A Fuzzy-Equality-Based Index of T-Transitivity for Fuzzy Relations

Abstract

In this paper, we introduce a novel index of transitivity for fuzzy relations based on fuzzy equality. The proposed index evaluates the supremal uniform fuzzy-equality proximity between a given relation and the family of exactly transitive relations induced by a continuous triangular norm. We establish several fundamental properties of the proposed index, including faithfulness, stability with respect to fuzzy equality, invariance under inverse relations, monotonicity under restriction, and bounds derived from transitive closures. Furthermore, we prove that the proposed index is attainable on finite universes. A detailed comparison with the traditional degree of transitivity demonstrates that the new index always provides an evaluation no smaller than the classical measure and can distinguish different aspects of approximate transitivity. Finally, we characterize the special case in which the two indices coincide for all fuzzy relations and show that this occurs precisely for the Gödel triangular norm.

1. Introduction

Fuzzy relations provide a standard framework for representing graded interactions, similarities, preferences, and dependencies in systems involving uncertainty or imprecision. Since the foundational work of Zadeh on fuzzy sets and similarity relations [1,2], fuzzy relational structures have been widely used in approximate reasoning, classification, clustering, decision making, information retrieval, and knowledge representation. Among their structural properties, transitivity is particularly important because it underlies fuzzy preorders, fuzzy equivalence relations, fuzzy orderings, and indistinguishability operators [3,4,5,6].

Let T be a triangular norm (t-norm). A fuzzy relation R on a universe X is T-transitive if $T\left(R(x,y),R(y,z)\right)\le R(x,z)$$(x,y,z\in X)$. Recent work continues to show the relevance of such conditions. He et al. [7] studied $T_L$-transitivity of fuzzy similarity measures and its connection with triangle-type inequalities. Armengol et al. [8] investigated T-generable indistinguishability operators and their use in feature selection and classification. Recasens [9] surveyed the development and applications of similarity relations more broadly.

In applications, however, fuzzy relations obtained from subjective assessments, incomplete information, or noisy observations need not satisfy transitivity exactly. This motivates graded evaluations of relational properties. Related approaches include extensional graded properties of fuzzy relations [10], similarity-based treatments of approximate fuzzy preorders and equivalences [3,4,6], and the degree of symmetry studied by Dan et al. [11].

For a left-continuous t-norm T, the traditional degree of T-transitivity is

$${\alpha }_T\left(R\right)=\underset{x,y,z\in X}{\bigwedge }\left(T\left(R(x,y),R(y,z)\right){\to }_{T}R(x,z)\right),$$

where ${\to }_T$ is the residual implication associated with T [12]. Boixader and Recasens [13] revisited this degree and characterized it through transformations that produce T-transitive relations below the original relation. Because ${\alpha }_T\left(R\right)$ is an infimum over all triples, it is a worst-constraint measure: its value records the least satisfied transitivity condition. This is appropriate when the aim is to detect the most serious local violation, but it does not directly quantify how uniformly close the entire relation is to the class of exactly T-transitive relations. The class of T-transitive relations itself has a rich structure. Aggregation of indistinguishability operators and transitive fuzzy relations has been studied in [14,15]; connections between fuzzy preorders and generalized distances were developed in [16]; and localization relations were introduced in [17]. These results motivate studying approximation to the whole class of transitive relations rather than evaluating only separate pointwise inequalities.

In this paper, we define a fuzzy-equality-based index of T-transitivity. Let $\mathcal{T}\left(X\right)$ denote the family of all T-transitive fuzzy relations on X, and let

$$E_{X^2}(R,S)=\underset{x,y\in X}{\bigwedge }\left(R(x,y){\leftrightarrow }_{T}S(x,y)\right),$$

where ${\leftrightarrow }_T$ is the bi-implication associated with T. For a continuous t-norm T, set

$$d_T\left(R\right)=\underset{S\in \mathcal{T}\left(X\right)}{\bigvee }{E}_{X^2}(S,R).$$

The quantity $d_T\left(R\right)$ is therefore the supremal uniform fuzzy-equality level between R and the exact class $\mathcal{T}\left(X\right)$. The optimization is global in the sense that it ranges over the whole class of T-transitive relations, but the criterion $E_{X^2}$ is itself maximin: it is determined by the least coordinatewise similarity. Hence $d_T$ is a uniform-approximation index, not an average or total measure of all entrywise modifications. In particular, a supremum need not be attained on an arbitrary universe.

The generic principle of measuring a property by similarity to objects satisfying it exactly is mathematically natural. The contribution claimed here is therefore not that approximation principle by itself, but its specific implementation for T-transitive fuzzy relations and the structural results that follow from it. We prove that $d_T$ is faithful to exact transitivity, establish its stability and behavior under natural relational operations, and show that the defining supremum is attained when X is finite. We then compare $d_T$ with the traditional degree ${\alpha }_T$. Although ${\alpha }_T\left(R\right)\le d_T\left(R\right)$ always holds for continuous T, strict inequality can occur. Finally, we characterize the exceptional continuous t-norm for which the two indices coincide for every fuzzy relation on every universe: this happens exactly for the Gödel t-norm.

