© Wuhan University 2026

0 Introduction

Fuzzy implication, as a generalization of Boolean implication in classical logic, serves as one of the fundamental connectives in fuzzy set theory and plays a significant role in approximate reasoning and fuzzy control systems^[1-5]. Consequently, based on the practical applications of fuzzy implication, it is necessary to construct and characterize a new implication model. Over the past decade, generating novel fuzzy implications has been a central theoretical focus, exemplified by the proposal of various types such as novel $\mathrm{e}$-threshold generated implications^[6], $(\phi ,k)$-implications^[7], $(h,l)$-implications^[8], $(D,N)$-implications^[9], and $(T,N)$-implications^[10]. According to the different construction methods, fuzzy implication generation approaches can be broadly categorized into three classes:

1) Implications generated via other essential fuzzy logical connectives, including fuzzy conjunctions,

disjunctions, and negations. These encompass $(S,N)$-, $R_{\mathrm{\top }}$-, $\mathrm{Q}\mathrm{L}$-, $(G,N)$-, and $(U^{\mathrm{2}},N)$-implications^[11-18].

2) Implications generated by additive or multiplicative generators of $t$-norms or $t$-conorms, along with their generalized variants. Examples include $f$-, $g$-, $h$-, $k$-, and $(\theta ,t)$-generated implications^[19-25].

3) Ordinal sum implications constructed through diverse ordinal sum operations^[26-34].

This paper focuses on methodologies for generating novel fuzzy implications via additive and multiplicative generators. To this end, we first review and analyze existing generator-based implication construction approaches to identify unresolved challenges. In Ref. [22], Yager introduced fuzzy implications generated by continuous additive generators of $t$-norms and $t$-conorms, namely $f$- and $g$-generated implications, and highlighted their critical role in fuzzy approximate reasoning. Subsequently, Balasubramaniam^[23] defined $h$-generated implications via continuous multiplicative generators of $t$-conorms. Building upon these foundations, Zhou^[24] observed that three of the four continuous generators of $t$-norms and $t$-conorms had been used for implication generation, and consequently proposed in 2021 the final category of $k$-generated implications—constructed from continuous multiplicative generators of $t$-norms—providing a complete characterization based on the law of importation and the flexible ordering property. Later, in 2022, Zhou^[25] further introduced $(\theta ,t)$-generated implications constructed via pairs of additive and multiplicative generators of $t$-norms combined with standard real addition, along with characterizations of two specialized subclasses.

Our research motivation stems from three primary aspects: Firstly, despite the introduction of numerous fuzzy implications, a vast majority remain unexplored, and significant open problems persist in discovering new models satisfying specific algebraic properties or characterizing subclasses of fuzzy implications. Consequently, the search for and characterization of novel classes of fuzzy implications constitute an ongoing pursuit. Furthermore, from both theoretical and applied perspectives, establishing a sufficient repertoire of fuzzy implications presents compelling interest, which explains its enduring status as a research focus for decades. Secondly, building upon the foundation of $(\theta ,t)$-generated implications and following a similar research trajectory, it is essential to elucidate whether a new class of fuzzy implications can be constructed using pairs of generators of $t$-conorms, and what novel theoretical contributions such a construction might yield. Thirdly, through this work, $(\xi ,s)$-generated implications provide a fundamentally new model of fuzzy implications, serving as a fundamental building block in the framework of constructing fuzzy implications via generators of $t$-norms and $t$-conorms.

In this paper, a new concept of fuzzy implication is proposed by means of multiplicative and additive generators of $t$-conorms. Then, we study the basic properties of $(\xi ,s)$-generated implications. Finally, the relationship between $(\xi ,s)$-generated implications and other common fuzzy implications is analyzed.

1 Preliminaries

In this section, we recall here only some fundamental definitions and results which shall be used in the paper. We begin with the definition of $t$-norms and $t$-conorms.

Definition 1^[35] A $t$-norm is a binary function $T:{[\mathrm{0,1}]}^{\mathrm{2}}\to [\mathrm{0,1}]$ such that $T$ is commutative, associative, nondecreasing in both variables, and has 1 as its neutral element. Dually, a $t$-conorm is a binary function $S:{[\mathrm{0,1}]}^{\mathrm{2}}\to [\mathrm{0,1}]$ such that $S$ is commutative, associative, non-decreasing in both variables, and has 0 as its neutral element.

