$.

(ii) A function $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ is said to be surjective if $Ran(\tilde{F})$ = $\tilde{B}$.

Theorem 15

Let $\tilde{F}$ : $(\tilde{A}, \rho ) \rightarrow (\tilde{B}, \tau )$ be a semi-proper function. Then the following holds.

• (i) If $\tilde{F}$ is a homomorphism then if $\tau$ is antireflexive then $\rho$ is so.

• (ii) If $\tilde{F}$ is a comorphism then, if $\tau$ is $\epsilon$-reflexive then $\rho$ is $\epsilon$-reflexive, if $\tau$ is $\omega$-reflexive then $\rho$ is so; and if $\tau$ is any kind of reflexive relation then $\rho$ is absolute reflexive.

• (iii) If $\tilde{F}$ is a cohomomorphism then if $\tau$ is symmetric then $\rho$ is symmetric.

• (iv) If $\tilde{F}$ is a cohomomorphism then if $\tau$ is transitive then $\rho$ is transitive.

• (v) If $\tilde{F}$ is a cohomomorphism and injective then if $\tau$ is antisymmetric then $\rho$ is antisymmetric.

• (vi) If $\tilde{F}$ is a homomorphism and satisfies $(\alpha )$ then if $\rho$ is reflexive (any) then $\tau$ is reflexive (absolute).

• (vii) If $\tilde{F}$ is a comorphism and satisfies $(\alpha )$ then if $\rho$ is antireflexive then $\tau$ is so.

• (viii) If $\tilde{F}$ is a cohomomorphism and satisfies $(\alpha )$ then if $\rho$ is symmetric then $\tau$ is so.

• (ix) If $\tilde{F}$ is a cohomomorphism and satisfies $(\alpha )$ then if $\rho$ is transitive then $\tau$ is so.

• (x) If $\tilde{F}$ is a cohomomorphism and satisfies $(\alpha ^{\prime })$ then if $\rho$ is antisymmetric then $\tau$ is so.

Proof

Let us prove some of the cases.

(ii) Let $\tilde{F}$ be comorphism and $\tau$ be $\epsilon$-reflexive. Then, for any $x \in A$ there is $y \in B$ such that $\tilde{F}(x, y)$ = $\tilde{A}(x)$. Now $\rho (x, x) \le \tilde{A}(x)$ = $\tilde{F}(x, y)$ = $\tilde{F}(x, y) \wedge \tilde{F}(x, y)$.

So, $\tau (y, y) \le \rho (x, x)$, i.e., $\epsilon \le \rho (x, x)$.

Let $\tau$ is $\omega$-reflexive, i.e., $\tau (y, y)$ = $\tilde{B}(y)$.

Now, $\rho (x, x) \le \tilde{A}(x)$ = $\tilde{F}(x, y) \wedge \tilde{F}(x, y)$ implies

$\tilde{B}(y)$ = $\tau (y, y) \le \rho (x, x) \le \tilde{A}(x)$. So, we have

$\tilde{A}(x)$ = $\tilde{F}(x, y) \le \tilde{B}(y)$ = $\tau (y, y) \le \rho (x, x) \le \tilde{A}(x)$.

That is $\rho (x, x)$ = $\tilde{A}(x)$, for any $x \in A$.

(iv) Let $\tilde{F}$ be a cohomomorphism and $\tau$ be transitive.

Now $\tilde{F}$ being a semi-proper function for $x, x^{\prime } \in A$ there exist $y, y^{\prime } \in B$, such that $\rho (x, x^{\prime }) \le \tilde{A}(x) \wedge \tilde{A}(x^{\prime })$ = $\tilde{F}(x, y) \wedge \tilde{F}(x^{\prime }, y^{\prime })$. Hence $\tau (y, y^{\prime })$ = $\rho (x, x^{\prime })$ and

similarly for $x^{\prime }, x^{\prime \prime } \in A$ there exist $y^{\prime }, y^{\prime \prime } \in B$, such that

$\rho (x^{\prime }, x^{\prime \prime }) \le \tilde{A}(x^{\prime }) \wedge \tilde{A}(x^{\prime \prime })$ = $\tilde{F}(x^{\prime }, y^{\prime }) \wedge \tilde{F}(x^{\prime \prime }, y^{\prime \prime })$. Hence $\tau (y^{\prime }, y^{\prime \prime })$ = $\rho (x^{\prime }, x^{\prime \prime })$.

