A Systematic Study on Distributivity of Threshold-Generated Implications over Uninorms

Abstract

The distributivity of implications over fuzzy operators is a desirable property for fuzzy systems and can be employed in the elimination of the explosion of if–then rules. In this paper, we try to explore the relationship between the distributivity over the uninorms-related fuzzy connectives and the distributivity over uninorms in the threshold generation method, i.e., the distributive equations $I(u,U_1(v,w))=U_2(I(u,v),I(u,w))$ and $I(U_1(u,v),w))=U_2(I(u,w),I(v,w))$ with I being the threshold-generated implication. Consequently, we find that if the uninorms are restricted to special classes, then the distributivity property by the first equation can be preserved between the original and threshold-generated implications; under certain constraints on the threshold-generated implication, the distributivity property by the second equation becomes trivial.

1. Introduction

Given the inherent incompleteness and uncertainty in our understanding of complex systems, one viable approach is to utilize expert knowledge expressed through natural language constructs, particularly if–then statements. Therefore, fuzzy implications [1,2], with which if–then statements can be well formalized, are essential for the operation of intelligent systems, approximate reasoning, and decision analysis.

1.1. Generated Fuzzy Implications

It is crucial to offer a broad range of fuzzy implications, as their adequacy depends on the specific conditional rule behavior they must model and the inference rules to be employed. These implications are commonly derived from fuzzy connectives like t-(co)norms and negations, with the R-, (S, N)-, QL-, and D-implications representing the well-established classes. Further, fuzzy implications could also be constructed using other binary operators, such as copulas, quasi-copulas, more general conjunctors [3], representable aggregation functions [4], uninorms [5,6,7,8], and overlap functions [9,10]. Based on generating functions, Yager [11] has introduced the f-,g-generated implications, which were extensively explored in [12,13,14,15,16]. Since then, various generation methods are proposed along the lines of Yager’s implications. Massanet and Torrens [17] introduced h-implications, which are found to have the validity of some axioms derived from classical fuzzy logic tautologies, especially those related fuzzy implications. Liu introduced (g,min)-implications [18] and (h,min)-implications [19], using g-generators and generalized h-generators, respectively. In another paper [20], leveraging the generalized additive generator h of a representable uninorm, she introduced the so-called $h^{-1}$-implications. A series of necessary and sufficient conditions, under which the three classes of implications satisfy several classical tautologies, is derived. Meanwhile, by an f-generator and an increasing function $g:[0,1]\to [0,1]$ satisfying $g\left(0\right)=0$ and $g\left(1\right)=1$, Xie and Liu [21] introduced $(f,g)$-implication which is extension of f-generated implication by Yager. With a g-generator and a u-operator of g, Zhang and Liu [22] developed the $(g,u)$-implications. With the interpretation of $\neg (A\bigwedge \neg B)$ and using two fuzzy negations (f and g) and a uninorm U, Hliněná et al. [23] generalized f-generated implication and developed fuzzy implications $I_{U,f,g}$.

Beyond the standard classes, new fuzzy implications can be generated from existing ones using various methods, including the conjugation, the min (or max) operation, the convex combination, and the composition of fuzzy implication(s).

Massanet and Torrens introduced two related construction techniques: the threshold generation method (in [24]), which uses an adequate transformation on the second variable of two original fuzzy implications, and the vertical threshold generation method (in [25]), which applies an adequate transformation on the first variable. Su et al. [26] proposed the ordinal sum of implications which is much similar to the ordinal sum of t-(co)norms [27]. In the very recent, Cheng et al. [28] constructed implications by vertical ordinal sum method and deformed vertical ordinal sum method.

1.2. Distributivity of Fuzzy Implications over Fuzzy Connectives

To handle the rule explosion in fuzzy reasoning, Combs and Andrews [29] took advantage of the axiom $(m\bigwedge n)\to r=(m\to r)\bigvee (n\to r)$ and asserted that the two-rule configurations, i.e., IRC and URC, are equivalent in fuzzy inference, which sparked intense discussion regarding the necessity for theoretical investigation before such equations could be properly employed. Subsequently, in the standard fuzzy sets theory, Trillas & Alsina [30] explored the fuzzy version of the above axiom:

$$I\left(T\right(u,v),w)=S\left(I\right(u,w),I(v,w\left)\right),$$

(1)

with t-norm T, t-conorm S, and fuzzy implication I, in addition to all solutions of T and S, given that I is a residual implication, an S-implication, and a $QL$-implication, respectively. Then, Balasubramaniam and Rao [31] explored the following three distributive equations closely related to Equation (1).

$$\begin{array}{c}I\left(S\right(u,v),w)=T\left(I\right(u,w),I(v,w\left)\right)\end{array}$$

(2)

$$\begin{array}{c}I(u,T_1(v,w))=T_2(I(u,v),I(u,w))\end{array}$$

(3)

$$\begin{array}{c}I(u,S_1(v,w))=S_2(I(u,v),I(u,w))\end{array}$$

(4)

with t-norms $T\mathrm{and}{T}_i (i=1,2)$, t-conorms $S\mathrm{and}{S}_i (i=1,2)$ and a fuzzy implication I. Actually, in the much earlier time, Turksen et al. proposed some new classes of implications with some new axioms, including Equation (3) and obtained necessary conditions for a fuzzy implication I to satisfy

$$I(u,T(v,w\left)\right)=T\left(I\right(u,v),I(u,w\left)\right)$$

(5)

with T is the product t-norm. Further, Baczyński focused on strict t-norm T and presented the sufficient and necessary conditions of a system of equations including Equation (5) and $I(u,I(v,w\left)\right)=I\left(T\right(u,v),w)$. Subsequently, a full characterization of Equation (5) and $I(u,v)=I\left(N\right(v),N(u\left)\right)$ was obtained step by step, which can be found in [32,33]. And then there is a flourish of investigations about the distributivity properties of fuzzy implications with respect to fuzzy operators, including the t-(co)norms [33,34,35,36,37,38,39,40], the uninorms [41,42]. Additionally, Baczyński tried to connect these works with the connectives in interval-valued fuzzy sets theory [34,43].

1.3. Main Motivations of the Work

Since uninorms are natural generalizations of t-(co)norm, the distributivity equations with respect to uninorms are also the focus of the research. Ruiz-Aguilera and Torrens [44,45,46] explored the aforementioned equation with uninorms. That is

$$I(U_1(u,v),w))=U_2(I(u,w),I(v,w))$$

(6)

$$I(u,U_1(v,w))=U_2(I(u,v),I(u,w)),$$

(7)

in which $U_i$ is a uninorm ($i=1,2$) and I is the strong implication and the residual implication, respectively. In [41], Baczyński presented a characterization for Equations (6) and (7) when $U_i$ is a given representable uninorm ($i=1,2$) and I is an operator like a fuzzy implication. Intrigued by those above, we consider the distributivity of the threshold-generated implications over uninorms.

As shown in [24,47], under some minimal necessary conditions, the threshold-generated implications preserve most established properties, particularly the distributivity properties over the t-norms and the t-conorms. Along the lines of thought, we intend to check the distributivity over uninorms when the involved implications are the threshold-generated implications. Specifically, we try to explore the relationship between the distributivity over the uninorm-related fuzzy connectives and the distributivity over uninorms with Equations (6) and (7). We refer to the four classes of uninorms, i.e., the uninorms in ${\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$, the representable uninorms, the idempotent uninorms, and the uninorms continuous in $]0,1[\times ]0,1[$. Consequently, we find that if the uninorms are restricted to special classes, then the distributivity property by Equation (7) can be preserved between the original and threshold-generated implications; under certain constraints on the threshold-generated implication, the distributivity property by Equation (6) becomes trivial. However, with the byproducts, some results related to h-implication are obtained.

1.4. Outline of the Work

In Section 2, some notations on the threshold-generated fuzzy implications and the structures of the uninorms are reviewed. In Section 3, we concentrate on Equation (7) and demonstrate the relationships between threshold generated implication and the original implications about the preservation on distributivity in three cases. Section 4 deals with Equation (6). At the end, some concluding remarks are presented.

2. Preliminary

For convenience, we list some notations that are used in the paper in Table 1.

Table 1. Main notations.

Notation	Description
I	Fuzzy implication
$I_f$	f-implication
$I_g$	g-implication
$I_h$	h-implication
$I_{I_1-I_2}$	e-threshold-generated implication
$I_E$	E-threshold-generated implication
T	t-norm
S	t-conorm
U	uninorm
${\mathcal{U}}_R$	Set of representable uninorms
${\mathcal{U}}_{cos}$	Set of uninorms continous in $]0,1[\times ]0,1[.$
${\mathcal{U}}_I$	Set of idempotent uninorms

T-norms and t-conorms are basic operators in information aggregating [48,49,50,51,52,53,54] in various applications. For the details about these operators, we recommend the monographs [2,27,55]. We only recall some of the notations and results that are used.

Definition 1

([1,2]). A function $I:[0,1]\times [0,1]\to [0,1]$ is a fuzzy implication (or implication), if the following holds:

$$\begin{array}{c}I(u,w)\ge I(v,w)\mathit{for}u\le v\end{array}$$

(8)

$$\begin{array}{c}I(u,v)\ge I(u,w)\mathit{for}v\ge w\end{array}$$

(9)

$$\begin{array}{c}I(0,0)=1,\end{array}$$

(10)

$$\begin{array}{c}I(1,1)=1,\end{array}$$

(11)

$$\begin{array}{c}I(1,0)=0.\end{array}$$

(12)

By Equations (8)–(12), we have $I(0,0)=I(0,1)=I(1,1)=1$. Therefore, a fuzzy implication is a natural extension of the classical one. We only recall the f-,g- and h-implications here.

Definition 2

([11,17]).

(i) 

Let $f:[0,1]\to [0,+\infty ]$ be a strictly decreasing and continuous operator satisfying $f\left(1\right)=0$, then $I_f:[0,1]\times [0,1]\to [0,1]$ given by $I_f(u,v)=f^{-1}\left(uf\left(v\right)\right)$, with the interpretation $0·+\infty =0$ is said to be an f-implication and f is an f-generator.

(ii) 

Let $g:[0,1]\to [0,+\infty ]$ be a strictly increasing and continuous function such that $g\left(0\right)=0$, then $I_g:[0,1]\times [0,1]\to [0,1]$ given by $I_g(u,v)=g^{-1}\left({\displaystyle \frac{1}{u}}g\left(v\right)\right)$, with the interpretation ${\displaystyle \frac{1}{0}}=+\infty ,+\infty ·0=0$, is said to be a g-implication and g is said to be a g-generator.

(iii) 

Fix an $e\in (0,1)$ and Let $h:[0,1]\to [-\infty ,+\infty ]$ be a strictly increasing and continuous function such that $h\left(0\right)=-\infty ,h\left(e\right)=0$ and $h\left(1\right)=+\infty$, then $I_h:[0,1]\times [0,1]\to [0,1]$ given by $I_h(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0;\hfill \\ h^{-1}\left(uh\left(v\right)\right),\hfill & u>0,v\le e;\hfill \\ h^{-1}\left({\displaystyle \frac{1}{u}}h\left(v\right)\right),\hfill & u>0,v>e;\hfill \end{array}\right.$ is said to be an h-implication and h is said to be an h-generator.

In fact, the f-generator, the g-generator, and the h-generator are the additive generators of some continuous Archimedean t-norms, some continuous Archimedean t-conorms, and some representable uninorms, respectively. For convenience, we list some of the most referred to implications in Table 2, some of which will be used in the rest of the paper.

Table 2. Basic Implications.