Example 1^[35] The following are basic dual pairs of $t$-norms and $t$-conorms, where $x,y\in [\mathrm{0,1}]$,

(i) Minimum and Maximum pair:$T_M(x,y)=\mathrm{m}\mathrm{i}\mathrm{n}(x,y);S_M(x,y)=\mathrm{m}\mathrm{a}\mathrm{x}(x,y),x,y\in [\mathrm{0,1}].$

(ii) Product and Probabilistic Sum pair:$T_P(x,y)=xy;S_P(x,y)=x+y-xy,x,y\in [\mathrm{0,1}].$

(iii) Łukasiewicz pair: $T_L\left(x,y\right)=\mathrm{m}\mathrm{a}\mathrm{x}(x+y-\mathrm{1,0});S_L(x,y)=\mathrm{m}\mathrm{i}\mathrm{n}(x+y,\mathrm{1}),x,y\in [\mathrm{0,1}].$

(iv) Drastic Product and Sum pair: $T_D(x,y)=\mathrm{0}$ for $x,y\in [\mathrm{0,1})$ and otherwise $T_D(x,y)=\mathrm{m}\mathrm{i}\mathrm{n}(x,y)$; $S_D(x,y)=\mathrm{1}$ for $x,y\in (\mathrm{0,1}]$ and otherwise $S_D(x,y)=\mathrm{m}\mathrm{a}\mathrm{x}(x,y)$.

Definition 2^[1] A fuzzy implication is a binary function $I:{\left[\mathrm{0,1}\right]}^{\mathrm{2}}\to \left[\mathrm{0,1}\right]$ satisfying the following conditions:

(I1) $I$ is non-increasing in its first variable.

(I2) $I$ is non-decreasing in its second variable.

(I3) $I(\mathrm{0,0})=I(\mathrm{1,1})=\mathrm{1}$, and $I(\mathrm{1,0})=\mathrm{0}$.

Definition 3^[1] Let $I$ be a fuzzy implication, $N$ a fuzzy negation and $T$ a $t$-norm. Then $I$ is said to satisfy

(i) the ordering property (OP), if $\forall x,y\in [\mathrm{0,1}],I(x,y)=\mathrm{1}\iff x\le y.$

(ii) the neutral property (NP), if $\forall y\in [\mathrm{0,1}],I(\mathrm{1},y)=y.$

(iii) the identity principle (IP), if $\forall x\in [\mathrm{0,1}],I(x,x)=\mathrm{1}.$

(iv) the exchange principle (EP), if $\forall x,y,z\in [\mathrm{0,1}],I(x,I(y,z))=I(y,I(x,z)).$

(v) the contraposition property with respect to $N(\mathrm{C}\mathrm{P}),\mathrm{i}\mathrm{f}\forall x,y\in [\mathrm{0,1}],I(x,y)=I(N(y),N(x)).$

(vi) the latter boundary condition (CB), if $\forall x,y\in [\mathrm{0,1}],I(x,y)\ge y.$

Definition 4^[1] A fuzzy negation is a nonincreasing unary function $N:[\mathrm{0,1}]\to [\mathrm{0,1}]$ that satisfies $N(\mathrm{0})=\mathrm{1}$ and $N(\mathrm{1})=\mathrm{0}$. Moreover, a fuzzy negation $N$ is called

(i) strict if it is continuous and strictly decreasing on $[\mathrm{0,1}]$.

(ii) strong if it is an involution, i.e., $N(N(x))=x$ for all $x\in [\mathrm{0,1}]$.

(iii) an $a$-normal CD-negation (continuous and strictly decreasing negation) if it is continuous on $\left[\mathrm{0,1}\right]$ and is strictly decreasing on $\left[a,\mathrm{1}\right]$, where $a=\mathrm{s}\mathrm{u}\mathrm{p}\{x\in [\mathrm{0,1}]|N(x)=\mathrm{1}\}$ (see Ref. [24]).

For a fuzzy implication $I$, the function $N_I:\left[\mathrm{0,1}\right]\to \left[\mathrm{0,1}\right]$ defined by $N_I(x)=I(x,\mathrm{0})$ for all $x\in [\mathrm{0,1}]$. Then $N_I$ is a fuzzy negation, called the natural negation of $I$.

Definition 5^[21] Let $\left[a,b\right]$ and $\left[c,d\right]$ be two closed subintervals of the extended real line $\overline{\mathbb{R}}=\left[-\mathrm{\infty },\mathrm{\infty }\right]$ and let $f:\left[a,b\right]\to \left[c,d\right]$ be a monotone function. Then the function $f^{(-\mathrm{1})}:[c,d]\to [a,b]$, defined for each $y\in \left[c,d\right]$ by

$f^{(-\mathrm{1})}(y)=\mathrm{s}\mathrm{u}\mathrm{p}\{x\in [a,b]|(f(x)-y)(f(b)-f(a))<\mathrm{0}\},$

is called the pseudo-inverse of $f$. If $f$ is strictly monotone and the range of $f$ is denoted as Ran($f$), then the restriction of $f^{(-\mathrm{1})}$ to Ran($f$), i.e., the function $f^{(-\mathrm{1})}{|}_{\mathrm{R}\mathrm{a}\mathrm{n}(f)}:\mathrm{R}\mathrm{a}\mathrm{n}(f)\to [a,b]$ is also strictly monotone. Moreover, in this case we have the following identities:

(i) $f\circ f^{(-\mathrm{1})}{|}_{\mathrm{R}\mathrm{a}\mathrm{n}(f)}=\mathrm{i}{\mathrm{d}}_{\mathrm{R}\mathrm{a}\mathrm{n}(f)}$.