Now let us fix any arbitrarily taken $x, x^{\prime \prime } \in A$. Then $\rho \circ \rho$ $(x, x^{\prime \prime })$ = $\sup _{x^{\prime } \in A}$ $[\rho (x, x^{\prime }) \wedge \rho (x^{\prime }, x^{\prime \prime })]$.

= $\sup _{y^{\prime } \in B}$ $[\tau (y, y^{\prime }) \wedge \tau (y^{\prime }, y^{\prime \prime })]$.

= $\tau \circ \tau$ $(y, y^{\prime \prime })$ $\le \tau (y, y^{\prime \prime })$ = $\rho (x, x^{\prime \prime })$.

(v) Let $\tilde{F}$ be cohomomorphism and injective. And let $\tau$ be antisymmetric.

Let us take any $x, x^{\prime } \in A$ such that $x \ne x^{\prime }$. Now $\tilde{F}$ being injective, it is a proper function. That is, there exist unique $y, y^{\prime } \in B$ such that $\tilde{F}(x, y)$ = $\tilde{A}(x)$ and $\tilde{F}(x^{\prime }, y^{\prime })$ = $\tilde{A}(x^{\prime })$.

Hence $\tilde{F}(x, y) \wedge \tilde{F}(x^{\prime }, y^{\prime })$ = $\tilde{A}(x) \wedge \tilde{A}(x^{\prime })$ $\ge \rho (x, x^{\prime })$. Then $\tau (y, y^{\prime })$ = $\rho (x, x^{\prime })$.

Now we claim $y \ne y^{\prime }$. If not then, $\tilde{F}$ being injective,

$\sup _{y \in B}$ $\tilde{F}(x, y)$ = $\tilde{F}(x, y)$ = $\tilde{F}(x^{\prime }, y)$ = $\sup _{y \in B}$ $\tilde{F}(x^{\prime }, y)$ implies $x$ = $x^{\prime }$.

Hence contradiction arises.

Now as $\tau$ is antisymmetric, either $\tau (y, y^{\prime }) \ne \tau (y^{\prime }, y)$, i.e., $\rho (x, x^{\prime }) \ne \rho (x^{\prime }, x)$

or $\tau (y, y^{\prime }) = \tau (y^{\prime }, y)$ = 0, i.e., $\rho (x, x^{\prime }) = \rho (x^{\prime }, x)$ = 0. That is, $\rho$ is antisymmetric.

(ix) Let $\tilde{F}$ be cohomomorphism and satisfy property $(\alpha )$. Let $\rho$ be transitive. Let $y, y^{\prime \prime } \in B$,

$\tau \circ \tau$ $(y, y^{\prime \prime })$ = $\sup _{y^{\prime } \in B}$ $[\tau (y, y^{\prime }) \wedge \tau (y^{\prime }, y^{\prime \prime })]$

= $\sup _{x^{\prime } \in A}$ $[\rho (x, x^{\prime }) \wedge \rho (x^{\prime }, x^{\prime \prime })]$

Since $\tilde{F}$ is cohomomorphism and satisfies $(\alpha )$

= $\rho \circ \rho$ $(x, x^{\prime \prime })$ $\le \rho (x, x^{\prime \prime })$

= $\tau (y, y^{\prime \prime })$

(x) Let $\tilde{F}$ be cohomomorphism and satisfy property $(\alpha ^{\prime })$. Let $y, y^{\prime } \in B$ such that $y \ne y^{\prime }$.

Now by property $(\alpha )$, for $y, y^{\prime } \in B$, there exist $x, x^{\prime } \in A$ such that $\tilde{F}(x, y)$ = $\tilde{A}(x)$ and $\tilde{F}(x^{\prime }, y^{\prime })$ = $\tilde{A}(x^{\prime })$.

Hence $\rho (x, x^{\prime })$ $\le \tilde{A}(x) \wedge \tilde{A}(x^{\prime })$ = $\tilde{F}(x, y)$ $\wedge$ $\tilde{F}(x^{\prime }, y^{\prime })$. Hence $\tau (y, y^{\prime })$ = $\rho (x, x^{\prime })$. Similarly, it can be proved that $\tau (y^{\prime }, y)$ = $\rho (x^{\prime }, x)$.