Name	Formula
ukasiewicz	$I_{LK}(u,v)=min(1,1-u+v)$
Reichenbach	$I_{RC}(u,v)=1-u+uv$
Kleene-Dienes	$I_{KD}(u,v)=max(1-u,v)$
Gödel	$I_{GD}(u,v)=\left\{\begin{array}{cc}1,\hfill & u\le v;\hfill \\ v,\hfill & u>v;\hfill \end{array}\right.$
Goguen	$I_{GG}(u,v)=\left\{\begin{array}{cc}1,\hfill & u\le v;\hfill \\ {\displaystyle \frac{v}{u}},\hfill & u>v;\hfill \end{array}\right.$
Rescher	$I_{RS}(u,v)=\left\{\begin{array}{cc}1,\hfill & u\le v;\hfill \\ 0,\hfill & u>v;\hfill \end{array}\right.$
Yager	$I_{YG}(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0andv=0;\hfill \\ v^u,\hfill & u>0orv>0;\hfill \end{array}\right.$
Weber	$I_{WB}(u,v)=\left\{\begin{array}{cc}1,\hfill & u<1;\hfill \\ v,\hfill & u=1;\hfill \end{array}\right.$
Foder	$I_{FD}(u,v)=\left\{\begin{array}{cc}1,\hfill & u\le v;\hfill \\ max(1-u,v)\hfill & u>v;\hfill \end{array}\right.$
Least	$I_{Lt}(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0orv=1;\hfill \\ 0\hfill & u>0andv<1;\hfill \end{array}\right.$
Greatest	$I_{Gt}(u,v)=\left\{\begin{array}{cc}1,\hfill & u<1orv>0;\hfill \\ 0\hfill & u=1andv=0;\hfill \end{array}\right.$

It is vital to provide plenty of suitable implications for various applications so that many methods to generate an implication from the given ones are proposed. Here, we only recall the threshold generation method.

Theorem 1

([24,47]). (Threshold generation method) Let $I_1,I_2$ be implications with $e\in \left]0,1\right[$. The binary operator $I_{I_1-I_2}:[0,1]\times [0,1]\to [0,1]$ by

$$I_{I_1-I_2}(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0;\hfill \\ e{I}_1(u,{\displaystyle \frac{v}{e}})\hfill & u>0\mathit{and}0\le v\le e;\hfill \\ e+(1-e)I_2(u,{\displaystyle \frac{v-e}{1-e}}),\hfill & u>0\mathit{and}v>e\hfill \end{array}\right.$$

(13)

is a fuzzy implication, which is said to be the e-threshold-generated implication from $I_1$ and $I_2$,

The properties of the threshold-generated implications can be found in [24,47]. In the last equation, $e\in ]0,1[$ is used; the method can be considered to be a single-valued threshold generation. The multi-valued version of the method can be found in [56].

Theorem 2

([56,57]). Let $n\in \mathbb{N}$ and ${\left\{I_k\right\}}_{0\le k\le n}$ be a set of implications, $E=\{e_i\mid i=1,2,\dots n,n\in \mathbb{N}\}$ with $0<e_1<e_2<\cdots <e_n<e_{n+1}=1$; then, the binary operator $I_E:[0,1]\times [0,1]\to [0,1]$ by

$$I_E(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0;\hfill \\ e_1{I}_0(u,{\displaystyle \frac{v}{e_1}})\hfill & u>0,0\le v\le e_1;\hfill \\ e_k+(e_{k+1}-e_k)I_k(u,{\displaystyle \frac{v-e_k}{e_{k+1}-e_k}}),\hfill & u>0,e_k<v\le e_{k+1}fork\ge 1,\hfill \end{array}\right.$$

(14)

is a fuzzy implication, which is said to be the E-generated implication.

Since the set E is finite, the extended threshold generation method can also be considered to be an iterative process of the single-valued threshold generation method. In what follows, we will also use the two-valued threshold-generated implications. For convenience, we denote the implications generated by single-valued (e) and the multi-valued threshold-generated implications by $I_{I_1-I_2}$ and $I_E$, respectively.

In the following, we will use the neutrality property (NP) of implication. Namely, for $u\in \left[0,1\right]$

$$I(1,u)=u,$$

(15)

where I is an implication.

As a well-established aggregating function, uninorm is a commutative, increasing, associative binary operator U in the unit square with the neutral element $e_u$ lying in $[0,1]$ and can be employed in various fields. A uninorm U is conjunctive (disjunctive) if $U(1,0)=0$ (1) and it has a combinational structure of a t-norm and a t-conorm [58]. Here, we list the structural results on some commonly used uninorms.

Theorem 3

([59]). A binary operator $U:[0,1]\times [0,1]\to [0,1]$ is a conjunctive (or disjunctive) uninorm with neutral element $e_u\in ]0,1]$ with $U(·,1)$ being continuous on $[0,e_u[$ if and only if there is a t-norm T and a t-conorm S such that

$$U(u,v)=\left\{\begin{array}{cc}{e}_{u}T({\displaystyle \frac{u}{e_u}},{\displaystyle \frac{v}{e_u}}),\hfill & (u,v)\in {[0,e_u]}^2;\hfill \\ e_u+(1-e_u)S({\displaystyle \frac{u-e_u}{1-e_u}},{\displaystyle \frac{v-e_u}{1-e_u}})\hfill & (u,v)\in {[e_u,1]}^2;\hfill \\ min(u,v)(ormax(u,v\left)\right),\hfill & else\hfill \end{array}\right.$$

(16)

The class of conjunctive and disjunctive uninorms are denoted by ${\mathcal{U}}_{min}$ and ${\mathcal{U}}_{max}$, respectively.

Theorem 4

([58]). Let U be a uninorm whose neutral element is $e_u\in ]0,1[$. There is a strictly increasing continuous function $h:[0,1]\to [-\infty ,+\infty ]$ satisfying $h\left(0\right)=-\infty ,h\left(e_u\right)=0$, and $h\left(1\right)=+\infty$ such that

$$U(u,v)=h^{-1}(h\left(u\right)+h\left(v\right))$$

(17)

for $(u,v)\in [0,1]\times [0,1]-\left\{\right(0,1),(1,0\left)\right\}$ if and only if

(i) U is strictly increasing and continuous on $]0,1[\times ]0,1[$;

(ii) there is an involutive negation N with the fixed point being $e_u$, such that

$$U(u,v)=N\left(U\right(N\left(u\right),N\left(v\right)\left)\right),$$

for $(u,v)\in [0,1]\times [0,1]-\left\{\right(0,1),(1,0\left)\right\}$

The uninorms in the above theorem are said to be representable and h is an additive generator. Denote the set of the representable uninorms by ${\mathcal{U}}_R$.

Theorem 5

([60]). Let U be a uninorm continuous in $]0,1[\times ]0,1[$ with the neutral element $e_u\in ]0,1[.$ Then one of the following holds:

(i) 

There are $\beta \in [0,e_u[\mathrm{and}\alpha \in [0,\beta ]$, two continuous t-norms $T_1$ and $T_2$, and a representable uninorm R such that the uninorm U can be characterized by

$$U(u,v)=\left\{\begin{array}{c}\alpha T_1({\displaystyle \frac{u}{\alpha }},{\displaystyle \frac{v}{\alpha }}),(u,v)\in {]0,\alpha [}^2;\hfill \\ \alpha +(\beta -\alpha )T_2({\displaystyle \frac{u-\alpha }{\beta -\alpha }},{\displaystyle \frac{v-\alpha }{\beta -\alpha }}),(u,v)\in {]\alpha ,\beta [}^2;\hfill \\ \beta +(1-\beta )R({\displaystyle \frac{u-\beta }{1-\beta }},{\displaystyle \frac{v-\beta }{1-\beta }}),(u,v)\in {]\beta ,1[}^2;\hfill \\ 1,min(u,v)\in ]\alpha ,1]andmax(u,v)=1;\hfill \\ \alpha or1,(u,v)\in \left\{\right(\alpha ,1),(1,\alpha \left)\right\};\hfill \\ min(u,v),else.\hfill \end{array}\right.$$

(18)

(ii) 

There are $\gamma \in ]e_u,1]\mathrm{and}\delta \in [\gamma ,1]$, two continuous t-conorms $S_1$ and $S_2$, and a representable uninorm R such that U can be characterized by

$$U(u,v)=\left\{\begin{array}{c}\gamma R({\displaystyle \frac{u}{\gamma }},{\displaystyle \frac{v}{\gamma }}),(u,v)\in {]0,\gamma [}^2;\hfill \\ \gamma +(\delta -\gamma )S_1({\displaystyle \frac{u-\gamma }{\delta -\gamma }},{\displaystyle \frac{v-\gamma }{\delta -\gamma }}),(u,v)\in {]\gamma ,\delta [}^2;\hfill \\ \delta +(1-\delta )S_2({\displaystyle \frac{u-\delta }{1-\delta }},{\displaystyle \frac{v-\delta }{1-\delta }}),(u,v)\in {]\delta ,1[}^2;\hfill \\ 0,max(u,v)\in [0,\delta [andmin(u,v)=0;\hfill \\ \delta or0,(u,v)\in \left\{\right(0,\delta ),(\delta ,0\left)\right\};\hfill \\ max(u,v),else.\hfill \end{array}\right.$$

(19)

Denote the set of these uninorms by ${\mathcal{U}}_{cos}$ and in particular the set of these uninorms with Equations (18) and (19), which are denoted by ${\mathcal{U}}_{cos,min}$ and ${\mathcal{U}}_{cos,max}$, respectively. Specifically, a uninorm U in ${\mathcal{U}}_{cos,min}$ (or ${\mathcal{U}}_{cos,max}$) will be represented by $U={⟨T_1,\alpha ,T_2,\beta ,(R,e_u)⟩}_{cos,min}$ (or $U={⟨(R,e_u),\gamma ,S_1,\delta ,S_2⟩}_{cos,max}$). It must be mentioned that ${\mathcal{U}}_R\subset {\mathcal{U}}_{cos}$ since the uninorm U reduces to the representable uninorm if $\beta =0$ in Equation (18) or $\delta =1$ in Equation (19).

Idempotency is one of the most referred properties of fuzzy operators in the literature. De Baets firstly provided a full characterization of the left-continuous idempotent uninorms and the right-continuous idempotent uninorms in [59]. Then Martin et al. characterized the idempotent uninorms in [61].

Theorem 6

([61]). U is an idempotent uninorm whose neutral element is $e_u\in [0,1]$ if and only if there is a decreasing function $g:[0,1]\to [0,1]$ satisfying $g\left(u\right)=1$ for $u<g\left(1\right)$, $g\left(e_u\right)=e_u,andg\left(u\right)=0$ for $u>g\left(0\right)$, such that $inf\{v\mid g(v)=g(u\left)\right\}\le g\left(g\right(u\left)\right)\le sup\{v\mid g(v)=g(u\left)\right\}$ for $u\in [0,1]$ and

$$U(u,v)=\left\{\begin{array}{c}min(u,v),v<g\left(u\right),orv=g\left(u\right)andu<g\left(g\right(u\left)\right);\hfill \\ max(u,v),v>g\left(u\right),orv=g\left(u\right)andu>g\left(g\right(u\left)\right);\hfill \\ min(u,v)ormax(u,v),v=g\left(u\right)andu=g\left(g\right(u\left)\right).\hfill \end{array}\right.$$

Denote the set of the idempotent uninorms by ${\mathcal{U}}_I$.