The remainder of the paper is organized as follows. Section 2 recalls only the t-norm, residual, and fuzzy-relation facts used later. Section 3 introduces $d_T$ and studies faithfulness, finite-universe attainment, stability, inverse invariance, restriction, and transitive closure. Section 4 compares $d_T$ with ${\alpha }_T$, gives strict-inequality examples, and proves the Gödel characterization. Section 5 summarizes the conclusions and indicates directions for future research.

2. Preliminaries

This section collects the notation and standard results used in the sequel. Further background on t-norms and residual structures can be found in ([18], Chapters 1 and 11) and (Chapters 1–2, [19]).

Definition 1 

([18], Chapter 1). A t-norm is a binary operation T:${[0,1]}^2\to [0,1]$ that is commutative, associative, increasing in each argument, and satisfies $T(x,1)=x$ for all $x\in [0,1]$. It is called left-continuous if, for each fixed $x\in [0,1]$,

$$T\left(x,\underset{i\in I}{\bigvee }{y}_i\right)=\underset{i\in I}{\bigvee }T(x,y_i),$$

and continuous if it is continuous as a function on ${[0,1]}^2$.

Example 1 

([18], Chapter 1). The three standard continuous t-norms used throughout the paper are

$$\begin{array}{cccc}{T}_M(x,y)\hfill & =min\{x,y\},\hfill & \hfill & \left(G\mathsf{ö}\mathit{del}\right),\hfill \\ T_P(x,y)\hfill & =xy,\hfill & \hfill & \left(\mathit{product}\right),\hfill \\ T_{\mathsf{Ł}}(x,y)\hfill & =max\{x+y-1,0\},\hfill & \hfill & \left(\mathsf{Ł}\mathit{ukasiewicz}\right).\hfill \end{array}$$

The only part of the ordinal sum representation needed later is the following consequence of the standard representation theorem.

Proposition 1 

([18], Chapter 5). Let T be a continuous t-norm. If $T\ne T_M$, then there exist $0\le a<b\le 1$, an order isomorphism ϕ:$[a,b]\to [0,1]$, and $T_{*}\in \{T_P,T_{\mathsf{Ł}}\}$ such that

$$T(u,v)={\varphi }^{-1}\left(T_{*}(\varphi \left(u\right),\varphi \left(v\right))\right),u,v\in [a,b].$$

The interval $[a,b]$ is a nontrivial Archimedean component of the ordinal sum representation of T.

Definition 2 

([18], Chapter 11). Let T be a left-continuous t-norm. Its residual implication is

$$x{\to }_{T}y=\bigvee \{t\in [0,1]:T(x,t)\le y\},x,y\in [0,1].$$

The associated bi-implication is

$$x{\leftrightarrow }_{T}y=\left(x{\to }_{T}y\right)\wedge \left(y{\to }_{T}x\right).$$

The residual implication is characterized by the adjointness relation

$$T(x,z)\le y⟺z\le x{\to }_{T}y.$$

(1)

We shall also use the standard facts

$$x{\to }_{T}y=1⟺x\le y,T(x{\to }_{T}y,y{\to }_{T}z)\le x{\to }_{T}z,$$

(2)

which follow from adjointness; see ([18], Chapter 11) and ([19], Chapter 2).

Example 2. 

For the t-norms in Example 1, the residual implications and bi-implications are

(i) 

$T=T_M$,

$$x{\to }_{M}y=\left\{\begin{array}{cc}1,\hfill & x\le y,\hfill \\ y,\hfill & x>y,\hfill \end{array}\right.x{\leftrightarrow }_{M}y=\left\{\begin{array}{cc}1,\hfill & x=y,\hfill \\ x\wedge y,\hfill & x\ne y;\hfill \end{array}\right.$$

(ii) 

$T=T_P$,

$$x{\to }_{P}y=\left\{\begin{array}{cc}1,\hfill & x\le y,\hfill \\ y/x,\hfill & x>y,\hfill \end{array}\right.x{\leftrightarrow }_{P}y=\left\{\begin{array}{cc}1,\hfill & x=y,\hfill \\ {\displaystyle \frac{x\wedge y}{x\vee y}},\hfill & x\ne y;\hfill \end{array}\right.$$

(iii) 

$T=T_{\mathsf{Ł}}$,

$$x{\to }_{\mathsf{Ł}}y=min\{1,1-x+y\},x{\leftrightarrow }_{\mathsf{Ł}}y=1-|x-y|.$$

Definition 3 

([1]). A fuzzy relation on a universe X is a mapping R:$X\times X\to [0,1]$. For fuzzy relations R and S, we write $R\le S$ if $R(x,y)\le S(x,y)$ for all $x,y\in X$.