(ii) $f^{(-\mathrm{1})}\circ f=\mathrm{i}{\mathrm{d}}_{\left[a,b\right]}$.

Definition 6^[35] Let $T$ and $S$ be a continuous Archimedean $t$-norm and $t$-conorm, respectively.

(i) An additive generator (AG) of $T$ is a strictly decreasing and continuous function$t:\left[\mathrm{0,1}\right]\to \left[\mathrm{0},\mathrm{\infty }\right]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}t(\mathrm{1})=\mathrm{0}\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}T(x,y)=t^{(-\mathrm{1})}(t(x)+t(y))\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x,y\in [\mathrm{0,1}].$

(ii) A multiplicative generator (MG) of $T$ is a strictly increasing and continuous function $\theta :[\mathrm{0,1}]\to [\mathrm{0,1}]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\theta (\mathrm{1})=\mathrm{1}\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}T(x,y)={\theta }^{(-\mathrm{1})}(\theta (x)\theta (y))\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x,y\in [\mathrm{0,1}].$

(iii) An additive generator of $S$ is a strictly increasing and continuous function$s:[\mathrm{0,1}]\to [\mathrm{0},\mathrm{\infty }]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}s(\mathrm{0})=\mathrm{0}\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}S(x,y)=s^{(-\mathrm{1})}(s(x)+s(y))\mathrm{f}\mathrm{o}\mathrm{r}x,y\in [\mathrm{0,1}].$

(iv) A multiplicative generator of $S$ is a strictly decreasing and continuous function $\xi :[\mathrm{0,1}]\to [\mathrm{0,1}]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\xi (\mathrm{0})=\mathrm{1}\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}S(x,y)={\xi }^{(-\mathrm{1})}(\xi (x)\xi (\mathrm{1}))\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x,y\in [\mathrm{0,1}].$

The relationships between additive and multiplicative generators of $t$-norms and $t$-conorms are illustrated in Fig. 1.

Fig. 1 The relationships between the additive and multiplicative generators of a $\mathit{t}$-norm T and its dual $\mathit{t}$-conorm S: a commutative diagram

Definition 7^{[22, 24-25]} (i) An $f$-generator is a strictly decreasing and continuous function $f:[\mathrm{0,1}]\to [\mathrm{0},\mathrm{\infty }]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}f(\mathrm{1})=\mathrm{0}.$ For an $f$-generator $f$, the function $I_f:{\left[\mathrm{0,1}\right]}^{\mathrm{2}}\to \left[\mathrm{0,1}\right],$ given by

$I_f(x,y)=f^{-\mathrm{1}}(xf(y)),x,y\in \left[\mathrm{0,1}\right],$

with the convention $\mathrm{0}\cdot \mathrm{\infty }=\mathrm{0}$, is a fuzzy implication, called $f$-generated implication.

(ii) A $g$-generator is a strictly increasing and continuous function$g:[\mathrm{0,1}]\to [\mathrm{0},\mathrm{\infty }]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}g(\mathrm{0})=\mathrm{0}.$ For a $g$-generator $g$, the function $I_g:{\left[\mathrm{0,1}\right]}^{\mathrm{2}}\to [\mathrm{0,1}]$, given by

$I_g(x,y)=g^{(-\mathrm{1})}(\frac{\mathrm{1}}{x}g(y)),x,y\in \left[\mathrm{0,1}\right],$

with the convention $\frac{\mathrm{1}}{\mathrm{0}}=\mathrm{\infty }$ and $\mathrm{\infty }·\mathrm{0}=\mathrm{\infty }$, is a fuzzy implication, called $g$-generated implication.

(iii) An $h$-generator is a strictly decreasing and continuous function $h:[\mathrm{0,1}]\to [\mathrm{0,1}]$ with $h(\mathrm{0})=\mathrm{1}$. For an $h$-generator $h$, the function $I_h:{\left[\mathrm{0,1}\right]}^{\mathrm{2}}\to \left[\mathrm{0,1}\right]$, given by

$I_h(x,y)=h^{(-\mathrm{1})}(xh(y)),x,y\in \left[\mathrm{0,1}\right],$

is a fuzzy implication, called $h$-generated implication.