3. On the Distributive Equation $\mathit{I}(\mathit{u},{\mathit{U}}_{\mathit{1}}(\mathit{v},\mathit{w}))={\mathit{U}}_{\mathit{2}}(\mathit{I}(\mathit{u},\mathit{v}),\mathit{I}(\mathit{u},\mathit{w}))$

Proposition 1.

Let $U_1,U_2$ be uninorms and $I_1,I_2$ be two fuzzy implications satisfying (NP), i.e., $I_i(1,v)=v$ for $v\in [0,1],(i=1,2)$. If the triple $(I_{I_1-I_2},U_1,U_2)$ satisfies Equation (7), then $U_1=U_2$.

Proof. 

According to Equation (13) and the neutral property of $I_1$ and $I_2$, we have for $v\in [0,1]$, $I_{I_1-I_2}(1,v)=v$. That is, $I_{I_1-I_2}$ satisfies (NP). Consequently, let $u=1$ and $I=I_{I_1-I_2}$ in Equation (7); we have $I_{I_1-I_2}(1,U_1(v,w))=U_2(I_{I_1-I_2}(1,v),I_{I_1-I_2}(1,w))$, which implies $U_1(v,w)=U_2(v,w)$ through the neutral property of $I_{I_1-I_2}$.□

Under the referred requirements, the two uninorms in Equation (7) are the same. Moreover, with the preservation of the neutral property [47] in the threshold generation method, we can easily get the neutral property of the threshold-generated implication if the neutral property of all original implications holds. Therefore, it is enough to consider

$$I(u,U(v,w\left)\right)=U\left(I\right(u,v),I(u,w\left)\right)$$

(20)

in which U is a uninorm with the neutral element being $e_u$ while I is a threshold-generated implication with the neutral property.

Proposition 2.

Assume $I_1$ and $I_2$ are fuzzy implications and $I_{I_1-I_2}$ satisfies Equation (20). For each $i\in \{1,2\}$, either $I_i$ is continuous except maybe at $(0,0)$, or there is $u_0\in ]0,1[$ such that $I_i(u_0,v)\ne v$ for $v\in ]0,1[$, then $U(e,e)=0$ or $U(e,e)=1$ or $U(e,e)=e$.

Proof. 

Let $u>0,v=w=e$ in Equation (20), then $I_{I_1-I_2}(u,U(e,e))=U(I_{I_1-I_2}(u,e),I_{I_1-I_2}$ $(u,e))=U(e,e)$.

(i)

For $U(e,e)\le e$, it follows that $e{I}_1(u,{\displaystyle \frac{U(e,e)}{e}})=U(e,e)$, i.e., $I_1(u,{\displaystyle \frac{U(e,e)}{e}})={\displaystyle \frac{U(e,e)}{e}}$.

(a)

If $I_1$ is continuous except maybe at $(0,0)$ and $U(e,e)>0$; then, there is $v_0={\displaystyle \frac{U(e,e)}{e}}$ such that $I_1(u,v_0)=v_0$ for $u>0$. Since $I_1$ is continuous at $(0,v_0)$ and $I_1(0,v_0)=1$, then $v_0={lim}_{u\to 0^{+}}{I}_1(u,v_0)=I_1(0,v_0)=1$, and consequently $U(e,e)=e$.

(b)

If there is $u_0\in ]0,1[$ such that $I_1(u_0,v)\ne v$ for $v\in ]0,1[$, then $v_0=0$ or $v_0=1$ with the fact that $I_1(u_0,v_0)=v_0$ and $v_0={\displaystyle \frac{U(e,e)}{e}}>0$. Thus $U(e,e)=e$ or $U(e,e)=0$.

(ii)

For $U(e,e)>e$, it follows that $e+(1-e)I_2(u,{\displaystyle \frac{U(e,e)-e}{1-e}})=U(e,e)$, i.e., $I_2(u,{\displaystyle \frac{U(e,e)-e}{1-e}})={\displaystyle \frac{U(e,e)-e}{1-e}}$, where ${\displaystyle \frac{U(e,e)-e}{1-e}}$ is denoted by $v_0$ and $v_0>0$.

(a)

If $I_2$ is continuous except maybe at $(0,0)$, and here there is $v_0>0$ such that $I_2(u,v_0)=v_0$ for $u>0$; then, we have $v_0=1$ with the fact that $I_2(0,v_0)=1$ and $I_2$ is continuous at the point $(0,v_0)$.

(b)

If there is $u_0\in ]0,1[$ such that $I_2(u_0,v)\ne v$ for $v\in ]0,1[$ and here we have $I_2(u_0,v_0)=v_0$, then $v_0=0$ or $v_0=1$; consequently, $U(e,e)=e$ or $U(e,e)=1$.

□

We provide some examples on solutions for Equation (20) for the uninorm U with $U(e,e)=0$ and $U(e,e)=1$, respectively.

Example 1.

(i) 

Let $e_u={\displaystyle \frac{2}{3}}$, $e={\displaystyle \frac{1}{3}}$ and the weak negation $N\left(u\right)=\left\{\begin{array}{cc}1,\hfill & u=0\hfill \\ {\displaystyle \frac{1}{2}},\hfill & 0<u\le {\displaystyle \frac{1}{2}};\hfill \\ 0\hfill & else;\hfill \end{array}\right.$, and the t-norm $T(u,v)=\left\{\begin{array}{cc}0,\hfill & u\le N\left(v\right);\hfill \\ min(u,v)\hfill & \mathit{else};\hfill \end{array}\right.$ and the uninorm $U(u,v)=\left\{\begin{array}{cc}{\displaystyle \frac{2}{3}}T({\displaystyle \frac{3u}{2}},{\displaystyle \frac{3v}{2}}),\hfill & (u,v)\in {[0,{\displaystyle \frac{2}{3}}]}^2;\hfill \\ max(u,v),\hfill & \mathit{else}.\hfill \end{array}\right.$. Assume $I_h$ be the h-implication with $h=ln{\displaystyle \frac{u}{1-u}}$, then it can be checked that $(I_{I_{YG}-I_h},U)$ satisfies Equation (20) with $U({\displaystyle \frac{1}{3}},{\displaystyle \frac{1}{3}})=0$.

(ii) 

Let $e_u={\displaystyle \frac{1}{3}}$, $e={\displaystyle \frac{2}{3}}$ and the function $f\left(u\right)=\left\{\begin{array}{cc}1,\hfill & 0\le u<{\displaystyle \frac{1}{2}}\hfill \\ {\displaystyle \frac{1}{2}},\hfill & {\displaystyle \frac{1}{2}}\le u<1;\hfill \\ 0,\hfill & u=1;\hfill \end{array}\right.$ and the t-conorm $S(u,v)=\left\{\begin{array}{cc}1,\hfill & u\ge f\left(v\right);\hfill \\ max(u,v),\hfill & \mathit{else};\hfill \end{array}\right.$ and the uninorm

$$U(u,v)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{3}}+{\displaystyle \frac{2}{3}}S({\displaystyle \frac{3u-1}{2}},{\displaystyle \frac{3v-1}{2}}),\hfill & (u,v)\in {[{\displaystyle \frac{1}{3}},1]}^2;\hfill \\ min(u,v),\hfill & \mathit{else}.\hfill \end{array}\right.$$

in which $U(e,e)=1$. Assume $I_h$ be the h-implication with $h=ln{\displaystyle \frac{u}{1-u}}$; then, it can be checked directly that $(I_{I_h-I_{YG}},U)$ satisfies Equation (20) with $U({\displaystyle \frac{2}{3}},{\displaystyle \frac{2}{3}})=1$.

In the sequel, we restrict the discussion to the case $U(e,e)=e$, where e is the value in the threshold generation method. In order to show the relationships between the threshold generation implication $I_{I_1-I_2}$ and the original implications $I_1$ and $I_2$ on the distributivity, we consider three cases, i.e., $e_u=e$, $e_u<e$, and $e_u>e$.

3.1. The Case $U(e,e)=e$ with $e=e_u$

Here, we consider Equation (20) using different classes of uninorms, viz, the uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$, the uninorm $U\in {\mathcal{U}}_{cos}$ including the representable uninorm, and the idempotent uninorm U. It can be obtained that the threshold generation method preserves the distributivity about Equation (20) for $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$ while the preservation of distributivity does not hold for the others.

3.1.1. The Uninorms $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$

Here, we consider the case that the uninorms $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$. For the case that $U(e,e)=e=e_u$, then $T(u,v)={\displaystyle \frac{U(eu,ev)}{e}}$ is a t-norm, and $S(u,v)={\displaystyle \frac{U(e+(1-e)u,e+(1-e\left)v\right)-e}{1-e}}$ is a t-conorm. The following result completely characterizes the solutions of Equation (20) with the uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$ and it guarantees the preservation of the distributivity property from the original implications to the threshold-generated implication.

Proposition 3.

Let $I_1,I_2$ be two fuzzy implications, the uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$ with the neutral element $e_u=e$, then the couple $(I_{I_1-I_2},U)$ satisfies Equation (20) if and only if the couple $(I_1,T)$ satisfies Equation (5) and

$$I_2(u,S(v,w))=S(I_2(u,v),I_2(u,w))$$

(21)

for $v,w>0$.

Proof. 

(⇐) If $u=0$, then $I_{I_1-I_2}(0,U(v,w))=1=U(1,1)=U(I_{I_1-I_2}(0,v),I_{I_1-I_2}(0,w))$. If $u>0$, then we consider, respectively, the cases as follows.

(i)

For $v,w\le e$, it holds that

$$I_{I_1-I_2}(u,U(v,w))=e{I}_1(u,{\displaystyle \frac{U(v,w)}{e}})=e{I}_1(u,T({\displaystyle \frac{v}{e}},{\displaystyle \frac{w}{e}}))$$

$$=eT(I_1(u,{\displaystyle \frac{v}{e}}),I_1(u,{\displaystyle \frac{w}{e}}))=eT({\displaystyle \frac{I_{I_1-I_2}(u,v)}{e}},{\displaystyle \frac{I_{I_1-I_2}(u,w)}{e}})=U(I_{I_1-I_2}(u,v),I_{I_1-I_2}(u,w)).$$

(ii)

For $v,w>e$, it holds that $U(v,w)>e$ and

$$I_{I_1-I_2}(u,U(v,w))=e+(1-e)I_2(u,{\displaystyle \frac{U(v,w)-e}{1-e}})=e+(1-e)I_2(u,S({\displaystyle \frac{v-e}{1-e}},{\displaystyle \frac{w-e}{1-e}}))$$

$$=e+(1-e)S(I_2(u,{\displaystyle \frac{v-e}{1-e}}),I_2(u,{\displaystyle \frac{w-e}{1-e}}))=e+(1-e)S({\displaystyle \frac{I_{I_1-I_2}(u,v)-e}{1-e}},{\displaystyle \frac{I_{I_1-I_2}(u,w)-e}{1-e}})$$

$$=U(I_{I_1-I_2}(u,v),I_{I_1-I_2}(u,w)).$$

(iii)

For $v\le e,w>e$, since $I_{I_1-I_2}$ is increasing with respect to the second variable, then there are two subcases to be considered: if $U\in {\mathcal{U}}_{min}$, then it holds that

$$I_{I_1-I_2}(u,U(v,w))=I_{I_1-I_2}(u,min(v,w))=I_{I_1-I_2}(u,v)$$

$$=min(I_{I_1-I_2}(u,v),I_{I_1-I_2}(u,w))=U(I_{I_1-I_2}(u,v),I_{I_1-I_2}(u,w));$$

and if $U\in {\mathcal{U}}_{max}$, then it holds that

$$I_{I_1-I_2}(u,U(v,w))=I_{I_1-I_2}(u,max(v,w))$$

$$=I_{I_1-I_2}(u,v)=max(I_{I_1-I_2}(u,v),I_{I_1-I_2}(u,w))=U(I_{I_1-I_2}(u,v),I_{I_1-I_2}(u,w)).$$

(iv)

For the case that $v>e,w\le e$, the equation can be checked similarly as (iii).