Unless otherwise stated, fuzzy relations considered in this paper are not assumed to be reflexive or symmetric.

Definition 4 

([13]). Let T be a t-norm. A fuzzy relation R on X is T-transitive if

$$T\left(R\right(x,y),R(y,z\left)\right)\le R(x,z)\mathit{for}\mathit{all}x,y,z\in X.$$

The family of all T-transitive fuzzy relations on X is denoted by $\mathcal{T}\left(X\right)$.

The reflexive and symmetric form of the numerical relations used below is chosen only for convenience of illustration; neither property is assumed in the general theory.

The exact condition can be sensitive to small perturbations. For instance, with the product t-norm,

$$R=\left(\begin{array}{ccc}1& 0.4& 0.801\\ 0.4& 1& 0.5\\ 0.801& 0.5& 1\end{array}\right),S=\left(\begin{array}{ccc}1& 0.4& 0.8\\ 0.4& 1& 0.5\\ 0.8& 0.5& 1\end{array}\right).$$

The relation R is not product-transitive because $T_P(0.801,0.5)=0.4005>0.4$, whereas S is product-transitive. This illustrates why a graded evaluation of transitivity is useful.

Definition 5 

([12]). For a fuzzy relation R on X and a left-continuous t-norm T, the degree of T-transitivity is

$${\alpha }_T\left(R\right)=\underset{x,y,z\in X}{\bigwedge }\left(T(R(x,y),R(y,z)){\to }_{T}R(x,z)\right).$$

Moreover, ${\alpha }_T\left(R\right)=1$ if and only if R is T-transitive.

Definition 6 

(Chapter 2, [20]). Let T be a left-continuous t-norm. The fuzzy-equality degree between two fuzzy relations R and S on X is

$$E_{X^2}(R,S)=\underset{x,y\in X}{\bigwedge }\left(R(x,y){\leftrightarrow }_{T}S(x,y)\right).$$

Thus $E_{X^2}(R,S)$ is a uniform equality degree between the two relations.

Definition 7 

([13]). Let R be a fuzzy relation on X, let T be a t-norm, and $c\in [0,1]$. Then $R_{(T,c)}$ is defined for all $x,y\in X$ by $R_{(T,c)}(x,y)=T(R(x,y),c)$.

Proposition 2 

([13]). Let T be a left-continuous t-norm and R a fuzzy relation on X. Then $R_{(T,{\alpha }_T\left(R\right))}$ is T-transitive.

Proposition 3 

([13]). Let T be a left-continuous t-norm and R a fuzzy relation on X. Then $E_{X^2}\left(R,R_{(T,{\alpha }_T\left(R\right))}\right)\ge {\alpha }_T\left(R\right)$.

The next elementary inequality will be used to establish stability of the new index.

Proposition 4. 

Let T be a left-continuous t-norm. For any fuzzy relations $R,Q,S$ on X, $T\left(E_{X^2}(R,Q),E_{X^2}(Q,S)\right)\le E_{X^2}(R,S)$.

Proof. 

For $a,b,c\in [0,1]$, Equation (2) gives $T(a{\to }_{T}b,b{\to }_{T}c)\le a{\to }_{T}c$ and, symmetrically, $T(c{\to }_{T}b,b{\to }_{T}a)\le c{\to }_{T}a$. Therefore $T(a{\leftrightarrow }_{T}b,b{\leftrightarrow }_{T}c)\le a{\leftrightarrow }_{T}c$.

Apply this pointwise with $a=R(x,y)$, $b=Q(x,y)$, and $c=S(x,y)$. Since $E_{X^2}(R,Q)$ and $E_{X^2}(Q,S)$ are bounded above by the corresponding pointwise bi-implications, monotonicity of T yields $T\left(E_{X^2}(R,Q),E_{X^2}(Q,S)\right)\le R(x,y){\leftrightarrow }_{T}S(x,y)$ for every $(x,y)\in X^2$. Taking the infimum over $X^2$ proves the result.  □

3. The Index of T-Transitivity of Fuzzy Relations Based on Continuous T-Norms

The equality degree $E_{X^2}(S,R)$ quantifies the least coordinatewise fuzzy equality between two relations. This suggests measuring the approximate transitivity of R by comparing it with the entire class of exactly T-transitive relations.

Definition 8. 

Let T be a continuous t-norm and let R be a fuzzy relation on X. The index of T-transitivity of R is

$$d_T\left(R\right)=\underset{S\in \mathcal{T}\left(X\right)}{\bigvee }{E}_{X^2}(S,R).$$

Definition 8 should be read as a supremal uniform-approximation index. The optimization ranges over the whole class $\mathcal{T}\left(X\right)$, while $E_{X^2}$ itself is a maximin criterion. Thus $d_T\left(R\right)$ does not measure the average or total number of changed entries. Moreover, on an arbitrary universe the supremum need not be attained.