(iv) A $k$-generator is a strictly increasing and continuous function $k:\left[\mathrm{0,1}\right]\to \left[\mathrm{0,1}\right]$ with $k(\mathrm{1})=\mathrm{1}$. For a $k$-generator $k$, the function $I_k:{\left[\mathrm{0,1}\right]}^{\mathrm{2}}\to \left[\mathrm{0,1}\right]$, given by

$I_k(x,y)=k^{(-\mathrm{1})}(\frac{\mathrm{1}}{x}k(y)),x,y\in \left[\mathrm{0,1}\right],$

with the convention $\frac{\mathrm{0}}{\mathrm{0}}=\mathrm{1}$ and $\frac{\mathrm{1}}{\mathrm{0}}=\mathrm{\infty }$, is a fuzzy implication, called $k$-generated implication.

(v) Let $\theta$ be a multiplicative generator of a $t$-norm and $t$ an additive generator of another (possibly the same) $t$-norm. Then, the pair $(\theta ,t)$ will be called a $(\theta ,t)$-generator if $t(\mathrm{0})\ge \theta ({\mathrm{1}}^{-})-\theta (\mathrm{0})$. For a $(\theta ,t)$-generator $(\theta ,t)$, define a binary function $I_{\theta ,t}:{\left[\mathrm{0,1}\right]}^{\mathrm{2}}\to \left[\mathrm{0,1}\right]$ by

$I_{\theta ,t}(x,y)={\theta }^{(-\mathrm{1})}(\mathrm{m}\mathrm{i}\mathrm{n}(t(x)+\theta (y),\mathrm{1})),x,y\in \left[\mathrm{0,1}\right],$

then, $I_{\theta ,t}$ is a fuzzy implication, called fuzzy implication additively generated by $\theta$ and $t$, or simply called $(\theta ,t)$-generated implication.

Theorem 1^[1] For a function $I:{\left[\mathrm{0,1}\right]}^{\mathrm{2}}\to \left[\mathrm{0,1}\right]$, the following statements are equivalent:

(i) $I$ is an $(S,N)$-implication with a continuous (strict, strong) fuzzy negation $N$.

(ii) $I$ satisfies (I1), (EP), and $N_I$ is a continuous (strict, strong) fuzzy negation.

2 $(\mathit{\xi },\mathit{s})$-Generated Implications and Basic Properties

In this section, we introduce a new class of fuzzy implications by means of a pair of additive and multiplicative generators of $t$-conorms and then study their basic algebraic properties.

Definition 8   Let $\xi :\left[\mathrm{0,1}\right]\to \left[\mathrm{0,1}\right]$ be a multiplicative generator of a $t$-conorm and $s:\left[\mathrm{0,1}\right]\to \left[\mathrm{0,1}\right]$ an additive generator of another (possibly the same) $t$-conorm. If the pair $(\xi ,s)$ will be called a $(\xi ,s)$-generator if $s({\mathrm{1}}^{-})+\xi (\mathrm{1})\ge \mathrm{1}$. For a $(\xi ,s)$-generator $(\xi ,s)$, define a binary function $I_{\xi ,s}:{\left[\mathrm{0,1}\right]}^{\mathrm{2}}\to \left[\mathrm{0,1}\right]$ by

$I_{\xi ,s}(x,y)={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1}))),x,y\in \left[\mathrm{0,1}\right],(\mathrm{1})$

then, $I_{\xi ,s}$ is a fuzzy implication, called fuzzy implication additively generated by $\xi$ and $s$, or simply called $(\xi ,s)$-generated implication.

Proof   For any $x_{\mathrm{1}},x_{\mathrm{2}}\in \left[\mathrm{0,1}\right]$ with $x_{\mathrm{1}}\le x_{\mathrm{2}}$, since $\xi$ is strictly decreasing, its pseudo-inverse ${\xi }^{(-\mathrm{1})}$ is also strictly decreasing. Additionally, as $s$ is strictly increasing, we have:

$s(x_{\mathrm{1}})+\xi (y)-s(\mathrm{1})\le s(x_{\mathrm{2}})+\xi (y)-s(\mathrm{1}).$

Consequently,

$\begin{array}{l}{I}_{\xi ,s}(x_{\mathrm{1}},y)\\ ={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(x_{\mathrm{1}})+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1})))\\ \ge {\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(x_{\mathrm{2}})+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1})))=I_{\xi ,s}(x_{\mathrm{2}},y),\end{array}$

which implies that $I_{\xi ,s}$​ satisfies (I1).

