Cross-Migrativity of Continuous T-Conorms over I h Implications

The cross-migrativity can be regarded as the weaker form of the commuting equation, which plays a crucial part in the framework of fuzzy connectives. This paper studies the cross-migrativity of continuous t-conorms over $I^h$ implications. We obtain full characterizations for the cross-migrativity of continuous t-conorms over $I^h$ implications.

• Cross-Migrativity
• Continuous T-Conorms
• Implications

1. Introduction

Cross migrativity is a recently studied property of binary operations defined on the unit interval. The scholars had been extensively investigated the cross-migrativity between conjunctive operators, such as t-norms [1] - [3] , overlap functions [4] [5] and uninorms [6] - [9] . Hence, the cross-migrativity of t-conorms over fuzzy implications [10] provides a new direction for the discussion of the relationships between disjunctive operators and fuzzy implications. Moreover, He and Fang [11] discussed the cross-migrativity of continuous t-conorms over generated implications, such as $\left(f,g\right)$-, $k$-, $h$- and $\left(\theta ,t\right)$-generated implications. However, there is a new fuzzy implication called the h-implications [12] , which are generated by an additive generator of a representable uninorm in a similar way of Yager’s f- and g-implications, that has not been discussed. Therefore, the $\alpha$-cross-migrativity of continuous t-conorms over $h$-implications should be studied more thoroughly in this context to remedy that defect.

This paper is organized as follows. Section 2 briefly reviews several basic notions and results. Section 3 focuses on characterizations of $\alpha$-cross-migrativity for continuous t-conorms over $I^h$ implications. Section 4 concludes our research.

2. Preliminaries

Definition 2.1 ( [13] ) A t-conorm $S$ is called [(i)]

i) Archimedean, if for all $u,w\in \left(0,1\right)$ there exists an $n\in \mathbb{N}$ such that $u_S^{\left[n\right]}>w$. If $S$ is continuous, then $S$ is Archimedean iff $S\left(u,u\right)>u$ for all $u\in \left]0,1\right[$.

ii) strict, if it is continuous and strictly monotone.

iii) nilpotent, if it is continuous and each $u\in \left(0,1\right)$ is a nilpotent element of $S$.

iv) positive, if $S\left(u,w\right)=1$, then either $u=1$ or $w=1$.

Theorem 2.2 ( [13] ) Let $S$ be a t-conorm. [(i)]

i) $S$ is continuous and Archimedean iff $S$ is either strict or nilpotent.

ii) If $S$ is continuous and Archimedean, then $S$ is positive iff $S$ is strict.

iii) $S$ is strict iff there exists a strictly decreasing bijection $\psi :\left[0,1\right]\to \left[0,1\right]$ such that $S\left(u,w\right)={\psi }^{-1}\left(\psi \left(u\right)\cdot \psi \left(w\right)\right)$ for all $u,w\in \left[0,1\right]$.

iv) $S$ is nilpotent iff there exists a strictly increasing bijection $\psi :\left[0,1\right]\to \left[0,1\right]$ such that $S\left(u,w\right)={\psi }^{-1}\left(\text{min}\left\{\psi \left(u\right)+\psi \left(w\right),1\right\}\right)$ for all $u,w\in \left[0,1\right]$.

v) $S$ is continuous iff there is a unique countable family ${\left(\left]a_{\lambda },e_{\lambda }\right[\right)}_{\lambda \in A}$ of pairwise disjoint open subintervals of $\left[0,1\right]$ and a family of continuous Archimedean t-conorms ${\left(S_{\lambda }\right)}_{\lambda \in A}$ such that

$S\left(u,w\right)=\{\begin{array}{ll}{a}_{\lambda }+\left(e_{\lambda }-a_{\lambda }\right)S_{\lambda }\left(\frac{u-a_{\lambda }}{e_{\lambda }-a_{\lambda }},\frac{w-a_{\lambda }}{e_{\lambda }-a_{\lambda }}\right),\hfill & \text{if}\left(u,w\right)\in {\left[a_{\lambda },e_{\lambda }\right]}^2;\hfill \\ \text{max}\left(u,w\right),\hfill & \text{otherwise}.\hfill \end{array}$

In this case, we will write $S={\left(\langle a_{\lambda },e_{\lambda },S_{\lambda }\rangle \right)}_{\lambda \in A}$.