Once the index is defined, several natural questions arise. First, is it faithful to exact transitivity? Second, is it stable when the underlying relation is replaced by a highly similar one? Third, how does it behave under basic operations such as inversion, restriction, and transitive closure? We answer these questions in turn, beginning with faithfulness.

Proposition 5. 

Let T be a continuous t-norm. A fuzzy relation R on X is T-transitive if and only if $d_T\left(R\right)=1$.

Proof. 

If R is T-transitive, then $R\in \mathcal{T}\left(X\right)$ and $E_{X^2}(R,R)=1$, so Definition 8 gives $d_T\left(R\right)=1$.

Conversely, suppose that $d_T\left(R\right)=1$, and fix $c\in [0,1)$. There exists $S_c\in \mathcal{T}\left(X\right)$ such that $E_{X^2}(S_c,R)>c$.

Hence, for every $u,v\in X$, $c<S_c(u,v){\leftrightarrow }_{T}R(u,v)$, so in particular $c\le R(u,v){\to }_T{S}_c(u,v)$, $c\le S_c(u,v){\to }_{T}R(u,v)$.

By adjointness, $T(c,R(u,v))\le S_c(u,v)$, $T(c,S_c(u,v))\le R(u,v)$.

For arbitrary $x,y,z\in X$, the transitivity of $S_c$ therefore yields

$$\begin{array}{cc}\hfill T\left(T(c,R(x,y\left)\right),T(c,R(y,z\left)\right)\right)& \le T(S_c(x,y),S_c(y,z))\hfill \\ \hfill & \le S_c(x,z).\hfill \end{array}$$

Applying the increasing map $T(c,·)$ gives

$$T\left(c,T\left(T(c,R(x,y\left)\right),T(c,R(y,z\left)\right)\right)\right)\le T(c,S_c(x,z))\le R(x,z).$$

By associativity and commutativity,

$$T\left(T\left(T\right(c,c),c),T\left(R\right(x,y),R(y,z\left)\right)\right)\le R(x,z).$$

Letting $c\to 1^{-}$ and using continuity of T, we obtain $T\left(R\right(x,y),R(y,z\left)\right)\le R(x,z)$. Since $x,y,z$ are arbitrary, R is T-transitive.  □

Although Definition 8 involves a supremum, the situation improves on finite universes.

Proposition 6. 

Let X be finite and let T be a continuous t-norm. For every fuzzy relation R on X, there exists $S^{*}\in \mathcal{T}\left(X\right)$ such that $d_T\left(R\right)=E_{X^2}(S^{*},R)$.

Proof. 

If $\left|X\right|=n$, identify all fuzzy relations on X with the compact cube ${[0,1]}^{n^2}$. For every fixed triple $(x,y,z)$, the set $\{S$:$T\left(S\right(x,y),S(y,z\left)\right)\le S(x,z)\}$ is closed because T is continuous. Since only finitely many triples are involved, $\mathcal{T}\left(X\right)$ is a closed subset of a compact cube and is therefore compact.

For a continuous t-norm, its residual implication is upper semicontinuous. Indeed, for each $\lambda \in [0,1]$, adjointness gives $\left\{\right(u,v)$:$u{\to }_{T}v\ge \lambda \}=\{(u,v):T(u,\lambda )\le v\}$, and the latter set is closed by continuity of T.

Hence $u{\leftrightarrow }_{T}v$, being the minimum of two upper semicontinuous functions, is upper semicontinuous. Since X is finite, $S⟼E_{X^2}(S,R)={min}_{x,y\in X}\left(S(x,y){\leftrightarrow }_{T}R(x,y)\right)$ is upper semicontinuous. It therefore attains its maximum on the compact set $\mathcal{T}\left(X\right)$, proving the claim.  □

Proposition 6 justifies speaking of a maximizing transitive approximation when X is finite. It does not, however, imply that a particular candidate displayed in an example is optimal unless that optimality is separately proved.

We next turn to stability under fuzzy equality.

Proposition 7. 

For any fuzzy relations $R,Q$ on X, $T\left(E_{X^2}(R,Q),d_T\left(Q\right)\right)\le d_T\left(R\right)$. By symmetry, $T\left(E_{X^2}(R,Q),d_T\left(R\right)\right)\le d_T\left(Q\right)$.

Proof. 

For any $S\in \mathcal{T}\left(X\right)$, Proposition 4 gives $T\left(E_{X^2}(R,Q),E_{X^2}(Q,S)\right)\le E_{X^2}(R,S)\le d_T\left(R\right)$.

Taking the supremum over $S\in \mathcal{T}\left(X\right)$ and using continuity of T in its second argument yields $T\left(E_{X^2}(R,Q),d_T\left(Q\right)\right)\le d_T\left(R\right)$.