To sum up, the couple ($I_{I_1-I_2},U$) satisfies Equation (20).

(⇒) Let $u>0,v,w\in [0,1]$, then $I_{I_1-I_2}(u,U(ev,ew))=U(I_{I_1-I_2}(u,ev),I_{I_1-I_2}(u,ew))$. By Equations (1) and (3), it holds that

$$e{I}_1(u,T(v,w))=I_{I_1-I_2}(u,eT(v,w))=U(e{I}_1(u,v),e{I}_1(u,w))=eT(I_1(u,v),I_1(u,w)).$$

Namely, $I_1(u,T(v,w))=T(I_1(u,v),I_1(u,w))$. If $u=0,v,w\in [0,1]$, $I_1(0,T(v,w))=1=T(1,1)=T(I_1(0,v),I_1(0,w))$. Thus $(I_1,T)$ satisfies Equation (5).

Now we turn to the check of Equation (21). For $u,v,w>0$, it holds that

$$I_{I_1-I_2}(u,U(e+(1-e)v,e+(1-e)w))=U(I_{I_1-I_2}(u,e+(1-e)v),I_{I_1-I_2}(u,e+(1-e)w)),$$

where $e+(1-e)v>e,e+(1-e)w>e$. By Equations (1) and (3), we have

$$e+(1-e)I_2(u,S(v,w))=I_{I_1-I_2}(u,e+(1-e)S(v,w))$$

$$=U(e+(1-e)I_2(u,v),e+(1-e)I_2(u,w))=e+(1-e)S(I_2(u,v),I_2(u,w)).$$

Consequently, $I_2(u,S(v,w))=S(I_2(u,v),I_2(u,w))$ for $u,v,w>0$. If $u=0,v,w>0$, then $I_2(0,S(v,w))=1=S(1,1)=S(I_2(0,v),I_2(0,w))$. Therefore, Equation (21) holds for $v,w>0$.□

The next result provides some solutions to Equation (20) with the implication I being an h-implication.

Proposition 4.

Assume the function h is an h-generator, and $U=T_M$, or $U=S_M$, or U are one of the following uninorms:

$$\begin{array}{c}{U}_1(u,v)=\left\{\begin{array}{cc}{h}^{-1}(h\left(u\right)+h\left(v\right)),\hfill & (u,v)\in {[0,e]}^2;\hfill \\ max(u,v),\hfill & else\hfill \end{array}\right.\end{array}$$

(22)

$$\begin{array}{c}{U}_2(u,v)=\left\{\begin{array}{cc}{h}^{-1}(h\left(u\right)+h\left(v\right)),\hfill & (u,v)\in {[e,1]}^2;\hfill \\ min(u,v),\hfill & else\hfill \end{array}\right.\end{array}$$

(23)

$$\begin{array}{c}{U}_3(u,v)=\left\{\begin{array}{cc}{h}^{-1}(h\left(u\right)+h\left(v\right)),\hfill & (u,v)\in {[0,e]}^2\cup {[e,1]}^2;\hfill \\ max(u,v),\hfill & else\hfill \end{array}\right.\end{array}$$

(24)

$$\begin{array}{c}{U}_4(u,v)=\left\{\begin{array}{cc}{h}^{-1}(h\left(u\right)+h\left(v\right)),\hfill & (u,v)\in {[0,e]}^2\cup {[e,1]}^2;\hfill \\ min(u,v),\hfill & else\hfill \end{array}\right.\end{array}$$

(25)

then $(I^h,U)$ satisfies Equation (20).

Proof. 

Trivially, due to the increasingness with respect to the second variable of $I^h$, $(I^h,U)$ satisfies Equation (20) with $U=T_M$ or $U=S_M$. Since $U_1$ is a t-conorm and $U_2$ is a t-norm, Propositions 26 and 32 in [47] $(I^h,U_1)$ and $(I^h,U_2)$, respectively, satisfy Equation (20).

For the other cases, $U_3\in {\mathcal{U}}_{max}$ and $U_4\in {\mathcal{U}}_{min}$. Through Remark 3 in [24], $I^h=I_{I_f-I_g}$ and the generators for the underlying f- and the g-implications which generated the $I^h$ can be given by $f\left(u\right)=-h\left(eu\right)$ and $g\left(u\right)=h(e+(1-e\left)u\right)$, respectively. Let $T(u,v)=f^{-1}(f\left(u\right)+f\left(v\right))$, then T is a t-norm with its additive generator f, and for $(u,v)\in {[0,e]}^2$ $U(u,v)=eT({\displaystyle \frac{u}{e}},{\displaystyle \frac{v}{e}})=h^{-1}(h\left(u\right)+h\left(v\right))$. It can be checked that

$$I_f(u,T(v,w))=T(I_f(u,v),I_f(u,w))=f^{-1}(uf\left(v\right)+wf\left(w\right)),$$

i.e., $(I_f,T)$ satisfies Equation (20). Let $S(u,v)=g^{-1}(g\left(u\right)+g\left(v\right))$, then S is a t-conorm with additive generator g, and for $(u,v)\in {[e,1]}^2$ $U(u,v)=e+(1-e)S({\displaystyle \frac{u-e}{1-e}},{\displaystyle \frac{v-e}{1-e}})=h^{-1}(h\left(u\right)+h\left(v\right))$. Thus, we have

$$I_g(u,S(v,w))=S(I_g(u,v),I_g(u,w))=g^{-1}\left({\displaystyle \frac{g\left(v\right)+g\left(w\right)}{u}}\right),$$

i.e., $(I_g,S)$ satisfies Equation (20). In particular, $(I_g,S)$ satisfies Equation (21) for $v,w>0$.

To sum up, $(I^h,U)$ satisfies Equation (20) by Proposition 3. □

3.1.2. The Other Classes of Uninorms

For the rest of uninorms, the preservation of the distributivity may not hold. Here, we show the results with several examples or propositions.

Generally, the threshold-generated implications do not satisfy Equation (20) with the uninorm $U\in {\mathcal{U}}_{cos}$ continuous in $]0,1[\times ]0,1[$, even if the associated t-norms or t-conorms satisfy the distributivity equations.

Example 2.

Assume $U={⟨T_P,{\displaystyle \frac{1}{4}},T_{LK},{\displaystyle \frac{1}{2}},(R,{\displaystyle \frac{3}{4}})⟩}_{cos,min}$, where R is a representable uninorm with additive generator $h=ln{\displaystyle \frac{u}{1-u}}$. Let $E=\{{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}\}$, $I_0=I_{YG}$, $I_1(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0\hfill \\ v,\hfill & else;\hfill \end{array}\right.$, $I_2$ and $I_3$ be given by Equation (26) and Equation (27), respectively. The E-generated implication $I_E$ be generated by the implications $I_0,I_1,I_2$ and $I_3$. Actually, $I_0,I_1,I_2$ satisfy Equation (3) with respect to $T_P,T_{LK},T$, respectively, and $I_3$ satisfies Equation (4) with respect to S, in which T and S are the underlyig t-norm and t-conorm of R. However, it holds that $I_E(0.1,U(0.6,0.8))\approx 0.73$ and $U(I_E(0.1,0.6),I_E(0.1,0.8))\approx 0.93$, which illustrates the couple $(I_E,U)$ does not satisfy Equation (20). This result is mainly originated in the special structureof the underlying representable uninorm R in the left-upper part and right-lower part of the unit square, where the uninorm is not defined by the commonly used $min\mathit{or}max$.

Specially, the threshold-generated implications may not satisfy Equation (20) with a representable uninorm U even if the pertaining t-norm and t-conorm satisfy the distributivity equations.

Example 3.

Let the uninorm U with the additive generator being $h=ln{\displaystyle \frac{u}{1-u}}$ and the ${\displaystyle \frac{1}{2}}$-generated implication $I_{I_1-I_2}$ be generated by the implications $I_1,I_2$, where

$$I_1(u,v)=\left\{\begin{array}{cc}1,\hfill & u=v=0;\hfill \\ f^{-1}\left({\displaystyle \frac{u}{e}}f\left(v\right)\right),\hfill & else\hfill \end{array}\right.$$

(26)

and

$$I_2(u,v)=\left\{\begin{array}{cc}1,\hfill & u=v=0;\hfill \\ g^{-1}\left({\displaystyle \frac{e}{u}}g\left(v\right)\right),\hfill & else\hfill \end{array}\right.$$

(27)

with $f\left(u\right)=-h\left(eu\right)$ and $g\left(u\right)=h(e+(1-e\left)u\right)$, then $(I_1,T)$ satisfies Equation (3) and $(I_2,S)$ satisfies Equation (4) with T and S generated by f and g, respectively. However the couple $(I_{I_1-I_2},U)$ does not satisfy Equation (20), since $I_{I_1-I_2}(0.1,U(0.1,0.6))\approx 0.41$ and $U(I_{I_1-I_2}(0.1,0.1),I_{I_1-I_2}(0.1,0.6))\approx 0.83$.

Now we turn to the h-implications and the representable uninorms.

Proposition 5.

Let h be an h-generator with $h\left(e\right)=0$ and U be a representable uninorm given by

$$U(u,v)=\left\{\begin{array}{cc}1or0,\hfill & (u,v)=(0,1)\mathit{or}(1,0);\hfill \\ h^{-1}(h\left(u\right)+h\left(v\right)),\hfill & else\hfill \end{array}\right.$$

(28)

then $(I^h,U)$ does not satisfy Equation (20).

Proof. 

Since $I^h(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0;\hfill \\ h^{-1}\left(uh\left(v\right)\right),\hfill & u>0,v\le e\hfill \\ h^{-1}\left({\displaystyle \frac{1}{u}}h\left(v\right)\right),\hfill & u>0,v>e\hfill \end{array}\right.$, then for $u>0$, the vertical sections $I^h(u,·)$ have the form

$$I^h(u,v)=\left\{\begin{array}{cc}0,\hfill & v=0;\hfill \\ h^{-1}\left(uh\left(v\right)\right),\hfill & 0<v\le e;\hfill \\ h^{-1}\left({\displaystyle \frac{1}{u}}h\left(v\right)\right),\hfill & e<v<1;\hfill \\ 1,\hfill & v=1;\hfill \end{array}\right.$$

which can be rewritten as $I^h(u,v)=\left\{\begin{array}{cc}0,\hfill & v=0;\hfill \\ h^{-1}\left(g_u\left(h\left(v\right)\right)\right),\hfill & 0<v<1;\hfill \\ 1,\hfill & v=1;\hfill \end{array}\right.$ with the function $g_u:R\to R$ is given by $g_u\left(t\right)=\left\{\begin{array}{cc}u{h}^{-1}\left(t\right),\hfill & t\le 0;\hfill \\ {\displaystyle \frac{1}{u}}{h}^{-1}\left(t\right),\hfill & t>0;\hfill \end{array}\right.$ Then, we can get that $g_u\left(0\right)=ue$, hence $g_u(0+0)\ne g_u\left(0\right)+g_u\left(0\right)$. Namely, $g_u$ is not an additive function. Thus, by Theorem 5.1 in [41], $(I^h,U)$ does not satisfy Equation (20). □

Here we provide an example for the above result.

Example 4.