Definition 2.3 ( [12] ) Let $e\in \left]0,1\right[$, $h:\left[0,1\right]\to \left[-\infty ,+\infty \right]$ be a strictly increasing and continuous function with $h\left(0\right)=-\infty$, $h\left(e\right)=0$ and $h\left(1\right)=+\infty$. Then function $I:{\left[0,1\right]}^2\to \left[0,1\right]$ defined as

$I\left(u,w\right)=\{\begin{array}{ll}1,\hfill & \text{if}u=0;\hfill \\ h^{-1}\left(u\cdot h\left(w\right)\right),\hfill & \text{if}u>0\text{and}w\le e;\hfill \\ h^{-1}\left(\frac{1}{u}\cdot h\left(w\right)\right),\hfill & \text{if}u>0\text{and}w>e;\hfill \end{array}$

is called an $h$-implication. The function $h$ itself is called an $h$-generator (with respect to $e$) of the implication function $I$ defined as above. We write it in this case $I^h$ instead of $I$.

Definition 2.4 ( [10] ) Consider $\alpha \in \left[0,1\right]$, $I$ be a fuzzy implication. A t-conorm $S$ is said to be $\alpha$-cross-migrative with respect to $I$, if for all $u,w\in \left[0,1\right]$,

$I\left(u,S\left(\alpha ,w\right)\right)=S\left(w,I\left(u,\alpha \right)\right).$(1)

3. Cross-Migrativity of Continuous T-Conorms over I h Implications

In this section, we characterize the $\alpha$-cross-migrativity of continuous t-conorms over $I^h$ implications.

Notice that Equation (1) is true for $\alpha =1$. Thus, we only consider the case $\alpha \in \left[0,1\right[$. Firstly, we discuss $\alpha =0$.

Theorem 3.1. Let $S$ be a t-conorm and $I^h$ be an $h$-generated implication. Then $\left(S,I^h\right)$ is not 0-cross-migrative.

Proof. Suppose $\left(S,I^h\right)$ is 0-cross-migrative, we have for all $u,w\in \left[0,1\right]$,

$I\left(u,w\right)=I\left(u,S\left(0,w\right)\right)=S\left(w,I\left(u,0\right)\right).$

The above equation is true for $u=0$. Thus, we only discuss the case $u>0$. By Definition 2.3, we obtain for all $u\ne 1$,

$h^{-1}\left(u\cdot h\left(w\right)\right)=S\left(w,h^{-1}\left(u\cdot h\left(0\right)\right)\right)=S\left(w,h^{-1}\left(-\infty \right)\right)=S\left(w,0\right)=w$ for all $w\le e$;

$h^{-1}\left(\frac{1}{u}\cdot h\left(w\right)\right)=S\left(w,h^{-1}\left(u\cdot h\left(0\right)\right)\right)=S\left(w,h^{-1}\left(-\infty \right)\right)=S\left(w,0\right)=w$ for all $w>e$.

By the monotonicity of $h^{-1}$, we have $u=1$, which is a contradiction.

In the equal, we discuss $\alpha \in \left]0,e\right[$.

Theorem 3.2. Take $\alpha \in \left]0,e\right[$. Then $S$ be $\alpha$-cross-migrative over $I^h$ iff

$S\left(u,w\right)=\{\begin{array}{ll}{h}^{-1}\left(\frac{h\left(w\right)}{h\left(\alpha \right)}\cdot h\left(S\left(\alpha ,u\right)\right)\right),\hfill & \text{if}S\left(\alpha ,u\right)\le e\text{and}w\in \left[\alpha ,e\right[;\hfill \\ h^{-1}\left(\frac{h\left(\alpha \right)}{h\left(w\right)}\cdot h\left(S\left(\alpha ,u\right)\right)\right),\hfill & \text{if}S\left(\alpha ,u\right)>e\text{and}w\in \left[\alpha ,e\right[.\hfill \end{array}$(2)

Proof. ( $\Rightarrow$). By the Definition 2.4, we have for all $u,v\in \left[0,1\right]$,

$I\left(v,S\left(\alpha ,u\right)\right)=S\left(u,I\left(v,\alpha \right)\right).$

Consider $v=0$. Then it is obvious. By Definition 2.3, one obtains

$h^{-1}\left(v\cdot h\left(S\left(\alpha ,u\right)\right)\right)=S\left(u,h^{-1}\left(v\cdot h\left(\alpha \right)\right)\right)$ for all $v\in \left]0,1\right]$ and $S\left(\alpha ,u\right)\le e$;

$h^{-1}\left(\frac{1}{v}\cdot h\left(S\left(\alpha ,u\right)\right)\right)=S\left(u,h^{-1}\left(v\cdot h\left(\alpha \right)\right)\right)$ for all $v\in \left]0,1\right]$ and $S\left(\alpha ,u\right)>e$.