As $y \ne y^{\prime }$, and $\tilde{F}(x, y)$ = $\tilde{A}(x)$, $\tilde{F}(x^{\prime }, y^{\prime })$ = $\tilde{A}(x^{\prime })$, by property $(\alpha ^{\prime })$, $x \ne x^{\prime }$.

Hence either $\rho (x, x^{\prime }) \ne \rho (x^{\prime }, x)$, i.e., $\tau (y, y^{\prime }) \ne \tau (y^{\prime }, y)$

or $\rho (x, x^{\prime }) = \rho (x^{\prime }, x)$ = 0, i.e., $\tau (y, y^{\prime }) = \tau (y^{\prime }, y)$ = 0. That is, $\tau$ is antisymmetric. $\square$

Corollary 3

If a semi-proper function $\tilde{F}$ from ($\tilde{A}, \rho$) to ($\tilde{B}, \tau$) is a cohomomorphism, then if $\tau$ is a $\epsilon$-similitude, $\rho$ is so, and if $\tau$ is a $\omega$-similitude, $\rho$ is also a $\omega$-similitude.

Proposition 11

Let $\tilde{F}$ : $(\tilde{A}, \rho ) \rightarrow (\tilde{A}, \tau )$ be a near identity on $\tilde{A}$ and also a homomorphism. Then $\rho \subseteq \tau$.

Theorem 16

Let $\rho$ and $\tau$ be two fuzzy relations on $\tilde{A}$ and $\rho \subseteq \tau$. Then there exists a near identity on $\tilde{A}$ that is also a homomorphism from $(\tilde{A}, \rho )$ to $(\tilde{A}, \tau )$.

Theorem 17

Let $\tilde{F}$ : $\tilde{A} \rightarrow \tilde{B}$ be a function and $\rho$ and $\tau$ be two fuzzy relations on $\tilde{A}$ and $\tilde{B}$, respectively.

• (i) If $\tilde{F}$ is surjective and $\gamma$-homomorphism for $\gamma$ $<$ $\inf _{y \in B} \tilde{B}(y)$ then $\rho$ is reflexive implies $\tau$ is absolute reflexive.

• (ii) If $\tilde{F}$ is $\gamma$-homomorphism for $\gamma$ $<$ $\inf _{x \in A} \tilde{A}(x)$ then $\tau$ is antireflexive implies $\rho$ is antireflexive.

• (iii) If $\tilde{F}$ is surjective and $\gamma$-cohomomorphism for $\gamma$ $<$ $\inf _{y \in B} \tilde{B}(y)$ then $\tau$ is antireflexive when $\rho$ is so.

• (iv) If $\tilde{F}$ is a $\gamma$-comorphism for $\gamma$ $<$ $\inf _{x \in A} \tilde{A}(x)$ then $\rho$ is absolute reflexive when $\tau$ is so.

• (v) If $\tilde{F}$ is a surjective function and $\gamma$-cohomomorphism such that $\gamma$ $<$ $\inf _{x\in A, y \in B}$ $(\tilde{A}(x)$ $\wedge$ $\tilde{B}(y))$, then $\tau$ is symmetric iff $\rho$ is symmetric.

• (vi) If $\tilde{F}$ is an injective function satisfying property ($\alpha ^{\prime }$) and $\tilde{F}$ is a $\gamma$-cohomomorphism where $\gamma$ $<$ $\inf _{x\in A, y \in B}$ $(\tilde{A}(x)$ $\wedge$ $\tilde{B}(y))$, then $\rho$ is antisymmetric iff $\tau$ is antisymmetric.

• (vii) If $\tilde{F}$ is a surjective function and $\gamma$-cohomomorphism such that $\gamma$ $<$ $\inf _{x\in A, y \in B}$ $(\tilde{A}(x)$ $\wedge$ $\tilde{B}(y))$, then $\rho$ is transitive iff $\tau$ is transitive.

• (viii) If $\tilde{F}$ is a surjective function and $\gamma$-cohomomorphism such that $\gamma$ $<$ $\inf _{x\in A, y \in B}$ $(\tilde{A}(x)$ $\wedge$ $\tilde{B}(y))$, then $\rho$ is equivalence relation iff $\tau$ is equivalence relation.

Proof

Let us give a proof outline of (v) from these series of results.

$Ran(\tilde{F})(y)$ = $\tilde{B}(y)$ for any $y \in B$.

That is, $\tilde{B}(y)$ = $\sup _{x \in A}\tilde{F}(x, y)$ for any $y \in B$.