Let $I^h$ be an h-implication with the additive generator $h=ln{\displaystyle \frac{u}{1-u}}$ and U is the representable uninorm generated by the same generator. Then, it holds that $I^h(0.7,U(0.4,0.7))\approx 0.65$ and $U(I^h(0.7,0.4),I^h(0.7,0.7))\approx 0.72$, thus $(I^h,U)$ does not satisfy Equation (20).

In general, the threshold-generated implications may not satisfy Equation (20) with idempotent uninorm U. Particularly, the h-implications do not satisfy Equation (20) with an idempotent uninorms. The results can be shown by the following examples and proposition, respectively.

Example 5.

Assume the uninorm

$$U=\left\{\begin{array}{c}min(u,v),v\le 1-u;\hfill \\ max(u,v),else\hfill \end{array}\right.$$

whose neutral element is ${\displaystyle \frac{1}{2}}$, and the ${\displaystyle \frac{1}{2}}$-generated implication $I_{I_{RS}-I_{YG}}$ be generated by the implications $I_{RS}$ and $I_{YG}$. Although the couples ($I_{RS},T_M$) and ($I_{YG},S_M$) satisfy Equation (3) and Equation (4), respectively, the couple $(I_{I_{RS}-I_{YG}},U)$ does not satisfy Equation (20), due to the fact that $I_{I_{RS}-I_{YG}}(0.9,U(0.9,0.3))\approx 0.91>U(I_{I_{RS}-I_{YG}}(0.9,0.9),I_{I_{RS}-I_{YG}}(0.9,0.3))=0.$

Proposition 6.

Let $I^h$ be an h-implication generated by an h-generator h and the strong negation $N\left(u\right)$ be given by $N\left(u\right)=h^{-1}(-h\left(u\right))$.

(i) 

Let U be a left-continuous idempotent uninorm given by

$$U(u,v)=\left\{\begin{array}{cc}min(u,v),\hfill & v\le N\left(u\right);\hfill \\ max(u,v),\hfill & else\hfill \end{array}\right.$$

then Equation (20) does not hold for $(I^h,U)$.

(ii) 

Let U be a right-continuous idempotent uninorm given by

$$U(u,v)=\left\{\begin{array}{cc}max(u,v),\hfill & v\ge N\left(u\right);\hfill \\ min(u,v),\hfill & else\hfill \end{array}\right.$$

then Equation (20) does not hold for $(I^h,U)$.

Proof. 

It is enough to prove the first case, since the other one can be checked similarly. Let $0<v_0<e<w_0$ and $v_0\le N\left(w_0\right)$, then $I(u,v_0)<I(u,w_0)$ for $u>0$ and $h\left(v_0\right)\le -h\left(w_0\right)$, i.e., $-{\displaystyle \frac{h\left(w_0\right)}{h\left(v_0\right)}}\le 1$. Now assume $0<u_0<\sqrt{-{\displaystyle \frac{h\left(w_0\right)}{h\left(v_0\right)}}}$, then $u_0^2<-{\displaystyle \frac{h\left(w_0\right)}{h\left(v_0\right)}}$, i.e., $-u_{0}h\left(v_0\right)<{\displaystyle \frac{1}{u_0}}h\left(w_0\right)$. Then $N\left(I^h(u_0,v_0)\right)=h^{-1}(-u_{0}h\left(v_0\right))<h^{-1}\left({\displaystyle \frac{1}{u_0}}h\left(w_0\right)\right)=I^h(u_0,w_0)$. Meanwhile, it holds that $I^h(u_0,U(v_0,w_0))=I^h(u_0,min(v_0,w_0))=I^h(u_0,v_0)$ and $U(I^h(u_0,v_0),I^h(u_0,w_0))=max(I^h(u_0,v_0),I^h(u_0,w_0))=I(u_0,w_0)$, which implies that

$$I^h(u_0,U(v_0,w_0))<U(I^h(u_0,v_0),I^h(u_0,w_0)).$$

Therefore, $(I^h,U)$ does not satisfy Equation (20).□

The above results can also be exemplified as follows.

Example 6.

Assume that $I^h$ is an h-implication with additive generator $h=ln{\displaystyle \frac{u}{1-u}}$ and the related strong negation of the idempotent uninorm U is defined by $N\left(u\right)=h^{-1}(-h\left(u\right))=1-u$. Since $I^h(0.6,U(0.6,0.3))\approx 0.38<U(I^h(0.6,0.6),I^h(0.6,0.3))\approx 0.66$, then $(I^h,U)$ does not satisfy Equation (20).

3.2. The Case $U(e,e)=e$ with $e_u<e$

In this case, there are at least two proper idempotent elements of the involved uninorm U, then the uninorm cannot be representable. Moreover, by the idempotency of the idempotent uninorms, there is no need to consider the idempotent uninorms for this case. So, in this part, it is enough to discuss Equation (20) with the other two classes of uninorms, viz, the uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$, and the uninorm $U\in {\mathcal{U}}_{cos}$.

3.2.1. The Uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$

In this part, we consider the class of uninorms ${\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$. Then, for $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$ with $U(e,e)=e$ and $e_u<e$, $T(u,v)={\displaystyle \frac{U(e_{u}u,e_{u}v)}{e_u}}$ is a t-norm, and $S_1(u,v)={\displaystyle \frac{U(e_u+(e-e_u)u,e_u+(e-e_u)v)-e_u}{e-e_u}}$ and $S_2(u,v)={\displaystyle \frac{U(e+(1-e)u,e+(1-e\left)v\right)-e}{1-e}}$ are t-conorms, respectively. Hence U can be given by

$$U(u,v)=\left\{\begin{array}{c}{e}_{u}T({\displaystyle \frac{u}{e_u}},{\displaystyle \frac{u}{e_u}}),(u,v)\in {[0,e_u]}^2;\hfill \\ e_u+(e-e_u)S_1({\displaystyle \frac{u-e_u}{e-e_u}},{\displaystyle \frac{v-e_u}{e-e_u}}),(u,v)\in {[e_u,e]}^2;\hfill \\ e+(1-e)S_2({\displaystyle \frac{u-e}{1-e}},{\displaystyle \frac{v-e}{1-e}}),(u,v)\in {[e,1]}^2;\hfill \\ min(u,v)ormax(u,v),(u,v)\in [0,e_u]\times ]e_u,1]\hfill \\ \cup ]e_u,1]\times [0,e_u];\hfill \\ max(u,v),else\hfill \end{array}\right.$$

(29)

For this case, the next result completely characterizes the solutions of Equation (20) with the uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$ and it guarantees the transformation of the distributivity property from the original implications to the threshold-generated counterpart.

Proposition 7.

Assume $I_0,I_1,I_2$ are fuzzy implications, the set $E=\{e_u,e\}$, and the uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$ given by Equation (29), then the E-generated implication $I_E$ satisfies Equation (20) with the uninorm U if and only if the couple $(I_0,T)$ satisfies Equation (5) and

$$I_i(u,S_i(v,w))=S_i(I_i(u,v),I_i(u,w)),(i\in \{1,2\left\}\right)$$

(30)

for $v,w>0$.

Proof. 

(⇐): We only consider the case that $u>0$.

(i)

If $(v,w)\in {[0,e_u]}^2$, then $U(v,w)\in [0,e_u],$ $I_E(u,v)=e_u{I}_0(u,{\displaystyle \frac{v}{e_u}})\in [0,e_u]$ and $I_E(u,w)\in [0,e_u]$. Thus

$$I_E(u,U(v,w))=e_u{I}_0(u,{\displaystyle \frac{U(v,w)}{e_u}})=e_u{I}_0(u,{\displaystyle \frac{e_{u}T(\frac{v}{e_u},\frac{w}{e_u})}{e_u}})=e_u{I}_0(u,T({\displaystyle \frac{v}{e_u}},{\displaystyle \frac{w}{e_u}})$$

$$=e_{u}T(I_0(u,{\displaystyle \frac{v}{e_u}}),I_0(u,{\displaystyle \frac{w}{e_u}}))=e_{u}T({\displaystyle \frac{I_E(u,v)}{e_u}},{\displaystyle \frac{I_E(u,w)}{e_u}})=U(I_E(u,v),I_E(u,w)).$$

(ii)

If $(v,w)\in ]e_u{,e]}^2$, then $U(v,w)\in ]e_u,e],$ $I_E(u,v)=e_u+(e-e_u)I_1(u,{\displaystyle \frac{v-e_u}{e-e_u}})\in [e_u,e]$ and $I_E(u,w)\in ]e_u,e]$. Thus

$$I_E(u,U(v,w))=e_u+(e-e_u)I_1(u,{\displaystyle \frac{U(v,w)-e_u}{e-e_u}})$$

$$=e_u+(e-e_u)I_1(u,S_1({\displaystyle \frac{v-e_u}{e-e_u}},{\displaystyle \frac{w-e_u}{e-e_u}}))$$

$$=e_u+(e-e_u)S_1(I_1(u,{\displaystyle \frac{v-e_u}{e-e_u}}),I_1(u,{\displaystyle \frac{w-e_u}{e-e_u}}))$$

$$=e_u+(e-e_u)S_1({\displaystyle \frac{I_E(u,v)-e_u}{e-e_u}},{\displaystyle \frac{I_E(u,w)-e_u}{e-e_u}})=U(I_E(u,v),I_E(u,w)).$$

(iii)

If $(v,w)\in {]e,1]}^2$, then $U(v,w)\in ]e,1],$ $I_E(u,v)=e+(1-e)I_2(u,{\displaystyle \frac{v-e}{1-e}})\in [e,1]$ and $I_E(u,w)\in ]e,1]$. Thus

$$I_E(u,U(v,w))=e+(1-e)I_2(u,{\displaystyle \frac{U(v,w)-e}{1-e}})=e+(1-e)I_2(u,S_2({\displaystyle \frac{v-e}{1-e}},{\displaystyle \frac{w-e}{1-e}}))$$

$$=e+(1-e)S_2(I_2(u,{\displaystyle \frac{v-e}{1-e}}),I_2(u,{\displaystyle \frac{w-e}{1-e}}))=e+(1-e)S_2({\displaystyle \frac{I_E(u,v)-e}{1-e}},{\displaystyle \frac{I_E(u,w)-e}{1-e}})$$

$$=U(I_E(u,v),I_E(u,w)).$$

(iv)

If $(v,w)\in [0,e_u]\times ]e_u,1]$ (Similar to the case $(w,v)\in [0,e_u]\times ]e_u,1]$), then $I_E(u,v)\in [0,e_u]$ and $I_E(u,w)\in ]e,1]$.

(a)

If $U\in {\mathcal{U}}_{max}$, then $U(v,w)=max(v,w)$, then

$$U(I_E(u,v),I_E(u,w))=max(I_E(u,v),I_E(u,w))$$

$$=I_E(u,max(v,w))=I_E(u,U(v,w));$$

(b)

If $U\in {\mathcal{U}}_{min}$, then $U(v,w)=min(v,w)$, then

$$U(I_E(u,v),I_E(u,w))=min(I_E(u,v),I_E(u,w))$$

$$=I_E(u,min(v,w))=I_E(u,U(v,w)).$$

(v)

If $(v,w)\in ]e_u,e]\times ]e,1]$ (Similarly for the case $(w,v)\in ]e_u,e]\times ]e,1]$), then $I_E(u,v)\in ]e_u,e]$ and $I_E(u,w)\in ]e,1]$. Thus

$$U(I_E(u,v),I_E(u,w))=min(I_E(u,v),I_E(u,w))=I_E(u,min(v,w))=I_E(u,U(v,w)).$$

To sum up, the E-generated implication $I_E$ satisfies Equation (20) with the uninorm U.