The second inequality follows by interchanging R and Q.  □

Remark 1. 

If $E_{X^2}(R,Q)\ge \lambda$ and $d_T\left(Q\right)\ge \mu$, then Proposition 7 implies

$$d_T\left(R\right)\ge T(\lambda ,\mu ).$$

For the three standard continuous t-norms this becomes

$$\begin{array}{cc}T& \mathit{Lower}\mathit{bound}\mathit{for}{d}_T\left(R\right)\\ T_M& min\{\lambda ,\mu \}\\ T_P& \lambda \mu \\ T_{\mathsf{Ł}}& max\{\lambda +\mu -1,0\}.\end{array}$$

Thus a relation that is uniformly very similar to one with a high transitivity index must itself have a controlled lower bound on its index.

The next two propositions show that the index behaves naturally under inversion and restriction.

Proposition 8. 

For a fuzzy relation R on X, define $R^{-1}(x,y)=R(y,x)$. Then $d_T\left(R^{-1}\right)=d_T\left(R\right)$.

Proof. 

If $S\in \mathcal{T}\left(X\right)$, then for all $x,y,z\in X$,

$$\begin{array}{cc}\hfill T(S^{-1}(x,y),S^{-1}(y,z))& =T\left(S\right(y,x),S(z,y\left)\right)\hfill \\ \hfill & =T\left(S\right(z,y),S(y,x\left)\right)\hfill \\ \hfill & \le S(z,x)=S^{-1}(x,z),\hfill \end{array}$$

so $S^{-1}\in \mathcal{T}\left(X\right)$. Moreover, $E_{X^2}(S^{-1},R^{-1})=E_{X^2}(S,R)$.

Since $S\mapsto S^{-1}$ is a bijection of $\mathcal{T}\left(X\right)$, $d_T\left(R^{-1}\right)={\bigvee }_{S\in \mathcal{T}\left(X\right)}{E}_{X^2}(S,R)=d_T\left(R\right)$.  □

Proposition 9. 

Let $Y\subseteq X$ and denote by ${R|}_Y$ the restriction of R to $Y\times Y$. Then $d_T\left(R{|}_Y\right)\ge d_T\left(R\right)$.

Proof. 

For $S\in \mathcal{T}\left(X\right)$, the restriction ${S|}_Y$ is T-transitive on Y and

$$E_{Y^2}{\left(S\right|}_Y,R{|}_Y)=\underset{x,y\in Y}{\bigwedge }\left(S(x,y){\leftrightarrow }_{T}R(x,y)\right)\ge E_{X^2}(S,R).$$

Hence

$$d_T\left(R{|}_Y\right)\ge E_{Y^2}{\left(S\right|}_Y,R{|}_Y)\ge E_{X^2}(S,R).$$

Taking the supremum over $S\in \mathcal{T}\left(X\right)$ proves the result.  □

Thus every restriction provides an upper bound for the transitivity index of the full relation.

Finally, the upper transitive closure gives a concrete lower bound.

Proposition 10. 

Let ${cl}_T\left(R\right)=\bigwedge \{S\in \mathcal{T}\left(X\right)$:$R\le S\}$. Then ${cl}_T\left(R\right)$ is the smallest T-transitive fuzzy relation containing R, and $d_T\left(R\right)\ge E_{X^2}\left(R,{cl}_T\left(R\right)\right)$.

Proof. 

The constant-1 relation is T-transitive and contains R, so the defining family is nonempty. Let ${\left\{S_i\right\}}_{i\in I}\subseteq \mathcal{T}\left(X\right)$ and put $S={\bigwedge }_i{S}_i$ pointwise. For every fixed $j\in I$,

$$T(S(x,y),S(y,z))\le T(S_j(x,y),S_j(y,z))\le S_j(x,z).$$

Taking the infimum over j gives $T\left(S\right(x,y),S(y,z\left)\right)\le S(x,z)$, so arbitrary pointwise infima of T-transitive relations remain T-transitive. Therefore ${cl}_T\left(R\right)$ is the smallest T-transitive relation above R. Since it belongs to $\mathcal{T}\left(X\right)$,

$$d_T\left(R\right)\ge E_{X^2}\left({cl}_T\left(R\right),R\right)=E_{X^2}\left(R,{cl}_T\left(R\right)\right).$$

□

This result supplies an explicit lower bound whenever the upper T-transitive closure can be constructed or estimated.

4. The Relationship Between ${\mathit{d}}_{\mathit{T}}\left(\mathit{R}\right)$ and ${\mathit{\alpha }}_{\mathit{T}}\left(\mathit{R}\right)$

The two quantities evaluate different aspects of approximate transitivity. The degree ${\alpha }_T\left(R\right)$ is determined by the least satisfied triplewise transitivity condition, whereas $d_T\left(R\right)$ measures the best uniform fuzzy-equality level obtainable over the exact class $\mathcal{T}\left(X\right)$. We first show that the latter always dominates the former.