Now, $\gamma < \inf _{x\in A, y \in B} (\tilde{A}(x) \wedge \tilde{B}(y)) \le \inf _{y \in B}\tilde{B}(y)$ $\le \tilde{B}(y)$ = $\sup _{x \in A}\tilde{F}(x, y)$ for any $y \in B$.

That is, for any $y \in B$ there is some $x \in A$ such that $\gamma < \tilde{F}(x, y)$. That is, for any $y, y^{\prime }$ there are $x, x^{\prime }$ such that $\tilde{F}(x, y) \wedge \tilde{F}(x^{\prime }, y^{\prime }) > \gamma$. Hence $\tau (y, y^{\prime })$ = $\rho (x, x^{\prime })$. Similarly, we also obtain $\tau (y^{\prime }, y)$ = $\rho (x^{\prime }, x)$. Now, if $\rho$ is symmetric then it is immediate that $\tau$ is also so.

Conversely, $\gamma < \inf _{x\in A, y \in B} (\tilde{A}(x) \wedge \tilde{B}(y)) \le \inf _{x \in A}\tilde{A}(x)$

$\le \tilde{A}(x)$ = $\sup _{y \in B}\tilde{F}(x, y)$ for any $x \in A$.

Hence, for any $x \in A$ there is $y \in B$ such that $\gamma < \tilde{F}(x, y)$. Now following the same argument as above, we can show that $\rho$ is symmetric if $\tau$ is so. $\square$

In the last section, we notice some connections between homorelation/correlation, and the properties of a function defined in the sense of (Demirci and Recasens 2004; Gilles 1984; Klawonn 2000; Perfilieva 2011). We observe that the left hand side of the conditions for homorelation/correlation plays some crucial role in getting those identities characterizing function due to (Demirci and Recasens 2004; Gilles 1984; Klawonn 2000; Perfilieva 2011). Following observations may throw light on how these notions of homorelation/correlation, function, homomorphism/comorphism are linked in an integrated manner, and what is the connection with the notion of function (Demirci and Recasens 2004; Gilles 1984; Klawonn 2000; Perfilieva 2011) prevalent in the existing literature.

Observation 3

(i) Let $\tilde{F}: (\tilde{A}, \rho ) \rightarrow (\tilde{B}, \tau )$ be a semi-proper function, and $\rho$, $\tau$ are two fuzzy relations on $\tilde{A}$ and $\tilde{B}$, respectively. Then for each $x, x^{\prime } \in A$, there are some $y, y^{\prime } \in B$, such that $\tilde{F}(x, y)$ = $\tilde{A}(x)$, $\tilde{F}(x^{\prime }, y^{\prime })$ = $\tilde{A}(x^{\prime })$. So, $\rho (x, x^{\prime }) \le \tilde{A}(x) \wedge \tilde{A}(x^{\prime })$ = $\tilde{F}(x, y) \wedge \tilde{F}(x^{\prime }, y^{\prime })$.

That is, for a semi-proper function $\tilde{F}$, given any

$x, x^{\prime } \in A$, there are $y, y^{\prime } \in B$, such that

$\rho (x, x^{\prime }) \le \tilde{F}(x, y) \wedge \tilde{F}(x^{\prime }, y^{\prime })$.    ...(a)

It is to be noted that (a) is exactly the pre-condition for a fuzzy relation to be a homorelation/correlation/cohomorelation.

(ii) Now, if for $x, x^{\prime } \in A$, and $y \in B$, (a) holds, then $\rho (x, x^{\prime }) \le \tilde{F}(x, y) \wedge \tilde{F}(x^{\prime }, y) \le \tilde{F}(x, y), \tilde{F}(x^{\prime }, y)$.

That is, $\tilde{F}(x, y) \wedge \rho (x, x^{\prime })$ = $\rho (x, x^{\prime }) \le \tilde{F}(x^{\prime }, y)$.

Hence for any $x, x^{\prime }, y$ satisfying (a), the following holds. $\tilde{F}(x, y) \wedge \rho (x, x^{\prime }) \le \tilde{F}(x^{\prime }, y)$.    ...(b)

Similarly, following the argument made in Observation 2, we can show the following.