(⇒): We divide the proof into three cases.

(i)

Let $u>0,(v,w)\in [0,1]\times [0,1]$, then $I_E(u,U(e_{u}v,e_{u}w))=U(I_E(u,e_{u}v),I_E(u,e_{u}w)).$

$$LHS=I_E(u,e_{u}T(v,w))=e_u{I}_0(u,T(v,w));$$

$$RHS=U(e_u{I}_0(u,v),e_u{I}_0(u,w))=e_{u}T(I_0(u,v),I_0(u,w)).$$

Then it follows that $I_0(u,T(v,w))=T(I_0(u,v),I_0(u,w))$. In addition, if $u=0$, $I_0(0,T(v,w))=1=T(1,1)=T(I_0(0,v),I_0(0,w))$. Thus $(I_0,T)$ satisfies Equation (5).

(ii)

Let $u>0,(v,w)\in {]0,1]}^2$, then

$$I_E(u,U(e_u+(e-e_u)v,e_u+(e-e_u)w))=U(I_E(u,e_u+(e-e_u)v),I_E(u,e_u+(e-e_u)w)).$$

$$LHS=I_E(u,U(e_u+(e-e_u)v,e_u+(e-e_u)w))$$

$$=I_E(u,e_u+(e-e_u)S_1(v,w))=e_u+(e-e_u)I_1(u,S_1(v,w));$$

$$RHS=U(e_u+(e-e_u)I_1(u,v),e_u+(e-e_u)I_1(u,w))=e_u+(e-e_u)S_1(I_1(u,v),I_1(u,w)).$$

Then it follows that $I_1(u,S_1(v,w))=S_1(I_1(u,v),I_1(u,w))$. Moreover, the last equation holds trivially for $u=0$. Thus, we get Equation (30) for $i=1$.

(iii)

Equation (30) for $i=2$ can be checked similarly like (ii).

□

Remark 1.

For this case, if the initial implications satisfy the distributive equations with respect to the related t-norms or t-conorms, then the E-generated implication satisfies Equation (20). For instance, let $T=T_P,S_1=S_{LK},S_2=S_P$, $e_u={\displaystyle \frac{1}{3}}$, and $e={\displaystyle \frac{2}{3}}$ in Equation (31) and $I_E$ be the E-generated implication with the implications $I_0=I_{YG}$ and $I_1(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0\hfill \\ v,\hfill & else;\hfill \end{array}\right.$ and $I_2=I_{Lt}$ (the least implication), where the couple $(I_{YG},T_P)$ satisfies Equation (3) and the couples $(I_1,S_{LK})$ and $(I_2,S_P)$ satisfy Equation (4), then the E-generated implication $I_E$ satisfies Equation (20).

3.2.2. The Uninorm $U\in {\mathcal{U}}_{cos}$

In this part, we focus on the the uninorm $U\in {\mathcal{U}}_{cos}$. Since $e_u<e$, then U has the form as Equation (19), i.e., $U\in {\mathcal{U}}_{cos,max}$. The following result completely characterizes the solutions of Equation (20) for the uninorm $U\in {\mathcal{U}}_{cos,max}$ and it guarantees the transformation of the distributivity properties from the original implications to the threshold-generated counterpart.

Proposition 8.

Let $U\in {\mathcal{U}}_{cos,max}$, $E=\{\gamma ,\delta \}$ with $e=\gamma$ or $e=\delta$ and $I_E$ be the E-generated implication generated by the implications $I_0,I_1,I_2$ with $N_{I_0}\left(u\right)=I_0(u,0)=\left\{\begin{array}{cc}1,\hfill & u=0;\hfill \\ 0,\hfill & else\hfill \end{array}\right.$, and $I_1(u,v)>0$ and $I_2(u,v)>0$ for $v>0$. Then, the couple $(I_E,U)$ satisfies Equation (20) if and only if the couple $(I_0,R)$ satisfies Equation (20) and $I_i(u,S_i(v,w))=S_i(I_i(u,v),I_i(u,w))\left(i=1,2\right)$ for $v,w>0$.

Proof. 

(⇒): Assume U is given by Equation (19) and $(U,I_E)$ satisfies Equation (20).

Let $u>0$ and $(v,w)\in [0,1]\times [0,1]-\left\{\right(0,1),(1,0\left)\right\}$, then $\gamma v,\gamma w\in [0,\gamma ]$ and $I_E(u,U(\gamma v,\gamma w))=U(I_E(u,\gamma v),$ $I_E(u,\gamma w))$. By Equations (14) and (19), we can get $\gamma I_0(u,{\displaystyle \frac{U(\gamma v,\gamma w)}{\gamma }})=\gamma I_0(u,R(v,w))=U(\gamma I_0(u,v),\gamma I_0(u,w))=\gamma R(I_0(u,v),I_0(u,w))$. Additionally, on one hand, if R is conjunctive, i.e., $R(0,1)=0$, then $I_0(u,R(0,1))=I_0(u,0)=0$ and $R(I_0(u,0),I_0(u,1))=R(0,1)=0$; on the other hand, if R is disjunctive, i.e., $R(0,1)=1$, then $I_0(u,R(0,1))=I_0(u,1)=1$ and $R(I_0(u,0),I_0(u,1))=R(0,1)=1$. Thus, we have $I_0(u,R(v,w))=R(I_0(u,v),I_0(u,w))$ for $u>0$ and $v,w\in [0,1]$. Moreover, $I_0(0,R(v,w))=R(I_0(0,v),I_0(0,w))=1$. Hence, $(R,I_0)$ satisfies Equation (20).

Let $u>0$ and $v,w\in ]0,1]$, then $\gamma +(\delta -\gamma )v,\gamma +(\delta -\gamma )w\in ]\gamma ,\delta ]$ and $I_E(u,U(\gamma +(\delta -\gamma )v,\gamma +(\delta -\gamma )w))=U(I_E(u,\gamma +(\delta -\gamma )v),I_E(u,\gamma +(\delta -\gamma )w))$. In virtue of Equations (2) and (19), we can get $\gamma +(\delta -\gamma )I_1(u,{\displaystyle \frac{U(\gamma +(\delta -\gamma )v,\gamma +(\delta -\gamma \left)w\right)-\gamma }{\delta -\gamma }})=\gamma +(\delta -\gamma )I_1(u,S_1(v,w))=U(\gamma +(\delta -\gamma )I_1(u,v),\gamma +(\delta -\gamma )I_1(u,w))=\gamma +(\delta -\gamma )S_1(I_1(u,v),$ $I_1(u,w))$. Then, we have, for $v,w\in ]0,1]$, $I_1(u,S_1(v,w))=S_1(I_1(u,v),I_1(u,w))$, as a result of the additional fact $I_1(0,S_1(v,w))=S_1(I_1(0,v),I_1(0,w))=1$. Similarly, for $v,w\in ]0,1]$, we have $I_2(u,S_2(v,w))=S_2(I_2(u,v),I_2(u,w))$.

(⇐): Since $I_E(0,U(v,w))=U(I_E(0,v),I_E(0,w))=1$, then we consider the case $u>0$ in the following.

(i)

If $(v,w)\in {[0,\gamma ]}^2-\{(0,\gamma ),(\gamma ,0)\}$, then $U(v,w)\in [0,\gamma ]$ and consequently,

$$I_E(u,U(v,w))=\gamma I_0(u,{\displaystyle \frac{U(v,w)}{\gamma }})=\gamma I_0(u,R({\displaystyle \frac{v}{\gamma }},{\displaystyle \frac{w}{\gamma }}))$$

$$U(I_E(u,v),I_E(u,w))=U(\gamma I_0(u,{\displaystyle \frac{v}{\gamma }}),\gamma I_0(u,{\displaystyle \frac{w}{\gamma }}))=\gamma R(I_0(u,{\displaystyle \frac{v}{\gamma }}),I_0(u,{\displaystyle \frac{w}{\gamma }})).$$

Since $(R,I_0)$ satisfies Equation (20), then $I_E(u,U(v,w))=U(I_E(u,v),I_E(u,w))$. The last equation holds for $v=0,w=\gamma$ (similarly $v=\gamma ,w=0$), since $I_E(u,U(0,\gamma ))=I_E(u,0)=\gamma I_0\left(u\right)=0$ and $U(I_E(u,0),I_E(u,\gamma ))=U(\gamma I_0(u,0),\gamma$ $I_0(u,1))=U(0,\gamma )=0$.

(ii)

If $v,w\in ]\gamma ,\delta ]$, then $U(v,w)\in ]\gamma ,\delta ]$, which yields that $I_E(u,U(v,w))=\gamma +(\delta -\gamma )I_1(u,{\displaystyle \frac{U(v,w)-\gamma }{\delta -\gamma }})=\gamma +(\delta -\gamma )I_1(u,S_1({\displaystyle \frac{v-\gamma }{\delta -\gamma }},{\displaystyle \frac{w-\gamma }{\delta -\gamma }}))$ and $U(I_E(u,v),I_E(u,w))=U(\gamma +(\delta -\gamma )I_1(u,{\displaystyle \frac{v-\gamma }{\delta -\gamma }}),\gamma +(\delta -\gamma )I_1(u,{\displaystyle \frac{w-\gamma }{\delta -\gamma }}))=\gamma +(\delta -\gamma )S_1(I_1(u,{\displaystyle \frac{v-\gamma }{\delta -\gamma }}),I_1(u,{\displaystyle \frac{w-\gamma }{\delta -\gamma }}))$. Since ${\displaystyle \frac{v-\gamma }{\delta -\gamma }},{\displaystyle \frac{w-\gamma }{\delta -\gamma }}\in ]0,1]$ and $I_1(u,S_1(v,w))=S_1(I_1(u,v),I_1(u,w))$ for $v,w>0$, then we get that $I_E(u,U(v,w))=U(I_E(u,v),I_E(u,w))$.

(iii)

If $v,w\in ]\delta ,1]$, then we can check the equation similarly.

(iv)

If $v=0,w\in ]\gamma ,\delta [$ (similarly $w=0,v\in ]\gamma ,\delta [$), then $U(v,w)=0,{\displaystyle \frac{w-\gamma }{\delta -\gamma }}<1,$ and $I_E(u,w)=\gamma +(\delta -\gamma )I_1(u,{\displaystyle \frac{w-\gamma }{\delta -\gamma }})\in ]\gamma ,\delta [$, thus $I_E(u,U(v,w))=I_E(u,0)=\gamma I_0(u,0)=0=U(0,\gamma +(\delta -\gamma )I_1(u,{\displaystyle \frac{w-\gamma }{\delta -\gamma }}))=U(I_E(u,v),I_E(u,w))$.

(v)

If $v=0,w=\delta$ (similarly $w=0,v=\delta$), then $U(v,w)=0$ or $U(v,w)=\delta$. If $U(v,w)=0$, then $I_E(u,U(v,w))=U(I_E(u,v),I_E(u,w))=0$; if $U(v,w)=\delta$, then $I_E(u,U(v,w))=I_E(u,\delta )=\delta =U(0,\delta )=U(I_E(u,v),I_E(u,w))$.

(vi)

For the other cases, $U(v,w)=max(v,w)$ and

$$U(I_E(u,v),I_E(u,w))=max(I_E(u,v),I_E(u,w)),$$

then it can be directly checked that $I_E(u,U(v,w))=U(I_E(u,v),I_E(u,w))$.

□

Example 7.