Proposition 11. 

Let T be a continuous t-norm and let R be a fuzzy relation on X. Then ${\alpha }_T\left(R\right)\le d_T\left(R\right)$.

Proof. 

By Propositions 2 and 3, the relation $R_{(T,{\alpha }_T\left(R\right))}$ is T-transitive and

$$E_{X^2}\left(R,R_{(T,{\alpha }_T\left(R\right))}\right)\ge {\alpha }_T\left(R\right).$$

Consequently,

$$d_T\left(R\right)=\underset{S\in \mathcal{T}\left(X\right)}{\bigvee }{E}_{X^2}(S,R)\ge E_{X^2}\left(R_{(T,{\alpha }_T\left(R\right))},R\right)\ge {\alpha }_T\left(R\right).$$

□

The inequality can be strict, as the following examples show.

Example 3. 

Let $T=T_{\mathsf{Ł}}$, $X=\{x,y,z\}$, and

$$S=\left(\begin{array}{ccc}1& 0.27& 0.72\\ 0.27& 1& 0.54\\ 0.72& 0.54& 1\end{array}\right),R=\left(\begin{array}{ccc}1& 0.3& 0.8\\ 0.3& 1& 0.6\\ 0.8& 0.6& 1\end{array}\right).$$

The relation S is $T_{\mathsf{Ł}}$-transitive. Direct calculation gives

$${\alpha }_{T_{\mathsf{Ł}}}\left(R\right)=0.9,E_{X^2}(S,R)=0.92.$$

Hence

$${\alpha }_{T_{\mathsf{Ł}}}\left(R\right)=0.9<0.92\le d_{T_{\mathsf{Ł}}}\left(R\right).$$

Example 4. 

Let $T=T_P$, $X=\{x,y,z\}$, and

$$S=\left(\begin{array}{ccc}1& 0.2& 0.52\\ 0.2& 1& 0.38\\ 0.52& 0.38& 1\end{array}\right),R=\left(\begin{array}{ccc}1& 0.3& 0.8\\ 0.3& 1& 0.6\\ 0.8& 0.6& 1\end{array}\right).$$

The relation S is product-transitive, and

$${\alpha }_{T_P}\left(R\right)=0.625,E_{X^2}(S,R)={\displaystyle \frac{19}{30}}.$$

Therefore

$${\alpha }_{T_P}\left(R\right)=0.625<{\displaystyle \frac{19}{30}}\le d_{T_P}\left(R\right).$$

The strict inequalities in Examples 3 and 4 raise a natural structural question: does there exist a continuous t-norm for which ${\alpha }_T\left(R\right)=d_T\left(R\right)$ holds for every fuzzy relation? To answer this question for non-Gödel t-norms, we first record the behavior of the residual implication inside an Archimedean ordinal sum component.

Lemma 1. 

Let T be a continuous t-norm, and let $[a,b]$, ϕ, and $T_{*}\in \{T_P,T_{\mathsf{Ł}}\}$ be as in Proposition 1. For $u,v\in [a,b]$,

$$u{\to }_{T}v=\left\{\begin{array}{cc}1,\hfill & u\le v,\hfill \\ {\varphi }^{-1}\left(\varphi \left(u\right){\to }_{T_{*}}\varphi \left(v\right)\right),\hfill & u>v,\hfill \end{array}\right.$$

and

$$u{\leftrightarrow }_{T}v=\left\{\begin{array}{cc}1,\hfill & u=v,\hfill \\ {\varphi }^{-1}\left(\varphi \left(u\right){\leftrightarrow }_{T_{*}}\varphi \left(v\right)\right),\hfill & u\ne v.\hfill \end{array}\right.$$

Proof. 

If $u\le v$, then $u{\to }_{T}v=1$ by Equation (2). Suppose that $u>v$. Put

$$q={\varphi }^{-1}\left(\varphi \left(u\right){\to }_{T_{*}}\varphi \left(v\right)\right).$$

Because $\varphi \left(u\right)>\varphi \left(v\right)$, the residual value $\varphi \left(u\right){\to }_{T_{*}}\varphi \left(v\right)$ is strictly less than 1, hence $q<b$. For $t\in [a,b]$, the ordinal sum representation and adjointness for $T_{*}$ give

$$\begin{array}{cc}\hfill T(u,t)\le v& ⟺T_{*}(\varphi \left(u\right),\varphi \left(t\right))\le \varphi \left(v\right)\hfill \\ \hfill & ⟺\varphi \left(t\right)\le \varphi \left(u\right){\to }_{T_{*}}\varphi \left(v\right)\hfill \\ \hfill & ⟺t\le q.\hfill \end{array}$$

If $t<a$, then $T(u,t)=t\le a\le v$, while if $t>b$, the ordinal sum operation gives $T(u,t)=u>v$. Thus the supremum of all t satisfying $T(u,t)\le v$ is exactly q, proving the residual formula. The bi-implication formula follows because one of the two residuals equals 1 when $u\ne v$, while both equal 1 when $u=v$.  □

Proposition 12. 