Denote $w=h^{-1}\left(v\cdot h\left(\alpha \right)\right)$. Then one obtains

$\alpha =h^{-1}\left(h\left(\alpha \right)\right)\le w=h^{-1}\left(v\cdot h\left(\alpha \right)\right)<h^{-1}\left(0\cdot h\left(\alpha \right)\right)=e.$

That is $w\in \left[\alpha ,e\right[$. Then we have

$S\left(u,w\right)=\{\begin{array}{ll}{h}^{-1}\left(\frac{h\left(w\right)}{h\left(\alpha \right)}\cdot h\left(S\left(\alpha ,u\right)\right)\right),\hfill & \text{if}S\left(\alpha ,u\right)\le e\text{and}w\in \left[\alpha ,e\right[;\hfill \\ h^{-1}\left(\frac{h\left(\alpha \right)}{h\left(w\right)}\cdot h\left(S\left(\alpha ,u\right)\right)\right),\hfill & \text{if}S\left(\alpha ,u\right)>e\text{and}w\in \left[\alpha ,e\right[.\hfill \end{array}$

( $\Leftarrow$). It is obvious.

Proposition 3.3. Take $\alpha \in \left]0,e\right[$. If $S$ is $\alpha$-cross-migrative over $I^h$, then $S\left(u,u\right)>u$ for all $u\in \left]\alpha ,e\right[$.

Proof. We obtain for all $S\left(\alpha ,u\right)\le e$ and $w\in \left[\alpha ,e\right[$ by Theorem 3.2,

$h\left(\alpha \right)\cdot h\circ S\left(u,w\right)=h\left(w\right)\cdot h\circ S\left(\alpha ,u\right).$(3)

Suppose that there exists some $u_0\in \left]\alpha ,e\right[$ such that $S\left(u_0,u_0\right)=u_0$. Then we obtain for all $u\in \left[\alpha ,u_0\right]$,

$u_0=\text{max}\left(u,u_0\right)\le S\left(u,u_0\right)\le S\left(u_0,u_0\right)=u_0.$

Hence we have $S\left(\alpha ,u_0\right)=u_0<e$. Let $u=u_0$ and $w\in \left]\alpha ,u_0\right[$ in Equation (3). Then one obtains

$h\left(\alpha \right)\cdot h\left(u_0\right)=h\left(\alpha \right)\cdot h\circ S\left(u_0,w\right)=h\left(w\right)\cdot h\circ S\left(\alpha ,u_0\right)=h\left(w\right)\cdot h\left(u_0\right).$

Thus we have $h\left(w\right)=h\left(\alpha \right)$, i.e., $w=\alpha$, which is a contradiction. □

Lemma 3.4. Take $\alpha \in \left]0,e\right[$. If $S$ be $\alpha$-cross-migrative over $I^h$, then $S$ is not nilpotent.

Proof. Assume that $S$ is nilpotent, there exists a strictly increasing $\psi :\left[0,1\right]\to \left[0,1\right]$ such that $S\left(u,w\right)={\psi }^{-1}\left(\text{min}\left\{\psi \left(u\right)+\psi \left(w\right),1\right\}\right)$ for all $u,w\in \left[0,1\right]$. Hence there must exist $u_{\alpha }$ such that $\psi \left(\alpha \right)+\psi \left(u_{\alpha }\right)=1$. Choose $w_0\in \left]\alpha ,e\right[$ such that $u_0={\psi }^{-1}\left(1-\psi \left(w_0\right)\right)$. Then we obtain $\psi \left(u_0\right)+\psi \left(w_0\right)=1$ and

$u_0={\psi }^{-1}\left(1-\psi \left(w_0\right)\right)<{\psi }^{-1}\left(1-\psi \left(\alpha \right)\right)=u_{\alpha }.$