For any $y_{\mathrm{1}},y_{\mathrm{2}}\in \left[\mathrm{0,1}\right]$ with $y_{\mathrm{1}}\le y_{\mathrm{2}}$, the strict monotonicity of $\xi$ implies:

$s(x)+\xi (y_{\mathrm{1}})-s(\mathrm{1})\ge s(x)+\xi (y_{\mathrm{2}})-s(\mathrm{1}).$

Thus,

$\begin{array}{l}{I}_{\xi ,s}(x,y_{\mathrm{1}})\\ ={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(x)+\xi (y_{\mathrm{1}})-s(\mathrm{1}),\xi (\mathrm{1})))\\ \le {\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}\left(s(x)+\xi (y_{\mathrm{2}})-s(\mathrm{1}),\xi (\mathrm{1})\right))=I_{\xi ,s}(x,y_{\mathrm{2}}),\end{array}$

proving that $I_{\xi ,s}$ satisfies (I2).

Next, we verify boundary conditions:

$\begin{array}{l}{I}_{\xi ,s}(\mathrm{0,0})={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(\mathrm{0})+\xi (\mathrm{0})-s(\mathrm{1}),\xi (\mathrm{1})))={\xi }^{(-\mathrm{1})}(\xi (\mathrm{1}))=\mathrm{1},\\ I_{\xi ,s}(\mathrm{1,1})={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(\mathrm{1})+\xi (\mathrm{1})-s(\mathrm{1}),\xi (\mathrm{1})))={\xi }^{(-\mathrm{1})}(\xi (\mathrm{1}))=\mathrm{1},\\ I_{\xi ,s}(\mathrm{1,0})={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(\mathrm{1})+\xi (\mathrm{0})-s(\mathrm{1}),\xi (\mathrm{1})))={\xi }^{(-\mathrm{1})}(\xi (\mathrm{0}))=\mathrm{0},\end{array}$

hence $I_{\xi ,s}$​ satisfies (I3).

Remark 1   (i) To align Definition 8 with the preliminary conventions, we designate $\xi$ as the $\xi$-generator and $s$ as the $s$-generator in the $(\xi ,s)$-generated implications. Notably, when $\xi$ or $s$ is discontinuous, the $(\xi ,s)$-generated implication need not be discontinuous (see Example 2 (iii)). Even if $\xi$ and $s$ are both continuous, there may exist discontinuous $(\xi ,s)$-generated implications (see Example 2 (iv)).

(ii) Our assumption $s({\mathrm{1}}^{-})+\xi (\mathrm{1})\ge \mathrm{1}$ ensures $I_{\xi ,s}(\mathrm{0,0})=\mathrm{1}$.

(iii) When $\xi (\mathrm{1})=\mathrm{0}$ and $s(\mathrm{1})=\mathrm{1}$, there exists a particular $(\xi ,s)$-generated implication

$I_{\xi ,s}\left(x,y\right)={\xi }^{\left(-\mathrm{1}\right)}\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(s\left(x\right)+\xi \left(y\right)-\mathrm{1,0}\right)\right),x,y\in \left[\mathrm{0,1}\right].$

(iv) By definition, $(\xi ,s)$-generated implications are different in general from any of $f$-, $g$-, $h$-, or $k$ -generated implications. We will discuss their relations in the following section.

Example 2 (i) Let $s(x)=x^a$ $(a>\mathrm{0})$ and $\xi (x)=\mathrm{1}-x$, which are the multiplicative generators of $S_P$​. The corresponding $(\xi ,s)$-generated implication is

$\begin{array}{l}{I}_{\xi ,s}(x,y)={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1})))\\ ={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(x^a-y,\mathrm{0}))\\ =\mathrm{1}-\mathrm{m}\mathrm{a}\mathrm{x}(x^a-y,\mathrm{0})\\ =\mathrm{m}\mathrm{i}\mathrm{n}(\mathrm{1}-x^a+y,\mathrm{1}).\end{array}$

When $a=\mathrm{1}$, this $(\xi ,s)$-generated implication reduces to the well-known Łukasiewicz implication $I_{\mathrm{L}\mathrm{K}}$. More generally, for continuous $(\xi ,s)$-generators satisfying $\xi =\mathrm{1}-s$ and $s(\mathrm{1})=\mathrm{1}$, the implication $I_{\xi ,s}$​ corresponds to the $s$-conjugate of $I_{\mathrm{L}\mathrm{K}}$, denoted $(I_{\mathrm{L}\mathrm{K}}{)}_s$​,