(i) 

Let R be the (conjunctive or disjunctive) representable uninorm with its additive generator being $h\left(u\right)=ln{\displaystyle \frac{u}{1-u}}$; $I_0(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0\hfill \\ v,\hfill & else;\hfill \end{array}\right.$; $I_1=I_2=I^h$; $S_1=S_D$ and $S_2(u,v)=\left\{\begin{array}{cc}{h}^{-1}\left({\displaystyle \frac{h\left(u\right)h\left(v\right)}{h\left(u\right)+h\left(v\right)}}\right),\hfill & (u,v)\in {[0,{\displaystyle \frac{1}{2}}]}^2\hfill \\ h^{-1}(h\left(u\right)+h\left(v\right)),\hfill & (u,v)\in {[{\displaystyle \frac{1}{2}},1]}^2\hfill \\ max(u,v),\hfill & else\hfill \end{array}\right.$,

According to Example 5.9. in [41] and Propositions 31 and 32 in [47], $(I_0,R)$ satisfies Equation (20) and $(I_i,S_i)$ satisfies Equation (4) for $i=1,2$. Thus the E-generated implication $I_E$ by $I_0,I_1,I_2$ with $E=\{{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}\}$ satisfies Equation (20) with $U={⟨(R,{\displaystyle \frac{1}{4}}),{\displaystyle \frac{1}{2}},S_1,{\displaystyle \frac{3}{4}},S_2⟩}_{cos,max}$.

(ii) 

If, in (i), we only put $I_0(u,v)=\left\{\begin{array}{c}1,u<1orv=1\hfill \\ {\displaystyle \frac{v^2}{2v^2-2v+1}},else\hfill \end{array}\right.$, for which $(R,I_0)$ satisfies Equation (20) but $N_{I_0}\left(u\right)\ne \left\{\begin{array}{cc}1,\hfill & u=0\hfill \\ 0,\hfill & else\hfill \end{array}\right.$ then Equation (20) does not hold since $I_E(0.2,U(0,0.6))=0.5<U(I_E(0.2,0),I_E(0.2,0.6))\approx 0.62.$

(iii) 

If, in (i) we put $I_1=I_{RS}$ and $U={⟨(R,{\displaystyle \frac{1}{4}}),{\displaystyle \frac{1}{2}},S_M,{\displaystyle \frac{3}{4}},S_2⟩}_{cos,max}$ with $U(0,{\displaystyle \frac{3}{4}})={\displaystyle \frac{3}{4}}$, for which $(I_{RS},S_M)$ satisfies Equation (20) with $U=S_M$ and $I=I_{RS}$ but it does not hold that $I_1(u,v)>0$ for $v>0$. Then $I_E(0.7,U(0.7,0))=0\ne {\displaystyle \frac{3}{4}}=U(I_E(0.7,0.7),I_E(0.7,0))$.

3.3. The Case $U(e,e)=e$ with $e_u>e$

In this part, we discuss Equation (20) with the two classes of uninorms, viz, the uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$ and the uninorm $U\in {\mathcal{U}}_{cos}$, just as the last part.

3.3.1. The Uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$

Here, we consider the classes of uninorms ${\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$. For the case $U(e,e)=e$ with $e_u>e$, the uninorm U has at least two idempotent elements $e,e_u$; then, for $(u,v)\in [0,1]\times [0,1]$, $S(u,v)={\displaystyle \frac{U(e_u+(1-e_u)u,e_u+(1-e_u)v)-e_u}{1-e_u}}$ is a t-conorm and $T_1(u,v)={\displaystyle \frac{U(eu,ev)}{e}}$ and $T_2(u,v)={\displaystyle \frac{U(e+(e_u-e)u,e+(e_u-e)v)-e}{e_u-e}}$ are two t-norms, respectively. Hence, U can be given by

$$U(u,v)=\left\{\begin{array}{c}e{T}_1({\displaystyle \frac{u}{e}},{\displaystyle \frac{v}{e}}),(u,v)\in {[0,e]}^2;\hfill \\ e+(e_u-e)T_2({\displaystyle \frac{u-e}{e_u-e}},{\displaystyle \frac{v-e}{e_u-e}}),(u,v)\in {[e,e_u]}^2;\hfill \\ e_u+(1-e_u)S({\displaystyle \frac{u-e_u}{1-e_u}},{\displaystyle \frac{v-e_u}{1-e_u}}),(u,v)\in {[e_u,1]}^2;\hfill \\ min(u,v)ormax(u,v),\hfill & \\ (u,v)\in [0,e_u]\times \left]e_u,1\right]\cup \left]e_u,1\right]\times [0,e_u];\hfill \\ min(u,v),else\hfill \end{array}\right.$$

(31)

The following result completely characterizes the solutions of Equation (20) with the uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$ and it guarantees the transformation of this distributivity properties from the original implications to the threshold-generated counterpart.

Proposition 9.

Let $I_0,I_1,I_2$ be fuzzy implications, the set $E=\{e,e_u\}$, and the uninorm $U\in {\mathcal{U}}_{min}$ or ${\mathcal{U}}_{max}$ be given by Equation (31) with $T_2$ having no zero divisors; then, the E-generated implication $I_E$ satisfies Equation (20) with the uninorm U if and only if the couple $(I_0,T_1)$ satisfies Equation (5) with $U=T_1$ and

$$\begin{array}{c}{I}_1(u,T_2(v,w))=T_2(I_1(u,v),I_1(u,w))\end{array}$$

(32)

$$\begin{array}{c}{I}_2(u,S(v,w))=S(I_2(u,v),I_2(u,w))\end{array}$$

(33)

for $v,w>0$.

Proof. 

Since the similar structure of the uninorms given by Equations (29) and (31), by exploring the relationships between Equation (20) and the properties of the scaled t-norms $T_1$ and $T_2$ and t-conorm S for the uninorm by Equation (31), the results can be checked in a way similar to that of Proposition 7. □

3.3.2. The Uninorm $U\in {\mathcal{U}}_{cos}$

Here, we consider the the uninorm $U\in {\mathcal{U}}_{cos}$. Since $e_u>e$, U has the form as Equation (18), i.e., $U\in {\mathcal{U}}_{cos,min}$. The next result completely characterizes the solutions of Equation (20) for the uninorm $U\in {\mathcal{U}}_{cos,min}$ and it guarantees the transformation of the distributivity properties from the original implications to the threshold-generated counterpart.

Proposition 10.

Let $U\in {\mathcal{U}}_{cos,min}$ given by Equation (18) and $E=\{\alpha ,\beta \}$, $I_E$ be the E-generated implication generated by the implications $I_0,I_1,I_2$ with $I_0(u,v)<1$ for $u>0\mathrm{and}v<1$, and $I_1(u,v)>0$ for every $v>0$, and the pertaining t-norm $T_2$ having no zero divisors. Then the couple $(I_E,U)$ satisfies Equation (20) if and only if the couple $(I_0,T_1)$ satisfies Equation (5) and

$$\begin{array}{c}{I}_1(u,T_2(v,w))=T_2(I_1(u,v),I_1(u,w))\end{array}$$

(34)

$$\begin{array}{c}{I}_2(u,R(v,w))=R(I_2(u,v),I_2(u,w))\end{array}$$

(35)

for $v,w>0$.

Proof. 

(⇒): Assume $(I_E,U)$ satisfies $I_E(u,U(v,w))=U(I_E(u,v),I_E(u,w))$ for every $u,v,w\in [0,1]$.

If $u>0$ and $(v,w)\in [0,1]\times [0,1]$, then

$$I_E(u,U(\alpha v,\alpha w))=\alpha I_0(u,{\displaystyle \frac{U(\alpha v,\alpha w)}{\alpha }})=\alpha I_0(u,T_1(v,w)),$$

$$U(I_E(u,\alpha v),I_E(u,\alpha w))=U(\alpha I_0(u,v),\alpha I_0(u,w))=\alpha T_1(I_0(u,v),I_0(u,w)).$$

With Equation (20), we have $I_0(u,T_1(v,w))=T_1(I_0(u,v),I_0(u,w))$. Moreover, $I_0(0,T_1(v,w))$ $=T_1(I_0(0,v),I_0(0,w))=1$, thus $(I_0,T_1)$ satisfies Equation (5).

By exploring the relationships between Equation (20) and the properties of the scaled t-norms $T_2$ and uninorm R for the uninorm U by Equation (18), the other two equations can be obtained similarly as Proposition 7.

(⇐): Since $I_E(u,U(v,w))=U(I_E(u,v),I_E(u,w))=1$ for $u=0$, then we only focus on $u>0$ in what follows.

(i)

If $v,w\in [0,\alpha ]$, then

$$I_E(u,U(v,w))=\alpha I_0(u,{\displaystyle \frac{U(v,w)}{\alpha }})=\alpha I_0(u,T_1({\displaystyle \frac{v}{\alpha }},{\displaystyle \frac{w}{\alpha }}))$$

$$=\alpha T_1(I_0(u,{\displaystyle \frac{v}{\alpha }}),I_0(u,{\displaystyle \frac{w}{\alpha }}))=U(I_E(u,v),I_E(u,w))$$

since $(I_0,T_1)$ satisfies Equation (20).

(ii)

If $v,w\in ]\alpha ,\beta ]$, then

$$I_E(u,U(v,w))=\alpha +(\beta -\alpha )I_1(u,{\displaystyle \frac{U(v,w)-\alpha }{\beta -\alpha }})$$

$$=\alpha +(\beta -\alpha )I_1(u,T_2({\displaystyle \frac{v-\alpha }{\beta -\alpha }},{\displaystyle \frac{w-\alpha }{\beta -\alpha }}))=\alpha +(\beta -\alpha )T_2(I_1(u,{\displaystyle \frac{v-\alpha }{\beta -\alpha }}),I_1(u,{\displaystyle \frac{w-\alpha }{\beta -\alpha }}))$$

from the fact that $I_1(u,T_2(v,w))=T_2(I_1(u,v),I_1(u,w))$ for $v,w>0$.

(iii)

If $v,w\in ]\beta ,1[$, then Equation (20) can be checked similarly from the fact that $I_2(u,S(v,w))=S(I_2(u,v),I_2(u,w))$ holds for every $v,w>0$.

(iv)

If $v=1$ (similarly $w=1$), then the following subcases need be considered.

(a)

If $w\in ]\alpha ,1]$, then $U(v,w)=1$ with Equation (18), $I_E(u,v)=I_E(u,1)=1$ and $I_E(u,w)\in ]\alpha ,1]$ with the fact $I_1(u,v)>0$ for $v>0$, hence $I_E(u,U(v,w))=1=U(I_E(u,v),I_E(u,w))$ with Equations (14) and (18).

(b)

If $w=\alpha$, then we have

$$I_E(u,U(v,w))=I_E(u,U(1,\alpha ))=1=U(1,\alpha )=U(I_E(u,1),I_E(u,\alpha ))$$

for the case $U(1,\alpha )=1$; and

$$I_E(u,U(v,w))=I_E(u,U(1,\alpha ))=\alpha =U(1,\alpha )=U(I_E(u,1),I_E(u,\alpha ))$$

for the case $U(1,\alpha )=\alpha$.

(c)

If $w\in [0,\alpha [$, then it holds that $I_E(u,w)<\alpha$ with the fact that $I_0(u,v)<1$ for $u>0,v<1$ and then $I_E(u,U(v,w))=I_E(u,U(1,w))=I_E(u,min(1,w))=I_E(u,w)=U(1,I_E(u,w))$.

(v)

For the other cases, we have $U(v,w)=max(v,w)$ and $U(I_E(u,v),I_E(u,w))=max(I_E(u,v),I_E(u,w))$, thus $I_E(u,U(v,w))=U(I_E(u,v),I_E(u,w))$.