Thus $u_0\in \left]0,u_{\alpha }\right[$. Hence we have $S\left(\alpha ,u_0\right)<1$. Let $u_0\in \left]0,u_{\alpha }\right[$ and $w_0\in \left]\alpha ,e\right[$ in Equation (2). Then one obtains

$1=S\left(u_0,w_0\right)=\{\begin{array}{ll}{h}^{-1}\left(\frac{h\left(w_0\right)}{h\left(\alpha \right)}\cdot h\left(S\left(\alpha ,u_0\right)\right)\right)\le h^{-1}\left(\frac{h\left(w_0\right)}{h\left(\alpha \right)}\cdot h\left(e\right)\right)=e<1,\hfill & \text{if}S\left(\alpha ,u_0\right)\le e;\hfill \\ h^{-1}\left(\frac{h\left(\alpha \right)}{h\left(w_0\right)}\cdot h\left(S\left(\alpha ,u_0\right)\right)\right)<h^{-1}\left(\frac{h\left(w_0\right)}{h\left(\alpha \right)}\cdot h\left(1\right)\right)=1,\hfill & \text{if}S\left(\alpha ,u_0\right)>e;\hfill \end{array}$

which is a contradiction. □

Theorem 3.5. Take $\alpha \in \left]0,e\right[$, $S$ be a continuous t-conorm. Then $S$ be $\alpha$-cross-migrative over $I^h$ iff one of the items holds. [(i)]

i) $S$ is strict. There exists a strictly decreasing bijection $\psi :\left[0,1\right]\to \left[0,1\right]$ such that

$\rho \left(\psi \left(\alpha \right)\right)\cdot \rho \left(\psi \left(u\right)\cdot \psi \left(w\right)\right)=\rho \left(\psi \left(w\right)\right)\cdot \rho \left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)\le e$ and $w\in \left[\alpha ,e\right[$;

$\rho \left(\psi \left(w\right)\right)\cdot \rho \left(\psi \left(u\right)\cdot \psi \left(w\right)\right)=\rho \left(\psi \left(\alpha \right)\right)\cdot \rho \left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)>e$ and $w\in \left[\alpha ,e\right[$;

where $\rho :\left[0,\psi \left(\alpha \right)\right]\to \left[h\left(\alpha \right),+\infty \right]$ and $\rho \left(u\right)=h\circ {\psi }^{-1}\left(u\right)$.

ii) $\beta \in \left]0,\alpha \right[$ and $S=\left(\langle 0,\beta ,S_a\rangle ,\langle \beta ,1,S_b\rangle \right)$, where $\beta =\text{sup}\left\{u\in \left[0,1\right[|S\left(u,u\right)=u\right\}$. There exists a strictly decreasing bijection $\xi :\left[\beta ,1\right]\to \left[0,1\right]$ such that for all $u\in \left[\beta ,1\right]$ and $w\in \left[\alpha ,e\right[$,

$\rho \left(\xi \left(\alpha \right)\right)\cdot \rho \left(\xi \left(u\right)\cdot \xi \left(w\right)\right)=\rho \left(\xi \left(w\right)\right)\cdot \rho \left(\xi \left(\alpha \right)\cdot \xi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)\le e$;

$\rho \left(\xi \left(w\right)\right)\cdot \rho \left(\xi \left(u\right)\cdot \xi \left(w\right)\right)=\rho \left(\xi \left(\alpha \right)\right)\cdot \rho \left(\xi \left(\alpha \right)\cdot \xi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)>e$;

where $\rho :\left[0,\xi \left(\alpha \right)\right]\to \left[h\left(\alpha \right),+\infty \right]$ and $\rho \left(u\right)=h\circ {\xi }^{-1}\left(u\right)$.

iii) $\beta =\alpha$ and $S=\left(\langle 0,\beta ,S_a\rangle ,\langle \beta ,1,S_b\rangle \right)$, where $\beta =\text{sup}\left\{u\in \left[0,1\right[|S\left(u,u\right)=u\right\}$. Then