$\begin{array}{l}{I}_{\xi ,s}(x,y)\\ ={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1})))\\ ={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(x)+\mathrm{1}-s(y)-s(\mathrm{1}),\mathrm{1}-s(\mathrm{1})))\\ =\mathrm{s}\mathrm{u}\mathrm{p}\left\{z\in \left[\mathrm{0,1}\right]|\xi (z)>\mathrm{m}\mathrm{a}\mathrm{x}(s(x)+\mathrm{1}-s(y)-s(\mathrm{1}),\mathrm{1}-s(\mathrm{1}))\right\}\\ =\mathrm{s}\mathrm{u}\mathrm{p}\left\{z\in \left[\mathrm{0,1}\right]|s(z)<\mathrm{1}-\mathrm{m}\mathrm{a}\mathrm{x}(s(x)+\mathrm{1}-s(y)-s(\mathrm{1}),\mathrm{1}-s(\mathrm{1}))\right\}=\mathrm{s}\mathrm{u}\mathrm{p}\left\{z\in \left[\mathrm{0,1}\right]|s(z)<\mathrm{m}\mathrm{i}\mathrm{n}(s(\mathrm{1})-s(x)+s(y),s(\mathrm{1}))\right\}=s^{(-\mathrm{1})}(\mathrm{m}\mathrm{i}\mathrm{n}(s(\mathrm{1})-s(x)+s(y),s(\mathrm{1})))\\ =s^{(-\mathrm{1})}(\mathrm{m}\mathrm{i}\mathrm{n}(\mathrm{1}-s(x)+s(y),\mathrm{1}))\\ =(I_{\mathrm{L}\mathrm{K}}{)}_s(x,y).\end{array}$

(ii) Let $\lambda \in (-\mathrm{1},\mathrm{\infty })$. Consider the additive and multiplicative generators of the continuous Sugeno-Weber t-conorm $S_{\lambda }^{\mathrm{S}\mathrm{W}}$,

$s_{\lambda }(x)=\{\begin{array}{c}x,\lambda =\mathrm{0},\\ \frac{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda x)}{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda )},\lambda \in (-\mathrm{1,0})\bigcup (\mathrm{0},\mathrm{\infty }),\end{array}$

and

${\xi }_{\lambda }(x)=\{\begin{array}{c}{\mathrm{e}}^{-x},\lambda =\mathrm{0},\\ {\mathrm{e}}^{\frac{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda x)}{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda )}},\lambda \in (-\mathrm{1,0})\bigcup (\mathrm{0},\mathrm{\infty }),\end{array}$

When $\lambda =\mathrm{0}$, $I_{{\xi }_{\mathrm{0}},s_{\mathrm{0}}}^{\mathrm{S}\mathrm{W}}=I_{{\xi }_{\mathrm{\infty }},s_{\mathrm{\infty }}}^F$​. For $\lambda \in (-\mathrm{1,0})\bigcup (\mathrm{0},\mathrm{\infty }),\mathrm{t}\mathrm{h}\mathrm{e}({\xi }_{\lambda },s_{\lambda })-$Sugeno-Weber generated implication is

$I_{{\xi }_{\lambda },s_{\lambda }}^{\mathrm{S}\mathrm{W}}(x,y)=\{\begin{array}{c}\frac{{\mathrm{e}}^{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda )\mathrm{l}\mathrm{n}(\frac{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda x)}{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda )}+{\mathrm{e}}^{\frac{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda y)}{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda )}}-\mathrm{1})}-\mathrm{1}}{\lambda },x\ge A_{\lambda }(x,y),\\ \mathrm{1},\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}$

where $A_{\lambda }(x,y)=\frac{\mathrm{1}}{\lambda }\left({\mathrm{e}}^{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda )(\mathrm{e}+\mathrm{1}-{\mathrm{e}}^{\frac{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda y)}{\mathrm{l}\mathrm{n}(\mathrm{1}+\lambda )}})}-\mathrm{1}\right)$.

(iii) A discontinuous generator example. Let

$s(x)=\{\begin{array}{c}\mathrm{0},x=\mathrm{0},\\ x+\mathrm{1},\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}$

and $\xi (x)=\mathrm{1}-x$, the resulting $(\xi ,s)$-generated implication is $I_{\xi ,s}(x,y)=\mathrm{m}\mathrm{i}\mathrm{n}(\mathrm{1}-x+y,\mathrm{1}),$ which coincides with the Łukasiewicz implication $I_{\mathrm{L}\mathrm{K}}$. Notably, $I_{\xi ,s}$​ is continuous even though $s$ is discontinuous.

(iv) A discontinuous $(\xi ,s)$-generated implication example: Let $\xi (x)={\mathrm{e}}^{-x}$ (the multiplicative generator of $S_L$) and $s(x)=-\mathrm{l}\mathrm{n}(\mathrm{1}-x)$ (the additive generator of $S_P$​). For all $x,y\in \left[\mathrm{0,1}\right]$,

$\begin{array}{l}{I}_{\xi ,s}(x,y)={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1})))\\ ={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(-\mathrm{l}\mathrm{n}(\mathrm{1}-x)+{\mathrm{e}}^{-y}+\mathrm{l}\mathrm{n}\mathrm{0},{\mathrm{e}}^{-\mathrm{1}}))\\ =\{\begin{array}{c}\mathrm{1},x<\mathrm{1},\\ y,x=\mathrm{1},\end{array}\end{array}$

which corresponds to the Weber implication $I_{\mathrm{W}\mathrm{B}}$. This $(\xi ,s)$-generated implication $I_{\xi ,s}$​ is discontinuous despite the continuity of $\xi$ and $s$.