□

With the above proposition, we can see that, under some minimal conditions, the threshold-generated implication well preserves the distributivity over the fuzzy connectives.

Example 8.

(i) 

Let the uninorm $U={⟨T_P,{\displaystyle \frac{1}{4}},T_P,{\displaystyle \frac{1}{2}},(R,{\displaystyle \frac{3}{4}})⟩}_{cos,min}$, where R is a (conjunctive or disjunctive) representable uninorm with additive generator $h\left(u\right)=ln{\displaystyle \frac{u}{1-u}}$, $I_0=I_{YG},$ and $I_1(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0\hfill \\ v,\hfill & else;\hfill \end{array}\right.$, and $I_2(u,v)=\left\{\begin{array}{cc}1,\hfill & u=0\hfill \\ {\displaystyle \frac{v^2}{2v^2-2v+1}},\hfill & else;\hfill \end{array}\right.$ and $I_E$ be the E-generated implication by $I_0,I_1,I_2$ with $E=\{{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}}\}$. It holds that, $(I_0,T_P)$ and $(I_1,T_P)$ satisfy Equation (5) and the couple $(I_2,R)$ satisfies Equation (20) (see [41]) with $U=R$. We only check Equation (5) for the first couple as an illustration. If $u=0$ and $vw=0$, then it holds that $I_{YG}(u,T_P(v,w))=I_{YG}(u,vw)=1=T_P(I_{YG}(u,v),I_{YG}(u,w));$ If $u>0$ or $vw>0$, then $I_{YG}(u,T_P(v,w))=I_{YG}(u,vw)={\left(vw\right)}^u=v^u{w}^u=T_P(I_{YG}(u,v),I_{YG}(u,w))$. Then according to Propostion 10, $(I_E,U)$ satisfies Equation (20).

(ii) 

Let $U({\displaystyle \frac{1}{4}},1)=U(1,{\displaystyle \frac{1}{4}})={\displaystyle \frac{1}{4}}$ and $I_1=I_{Lt}$ in (i). It can be checked that the couple $(I_1,T_{LK})$ satisfies Equation (3) and there is $u_0\in [0,1],v_0>0$ such that $I_1(u_0,v_0)=0$. However, it can be obtained that $I_E(0.1,U(0.3,1))=1\ne 0.25=U(I_E(0.1,0.3),I_E(0.1,1))$.

(iii) 

Let $U({\displaystyle \frac{1}{4}},1)=U(1,{\displaystyle \frac{1}{4}})=1$ and $I_0=I_{Gt}$ in (i). It can be checked that the couple $(I_0,T_P)$ satisfies Equation (3) and there is $u_0>0\mathrm{and}{v}_0<0$ such that $I_0(u_0,v_0)=1$. However, it can be obtained that $I_E(0.1,U(0.1,1))=0.25\ne 1=U(I_E(0.1,0.1),I_E(0.1,1))$.

4. On the Equation $\mathit{I}({\mathit{U}}_{\mathit{1}}(\mathit{u},\mathit{v}),\mathit{w}))={\mathit{U}}_{\mathit{2}}(\mathit{I}(\mathit{u},\mathit{w}),\mathit{I}(\mathit{v},\mathit{w}))$

In this section, we focus on Equation (6); it turns out that Equation (6) reduces to the trivial case under some requirements.

According to [41], if the triple $(I,U_1,U_2)$ satisfies Equation (6), then $U_1$ is conjunctive (disjunctive) if and only if $U_2$ is disjunctive (conjunctive). Moreover, we have the following results.

Proposition 11.

If the triple $(I_{I_1-I_2},U_1,U_2)$ satisfies Equation (6), then $U_2(e,e)=e$; if the implications $I_1$ and $I_2$ satisfy (NP), then $U_2$ is idempotent.

Proof. 

The results can be checked by putting $w=e$ and $u=v=1$ in Equation (6), respectively. □

Proposition 12.

Let the implications $I_1$ and $I_2$ satisfy (NP), and there is some $w_0\in [0,1[$ such that $I_1(·,w_0)$ or $I_2(·,w_0)$ is one-to-one. If the triple $(I_{I_1-I_2},U_1,U_2)$ satisfies Equation (6), then $U_1$ is idempotent.

Proof. 

We can get the result by putting $w=w_0$ and $u=v$ in Equation (6). □

The idempotent uninorms are extensively investigated in [59,61]. T-(co)norms can be uniquely determined by their idempotency while uninorms cannot. However, from Theorem 6, for an idempotent uninorm U, $U(u,v)=min(u,v)$ for every $(u,v)\in [0,e_u[\times [0,e_u[$ and $U(u,v)=max(u,v)$ for $(u,v)\in ]e_u,1]\times ]e_u,1]$. Then we can derive the results as follows.

Theorem 7.

Assume $I_1$ and $I_2$ are two fuzzy implications satisfying (NP) and the following conditions are satisfied.

(i) 

$I_1$ and $I_2$ are strictly decreasing with respect to the first variable whenever the second variable belongs to $]0,1[$;

(ii) 

There is some $w_0\in [0,1[$ such that $I_1(·,w_0)$ or $I_2(·,w_0)$ is one-to-one;

(iii) 

$I_1(u,v),I_2(u,v)\in ]0,1[$ whenever $u>0$ and $v<1$;

(iv) 

$U_i$ is a uninorms with its neutral element being $e_{u_i}$ $(i=1,2)$.

Then the triple $(I_{I_1-I_2},U_1,U_2)$ satisfies Equation (6) if and only if $(U_1=S_M,U_2=T_M)$ or $(U_1=T_M,U_2=S_M)$.

Proof. 

(⇒) Assume $0<e_{u_1}<1$, we focus on two cases as follows:

For the case $e_{u_2}>0$, Let $u,v,w\in ]0,min(e,e_{u_1},e_{u_2})[$, then $e{I}_1(\min (u,v),{\displaystyle \frac{w}{e}}))=\min (e{I}_1(u,{\displaystyle \frac{w}{e}}),e{I}_1(v,{\displaystyle \frac{w}{e}}))$, which contradicts the decreasingness of $I_1$ in the first variable whenever $u\ne v$;

For the case $e_{u_2}=0$, $U_2=S_M=max$. Let $u,v,w\in ]min(e,e_{u_1}),1[$, then we can get a similar contradiction as above.

Therefore, $e_{u_1}=0$ or $e_{u_1}=1$. Thus, we proceed with the proof in cases as follows.

(i)

If $e_{u_1}=0$, i.e., $U_1=S_M$, then Equation (6) becomes

$$I_{I_1-I_2}(\max (u,v),w))=U_2(I_{I_1-I_2}(u,w),I_{I_1-I_2}(v,w)).$$

Assume $e_{u_2}<1$. Put $u,v,w\in ]\max (e,e_{u_2}),1[$, then we have

$$I_{I_1-I_2}(\max (u,v),w))=\max (I_{I_1-I_2}(u,w),I_{I_1-I_2}(v,w)),$$

i.e., $e+(1-e)I_2(\max (u,v),{\displaystyle \frac{w-e}{1-e}}))=\max (e+(1-e)I_2(u,{\displaystyle \frac{w-e}{1-e}}),(e+(1-e)I_2(v,{\displaystyle \frac{w-e}{1-e}}))$, which contradicts the decreasingness of $I_2$ in the first variable whenever $u\ne v$ in this case. Thus $e_{u_2}=1$, i.e., $U_2=T_M$.

(ii)

If $e_{u_1}=1$, i.e., $U_1=T_M$, then Equation (6) becomes

$$I_{I_1-I_2}(\min (u,v),w))=U_2(I_{I_1-I_2}(u,w),I_{I_1-I_2}(v,w)).$$

Assume $e_{u_2}>0$. Put $u,v,w\in ]0,\min (e,e_{u_2})[$, then it holds that $I_{I_1-I_2}(\min (u,v),w))=\min (I_{I_1-I_2}(u,w),$ $I_{I_1-I_2}(v,w))$, i.e.,

$$e{I}_1(\min (u,v),{\displaystyle \frac{w}{e}}))=\min (e{I}_1(u,{\displaystyle \frac{w}{e}}),\left(e{I}_1(v,{\displaystyle \frac{w}{e}})\right),$$

which contradicts the decreasingness of $I_1$ in the first variable whenever $u\ne v$ in this case. Thus $e_{u_2}=0$, i.e., $U_2=S_M$.

To sum up, $(U_1=S_M,U_2=T_M)$ or $(U_1=T_M,U_2=S_M)$.

(⇐) It can be checked with the increasingness of $I_{I_1-I_2}$ in the first variable if $(U_1=S_M,U_2=T_M)$ or $(U_1=T_M,U_2=S_M)$.□

Corollary 1.

If h is an h-generator and $U_1\mathrm{and}{U}_2$ are uninorms, then the triple $(I^h,U_1,U_2)$ satisfies Equation (6) if and only if $(U_1=S_M,U_2=T_M)$ or $(U_1=T_M,U_2=S_M)$.

Under the referred conditions, Equation (6) reduces to the trivial cases. Nevertheless, the problem with the original implications being free from the restricted conditions in the threshold generation is still open. However, the existing results show that [41] that if both $U_1\mathrm{and}{U}_2$ are representable uninorms, then for Equation (6) there are no solutions I which are fuzzy implications. Thus, we can get the following result trivially.

Proposition 13.

Assume $I_1$ and $I_2$ are fuzzy implications, and $U_1$ and $U_2$ be representable uninorms, then the triple $(I_{I_1-I_2},U_1,U_2)$ does not satisfy Equation (6). In particular, if h is an h-generator, then the triple $(I^h,U_1,U_2)$ does not satisfy Equation (6).

5. Concluding Remarks

The threshold-generated implications preserve most of the established properties, particularly the distributivity over t-norms and t-conorms, under certain minimal necessary conditions. In this paper, under some necessary conditions, the threshold-generated implications also preserve the distributivity over fuzzy connectives with respect to Equation (7) when the uninorm U is restricted to some commonly referred classes of uninorms (see Table 3, where √ indicates the preservation of distributivity, while × does not.). At the same time, the threshold-generated implications may not preserve the distributivity for some other classes of uninorms, which are illustrated by various examples and propositions. As byproducts, it is shown that the h-implications do not satisfy the distributive Equation (7) when the uninorm is the representable uninorm in Theorem 4 and the idempotent uninorm in Theorem 6, respectively. It is shown that Equation (6) reduces to the trivial case if the threshold-generated implication is restricted to the referred requirement in Theorem 7. However, Equation (6) with more general conditions on the implications still needs extensive investigation.

Table 3. Main results on the preservation of distributivity about Equation (20).

Cases	${\mathcal{U}}_{\mathit{min}}$ or ${\mathcal{U}}_{\mathit{max}}$	${\mathcal{U}}_{\mathit{cos}}$	${\mathcal{U}}_{\mathit{R}}$	${\mathcal{U}}_{\mathit{I}}$
$e_u=e$	√ (Proposition 3)	× (Example 2)	× (Example 3)	× (Example 5)
$e_u<e$	√ (Proposition 7)	√ (Proposition 8)	—	—
$e_u>e$	√ (Proposition 9)	√ (Proposition 10)	—	—

In recent studies, many constructing methods are developed to generate new classes of fuzzy implications, such as those methods in [9,28]. The distributivity properties of the generated fuzzy over fuzzy connective are interesting topics for future research. In addition, the balanced generators introduced by Kamila and Mirko [62] bring some novel insights on local properties of t-norms from the angle of geometry. Since many implications can be generated by the methods in [11,18] using the particular balanced generators, it is quite necessary to investigate some geometric properties of the generated implications.