Let T be a continuous t-norm. The following are equivalent:

(i) 

${\alpha }_T\left(R\right)=d_T\left(R\right)$ for every fuzzy relation R on every universe X;

(ii) 

$T=T_M$.

Proof. 

(ii)⇒(i).

Assume $T=T_M$. Proposition 11 gives ${\alpha }_{T_M}\left(R\right)\le d_{T_M}\left(R\right)$.

If R is $T_M$-transitive, both sides equal 1. Suppose that R is not $T_M$-transitive and put $a={\alpha }_{T_M}\left(R\right)$.

For every violated triple $(x,y,z)$, $min\left\{R\right(x,y),R(y,z\left)\right\}>R(x,z)$.

For the Gödel residuum, the satisfaction degree of such a triple is exactly its consequent $R(x,z)$, whereas every nonviolated triple contributes 1. Hence

$$a=\underset{\begin{array}{c}x,y,z\in X\\ min\left\{R\right(x,y),R(y,z\left)\right\}>R(x,z)\end{array}}{\bigwedge }R(x,z).$$

Let S be any $T_M$-transitive relation and consider an arbitrary violated triple. Set $r=R(x,z)$. We show that $E_{X^2}(S,R)\le r$.

If instead $E_{X^2}(S,R)>r$, then, because $R(x,y)>r$ and $R(y,z)>r$, the formula for the Gödel bi-implication implies $S(x,y)>r$, $S(y,z)>r$.

At $(x,z)$, where $R(x,z)=r$, the inequality $S(x,z){\leftrightarrow }_{M}r>r$ can hold only if $S(x,z)=r$: if $S(x,z)\ne r$, the Gödel bi-implication equals $min\left\{S\right(x,z),r\}\le r$. Consequently,

$$min\left\{S\right(x,y),S(y,z\left)\right\}>r=S(x,z),$$

contradicting the $T_M$-transitivity of S. Thus $E_{X^2}(S,R)\le r$ for every violated triple, so $E_{X^2}(S,R)\le a$. Taking the supremum over all $T_M$-transitive S gives

$$d_{T_M}\left(R\right)\le a={\alpha }_{T_M}\left(R\right).$$

Together with Proposition 11, this proves equality.

(i)⇒(ii). Suppose that $T\ne T_M$. By Proposition 1, T has a nontrivial Archimedean component $[a,b]$ isomorphic either to $T_P$ or to $T_{\mathsf{Ł}}$ via an order isomorphism $\varphi :[a,b]\to [0,1]$.

First suppose that the component is of product type. On $X=\{1,2,3\}$, let

$$R_0=\left(\begin{array}{ccc}1& 0.3& 0.8\\ 0.3& 1& 0.6\\ 0.8& 0.6& 1\end{array}\right),S_0=\left(\begin{array}{ccc}1& 0.2& 0.52\\ 0.2& 1& 0.38\\ 0.52& 0.38& 1\end{array}\right).$$

As in Example 4, $S_0$ is product-transitive and ${\alpha }_{T_P}\left(R_0\right)=0.625$, $E_{X^2}^{T_P}(S_0,R_0)={\displaystyle \frac{19}{30}}$.

Define lifted relations by $R(i,j)={\varphi }^{-1}\left(R_0(i,j)\right)$, $S(i,j)={\varphi }^{-1}\left(S_0(i,j)\right)$.

Because all entries lie in $[a,b]$, the component formula shows that S is T-transitive. By Lemma 1, nonviolated triples of $R_0$ contribute residual value 1 after lifting, while violated triples contribute the image under ${\varphi }^{-1}$ of their product-residual values. Since the minimum product-residual value is $0.625<1$, ${\alpha }_T\left(R\right)={\varphi }^{-1}\left(0.625\right)$.

Similarly, equal matrix entries contribute bi-implication 1 and unequal entries are transformed by ${\varphi }^{-1}$; hence $E_{X^2}(S,R)={\varphi }^{-1}\left({\displaystyle \frac{19}{30}}\right)$.

Therefore

$${\alpha }_T\left(R\right)={\varphi }^{-1}\left(0.625\right)<{\varphi }^{-1}\left({\displaystyle \frac{19}{30}}\right)=E_{X^2}(S,R)\le d_T\left(R\right).$$

If the component is of Łukasiewicz type, use

$$R_0=\left(\begin{array}{ccc}1& 0.3& 0.8\\ 0.3& 1& 0.6\\ 0.8& 0.6& 1\end{array}\right),S_0=\left(\begin{array}{ccc}1& 0.27& 0.72\\ 0.27& 1& 0.54\\ 0.72& 0.54& 1\end{array}\right).$$

By Example 3, $S_0$ is $T_{\mathsf{Ł}}$-transitive and ${\alpha }_{T_{\mathsf{Ł}}}\left(R_0\right)=0.9$, $E_{X^2}^{T_{\mathsf{Ł}}}(S_0,R_0)=0.92$.