Proposition 1   Let $({\xi }_{\mathrm{1}},s_{\mathrm{1}})$, $({\xi }_{\mathrm{2}},s_{\mathrm{2}})$, and $(\xi ,s)$ be any $(\xi ,s)$-generators. Then

(i) If $\xi$ is continuous, then $I_{\xi ,s_{\mathrm{1}}}=I_{\xi ,s_{\mathrm{2}}}$​​ if and only if $s_{\mathrm{1}}=s_{\mathrm{2}}$​.

(ii) If ${\xi }_{\mathrm{1}}$, ${\xi }_{\mathrm{2}}$​, $s_{\mathrm{1}}$ and $s_{\mathrm{2}}$​ are all continuous, then $I_{{\xi }_{\mathrm{1}},s_{\mathrm{1}}}=I_{{\xi }_{\mathrm{2}},s_{\mathrm{2}}}$​​if and only if ${\xi }_{\mathrm{2}}=\frac{s_{\mathrm{2}}(\mathrm{1})}{s_{\mathrm{1}}(\mathrm{1})}({\xi }_{\mathrm{1}}-{\xi }_{\mathrm{1}}(\mathrm{1})+s_{\mathrm{1}}(\mathrm{1}))+{\xi }_{\mathrm{2}}(\mathrm{1})-s_{\mathrm{2}}(\mathrm{1})$ and $s_{\mathrm{2}}=\frac{s_{\mathrm{2}}(\mathrm{1})}{s_{\mathrm{1}}(\mathrm{1})}{s}_{\mathrm{1}}$.

(iii) If ${\xi }_{\mathrm{1}}$, ${\xi }_{\mathrm{2}}$​ and $s$ are continuous with ${\xi }_{\mathrm{1}}(\mathrm{1})={\xi }_{\mathrm{2}}(\mathrm{1})$, then $I_{{\xi }_{\mathrm{1}},s}=I_{{\xi }_{\mathrm{2}},s}$ if and only if ${\xi }_{\mathrm{1}}={\xi }_{\mathrm{2}}$​.

Proof   (i) Assume $\xi$ is continuous. The equality $I_{s_{\mathrm{1}},\xi }(x,y)=I_{s_{\mathrm{2}},\xi }(x,y)$ holds if and only if for all $x,y\in [\mathrm{0,1}]$,

$\begin{array}{l}{\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{1}}(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1})))\\ ={\xi }^{(-\mathrm{1})}(\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{2}}(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1}))),\end{array}$

if and only if

$\begin{array}{l}{\xi }^{-\mathrm{1}}(\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{1}}(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1})))\\ ={\xi }^{-\mathrm{1}}(\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{2}}(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1}))),\end{array}$

if and only if

$\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{1}}(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1}))=\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{2}}(x)+\xi (y)-s(\mathrm{1}),\xi (\mathrm{1}))$

if and only if for all $x\in [\mathrm{0,1}]$, $s_{\mathrm{1}}(x)=s_{\mathrm{2}}(x).$

(ii) Let ${\xi }_{\mathrm{1}}$, ${\xi }_{\mathrm{2}}$​, $s_{\mathrm{1}}$ and $s_{\mathrm{2}}$​ are all continuous. Then $I_{{\xi }_{\mathrm{1}},s_{\mathrm{1}}}=I_{{\xi }_{\mathrm{2}},s_{\mathrm{2}}}$​​ implies for all $x,y\in [\mathrm{0,1}]$,

$\begin{array}{l}{\xi }_{\mathrm{1}}^{-\mathrm{1}}(\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{1}}(x)+{\xi }_{\mathrm{1}}(y)-s_{\mathrm{1}}(\mathrm{1}),{\xi }_{\mathrm{1}}(\mathrm{1})))\\ ={\xi }_{\mathrm{2}}^{-\mathrm{1}}(\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{2}}(x)+{\xi }_{\mathrm{2}}(y)-s_{\mathrm{2}}(\mathrm{1}),{\xi }_{\mathrm{2}}(\mathrm{1}))),\end{array}$

which is equivalent to

$\begin{array}{l}{\xi }_{\mathrm{2}}\circ {\xi }_{\mathrm{1}}^{-\mathrm{1}}(\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{1}}(x)+{\xi }_{\mathrm{1}}(y)-s_{\mathrm{1}}(\mathrm{1}),{\xi }_{\mathrm{1}}(\mathrm{1})))\\ =\mathrm{m}\mathrm{a}\mathrm{x}(s_{\mathrm{2}}(x)+{\xi }_{\mathrm{2}}(y)-s_{\mathrm{2}}(\mathrm{1}),{\xi }_{\mathrm{2}}(\mathrm{1})).\end{array}$