(iii) Let us assume $\tilde{F}$, the semi-proper function mentioned above, is a comorphism. Then, for any $x, y, y^{\prime }$ satisfying (a), $\tilde{F}(x, y) \wedge \tau (y, y^{\prime }) \le \tilde{F}(x, y^{\prime })$.    ...(c)

(iv) Let us assume $\tilde{F}$, the semi-proper function mentioned above, is a homomorphism, and $\rho$ is a near identity on $\tilde{A}$. Then, for any $x, y, y^{\prime }$ satisfying (a), we get the inequality $\tilde{F}(x, y) \wedge \tilde{F}(x, y^{\prime }) \le \tau (y, y^{\prime })$.    ...(d)

Let us emphasize the role of semi-proper function $\tilde{F}$ in this context. Observation 3(i) shows that because of the functionality property of $\tilde{F}$ for any $x, x^{\prime } \in A$, there exist $y, y^{\prime } \in B$ such that the pre-condition for $\tilde{F}$ being a homorelation/correlation/cohomorelation, viz., $\rho (x, x^{\prime }) \le \tilde{F}(x, y) \wedge \tilde{F}(x^{\prime }, y^{\prime })$ holds. This may not hold for arbitrary $x, x^{\prime }, y, y^{\prime }$.

5 Conclusion

The basic difference of the present approach from the existing approaches lies in accepting a fuzzy relation as a subset of the cartesian product of fuzzy sets, and in defining various properties of relations taking into consideration that the base set is fuzzy. The significance of this point of view has been stressed. Though the development, made in this paper, is based on [0, 1], and simple lattice meet and join operators, many of the results can be generalized in the context of t-norms too. The usefulness of lattice meet operator over t-norms, which fails to retain some important properties of lattice meet, may be considered in this context. In the definition of fuzzy function no involvement of fuzzy equivalence relation in the domain and range set is considered; rather, the functions have been defined in a straightforward manner following the tradition of standard set theory. Fuzzy homomorphisms/comorphisms have been introduced as genuine fuzzy concepts.

We conclude this paper with some open issues related to these notions.

• (i) We notice that two kinds of identities are introduced. The notion of identity in Definition 19, is somewhat restricted compared to the notion of near identity. But, from Note 6, Proposition 10, and Theorem 14 we can see the usefulness of the notion of identity over near identity. Generally, the concept of identity function is relative to a set; for each set $X$, there is exactly one $I_{X}$ with both domain and range set $X$. What happens if we relax this notion with respect to functions as well? The idea is as follows. Let $\tilde{F}: \tilde{A} \rightarrow \tilde{B}$ be a fuzzy function, and $\tilde{I}_{A}^{F}$ and $\tilde{I}_{B}^{F}$ be two fuzzy relation on $\tilde{A}$ and $\tilde{B}$, respectively, such that $\tilde{I}_{A}^{F}(x, x)$ = $\tilde{A}(x)$ and $\tilde{I}_{B}^{F}(y, y)$ = $\tilde{B}(y)$. And, for $x \ne x^{\prime }$ of $A$, $y \ne y^{\prime }$ of $B$, the following hold. $\tilde{I}_{A}^{F}(x, x^{\prime })$ = $\tilde{I}_{A}^{F}(x^{\prime }, x)$ $\le \tilde{F}(x, y) \wedge \tilde{F}(x^{\prime }, y)$, and $\tilde{I}_{B}(y, y^{\prime })$ = $\tilde{I}_{B}(y^{\prime }, y)$ $\le \tilde{F}(x, y) \wedge \tilde{F}(x, y^{\prime })$. Following this notion of identity function, may be called functional identity, a variant form of Proposition 10, i.e., $\tilde{F} \circ \tilde{I}_{A}^{F}$ = $\tilde{F}$, and $\tilde{I}_{B}^{F} \circ \tilde{F}$ = $\tilde{F}$, and some more important results can be obtained. This notion has some category theoretic flavor. Further development in this area is required.

• (ii) Though the notion of near identity is an elegant generalization of the concept of identity, it fails to give some necessary properties of identity. It would be interesting to check the results adding symmetry and/or transitivity to the existing notion of near identity.

• (iii) In $(\tilde{A}, \rho )$ and $(\tilde{B}, \tau )$, $\rho$ and $\tau$ are predecessors of fuzzy identities on $\tilde{A}$ and $\tilde{B}$, respectively. So, fuzzy function from $\tilde{A}$ to $\tilde{B}$ in the special case where $\rho$ and $\tau$ are near identities/identities may open an interesting area of investigation.