The same lifting argument gives

$${\alpha }_T\left(R\right)={\varphi }^{-1}\left(0.9\right)<{\varphi }^{-1}\left(0.92\right)=E_{X^2}(S,R)\le d_T\left(R\right).$$

Thus, whenever $T\ne T_M$, there exists a fuzzy relation for which ${\alpha }_T\left(R\right)<d_T\left(R\right)$. This contradicts (i), and therefore $T=T_M$.  □

The following numerical example illustrates what the inequality ${\alpha }_T\left(R\right)<d_T\left(R\right)$ can mean without claiming that a displayed transitive relation is an optimizer.

Example 5. 

Let $T=T_P$, $X=\{x,y,z\}$, and

$$R=\left(\begin{array}{ccc}1& 0.288& 0.75\\ 0.288& 1& 0.75\\ 0.75& 0.75& 1\end{array}\right).$$

Since $R(x,z)R(z,y)=0.75·0.75=0.5625>0.288=R(x,y)$,

R is not product-transitive. The most severe violated constraint gives

$${\alpha }_{T_P}\left(R\right)={\displaystyle \frac{0.288}{0.5625}}=0.512.$$

Now consider

$$S=\left(\begin{array}{ccc}1& {\displaystyle \frac{9}{25}}& {\displaystyle \frac{3}{5}}\\ {\displaystyle \frac{9}{25}}& 1& {\displaystyle \frac{3}{5}}\\ {\displaystyle \frac{3}{5}}& {\displaystyle \frac{3}{5}}& 1\end{array}\right).$$

The relation S is product-transitive because ${(3/5)}^2=9/25$. Moreover,

$$E_{X^2}(S,R)={\displaystyle \frac{4}{5}}.$$

Consequently,

$${\alpha }_{T_P}\left(R\right)=0.512<0.8=E_{X^2}(S,R)\le d_{T_P}\left(R\right).$$

This calculation proves a lower bound $d_{T_P}\left(R\right)\ge 0.8$; it does not prove that the displayed S is a maximizing transitive relation or that $d_{T_P}\left(R\right)=0.8$. Its role is only to show that simultaneously modifying several entries can produce an exactly product-transitive relation with uniform fuzzy-equality level $0.8$, already higher than the traditional degree $0.512$. The example therefore illustrates the difference between a worst violated constraint and a uniform approximation to the transitive class, without interpreting $E_{X^2}$ as the total amount of modification.

5. Conclusions

We introduced a fuzzy-equality-based index of T-transitivity for continuous t-norms. The index is best interpreted as a supremal uniform or maximin similarity to the exact class of T-transitive relations. The optimization ranges over the whole transitive class, but the equality functional is determined by the least coordinatewise similarity; accordingly, $d_T$ is not an average or total measure of the entries that must be changed.

We proved that $d_T\left(R\right)=1$ if and only if R is T-transitive, so the index is faithful to the exact property. We also established stability under fuzzy equality, invariance under inversion, monotonicity under restriction, and a lower bound obtained from the upper T-transitive closure. In addition, when the universe is finite, compactness of the transitive class together with upper semicontinuity of the equality functional shows that the supremum defining $d_T\left(R\right)$ is attained. For arbitrary universes, no general attainment claim is made.

The comparison with the traditional degree ${\alpha }_T\left(R\right)$ shows that ${\alpha }_T\left(R\right)\le d_T\left(R\right)$ for every continuous t-norm. The two quantities have different interpretations: ${\alpha }_T$ records the least satisfied triplewise transitivity constraint, whereas $d_T$ measures the best uniform fuzzy-equality level over exactly transitive relations. Product and Łukasiewicz examples demonstrate that the inequality may be strict. We further proved the characterization

$${\alpha }_T\left(R\right)=d_T\left(R\right)\mathrm{for}\mathrm{every}\mathrm{fuzzy}\mathrm{relation}\mathrm{on}\mathrm{every}\mathrm{universe}⟺T=T_M.$$

Thus the Gödel t-norm is the unique continuous t-norm for which the local worst-constraint degree and the uniform approximation index coincide universally.

The present contribution is theoretical. Future work may develop algorithms for evaluating or approximating $d_T\left(R\right)$ on finite universes, study their computational complexity, investigate uniqueness and structure of maximizing transitive approximations in cases where maximizers exist, and extend the construction to broader classes of residuated operations. The index may also provide a basis for future studies of noisy preference, similarity, clustering, or recommendation relations, but such applications require separate computational and empirical analysis.