Let $\alpha ={\xi }_{\mathrm{2}}\circ {\xi }_{\mathrm{1}}^{-\mathrm{1}}$,​ $\beta =s_{\mathrm{2}}\circ s_{\mathrm{1}}^{-\mathrm{1}}$, $u=s_{\mathrm{1}}(x)\in \left[\mathrm{0},s_{\mathrm{1}}(\mathrm{1})\right]$, $v={\xi }_{\mathrm{1}}(y)\in \left[{\xi }_{\mathrm{1}}(\mathrm{1}),\mathrm{1}\right]$, then $\alpha :\left[{\xi }_{\mathrm{1}}(\mathrm{1}),\mathrm{1}\right]\to \left[{\xi }_{\mathrm{2}}(\mathrm{1}),\mathrm{1}\right]$ and $\beta :\left[\mathrm{0},s_{\mathrm{1}}(\mathrm{1})\right]\to \left[\mathrm{0},s_{\mathrm{2}}(\mathrm{1})\right]$ are two strictly increasing bijections satisfying $\alpha (v)={\xi }_{\mathrm{2}}(y)$ and $\beta (u)=s_{\mathrm{2}}(x)$. The equation becomes

$\begin{array}{c}\alpha (\mathrm{m}\mathrm{a}\mathrm{x}(u+v-s_{\mathrm{1}}(\mathrm{1}),{\xi }_{\mathrm{1}}(\mathrm{1})))\\ =\mathrm{m}\mathrm{a}\mathrm{x}(\beta (u)+\alpha (v)-s_{\mathrm{2}}(\mathrm{1}),{\xi }_{\mathrm{2}}(\mathrm{1})),(\mathrm{2})\end{array}$

Since $\alpha ({\xi }_{\mathrm{1}}(\mathrm{1}))={\xi }_{\mathrm{2}}(\mathrm{1})$, equation (2) reduces to $\alpha (u+v-s_{\mathrm{1}}(\mathrm{1}))=\beta (u)+\alpha (v)-s_{\mathrm{2}}(\mathrm{1})$. Letting $v={\xi }_{\mathrm{1}}(\mathrm{1})$, then we have

$\beta (u)=\alpha (u+{\xi }_{\mathrm{1}}(\mathrm{1})-s_{\mathrm{1}}(\mathrm{1}))-{\xi }_{\mathrm{2}}(\mathrm{1})+s_{\mathrm{2}}(\mathrm{1})$

and consequently,

$\begin{array}{l}\beta (\mathrm{m}\mathrm{a}\mathrm{x}(u+v-s_{\mathrm{1}}(\mathrm{1}),s_{\mathrm{1}}(\mathrm{1})))\\ =\alpha (\mathrm{m}\mathrm{a}\mathrm{x}(u+v-s_{\mathrm{1}}(\mathrm{1}),s_{\mathrm{1}}(\mathrm{1}))+{\xi }_{\mathrm{1}}(\mathrm{1})-s_{\mathrm{1}}(\mathrm{1}))-{\xi }_{\mathrm{2}}(\mathrm{1})+s_{\mathrm{2}}(\mathrm{1})\\ =\alpha (\mathrm{m}\mathrm{a}\mathrm{x}(u+v-s_{\mathrm{1}}(\mathrm{1})+{\xi }_{\mathrm{1}}(\mathrm{1})-s_{\mathrm{1}}(\mathrm{1}),{\xi }_{\mathrm{1}}(\mathrm{1})))-{\xi }_{\mathrm{2}}(\mathrm{1})+s_{\mathrm{2}}(\mathrm{1})\\ =\mathrm{m}\mathrm{a}\mathrm{x}(\beta (u)+\alpha (v-s_{\mathrm{1}}(\mathrm{1})+{\xi }_{\mathrm{1}}(\mathrm{1}))-s_{\mathrm{2}}(\mathrm{1}),{\xi }_{\mathrm{2}}(\mathrm{1}))-{\xi }_{\mathrm{2}}(\mathrm{1})+s_{\mathrm{2}}(\mathrm{1})\\ =\mathrm{m}\mathrm{a}\mathrm{x}(\beta (u)+\alpha (v-s_{\mathrm{1}}(\mathrm{1})+{\xi }_{\mathrm{1}}(\mathrm{1}))-{\xi }_{\mathrm{2}}(\mathrm{1})+s_{\mathrm{2}}(\mathrm{1})-s_{\mathrm{2}}(\mathrm{1}),s_{\mathrm{2}}(\mathrm{1}))\\ =\mathrm{m}\mathrm{a}\mathrm{x}(\beta (u)+\beta (v)-s_{\mathrm{2}}(\mathrm{1}),s_{\mathrm{2}}(\mathrm{1})).\end{array}$