$S\left(u,w\right)=\{\begin{array}{ll}\alpha S_a\left(\frac{u}{\alpha },\frac{w}{\alpha }\right),\hfill & \text{if}\left(u,w\right)\in {\left[0,\alpha \right]}^2;\hfill \\ h^{-1}\left(\frac{h\left(u\right)\cdot h\left(w\right)}{h\left(\alpha \right)}\right),\hfill & \text{if}u\in \left[\alpha ,1\right],S\left(\alpha ,u\right)\le e\text{and}w\in \left[\alpha ,e\right[;\hfill \\ h^{-1}\left(\frac{h\left(u\right)\cdot h\left(\alpha \right)}{h\left(w\right)}\right),\hfill & \text{if}u\in \left[\alpha ,1\right],S\left(\alpha ,u\right)>e\text{and}w\in \left[\alpha ,e\right[;\hfill \\ \alpha +\left(1-\alpha \right)S_b\left(\frac{u-\alpha }{1-\alpha },\frac{w-\alpha }{1-\alpha }\right),\hfill & \text{if}\left(u,w\right)\in \left[\alpha ,1\right]\times \left[e,1\right];\hfill \\ \text{max}\left(u,w\right),\hfill & \text{otherwise}.\hfill \end{array}$

Proof. Necessity. Denote $\beta =\text{sup}\left\{u\in \left[0,1\right[|S\left(u,u\right)=u\right\}$. Then we have $u\in \left[0,\alpha \right]$ from Proposition 3.3. By the continuity of $S$, $\beta$ satisfies $S\left(\beta ,\beta \right)=\beta$. The conclusions are verified as follows.

Case (i) $\beta =0$.

We have $S\left(u,u\right)>u$ for all $u\in \left]0,1\right[$, that is, $S$ is Archimedean by Definition 2.1(i). By Theorem 2.2(i) and Lemma 3.4, $S$ is strict. Hence, there exists a strictly decreasing bijection $\psi :\left[0,1\right]\to \left[0,1\right]$ such that $S\left(u,w\right)={\psi }^{-1}\left(\psi \left(u\right)\cdot \psi \left(w\right)\right)$ for all $u,w\in \left[0,1\right]$. By Theorem 3.2, one obtains

$h\left(\alpha \right)\cdot h\circ {\psi }^{-1}\left(\psi \left(u\right)\cdot \psi \left(w\right)\right)=h\left(w\right)\cdot h\circ {\psi }^{-1}\left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)\le e$ and $w\in \left[\alpha ,e\right[$;

$h\left(w\right)\cdot h\circ {\psi }^{-1}\left(\psi \left(u\right)\cdot \psi \left(w\right)\right)=h\left(\alpha \right)\cdot h\circ {\psi }^{-1}\left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)>e$ and $w\in \left[\alpha ,e\right[$.

Let $\rho =h\circ {\psi }^{-1}$. Then one obtains

$\rho \left(\psi \left(\alpha \right)\right)\cdot \rho \left(\psi \left(u\right)\cdot \psi \left(w\right)\right)=\rho \left(\psi \left(w\right)\right)\cdot \rho \left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)\le e$ and $w\in \left[\alpha ,e\right[$;

$\rho \left(\psi \left(w\right)\right)\cdot \rho \left(\psi \left(u\right)\cdot \psi \left(w\right)\right)=\rho \left(\psi \left(\alpha \right)\right)\cdot \rho \left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)>e$ and $w\in \left[\alpha ,e\right[$.

Case (ii) $\beta \in \left]0,\alpha \right[$

By Theorem 2.2(v), $S$ is an ordinal sum t-conorm. Because $S_b$ is obviously Archimedean by Definition 2.1(i) and Proposition 3.3. Hence $S_b$ is strict by Theorem 2.2(i) and Lemma 3.4. There exists a strictly decreasing bijection $\psi :\left[0,1\right]\to \left[0,1\right]$ such that $S_b\left(u,w\right)={\psi }^{-1}\left(\psi \left(u\right)\cdot \psi \left(w\right)\right)$ for all $u,w\in \left[0,1\right]$. Denote $\xi :\left[\beta ,1\right]\to \left[0,1\right]$ with $\xi \left(u\right)=\psi \left(\frac{u-\beta }{1-\beta }\right)$ for all $u\in \left[\beta ,1\right]$. Then $\xi$ is a strictly increasing bijection and $S\left(u,w\right)={\xi }^{-1}\left(\xi \left(u\right)\circ \xi \left(w\right)\right)$ for all $u,w\in \left[\beta ,1\right]$. Hence, we have for all $u\in \left[\beta ,1\right]$,

$h\left(\alpha \right)\cdot h\left({\xi }^{-1}\left(\xi \left(u\right)\circ \xi \left(w\right)\right)\right)=h\left(w\right)\cdot h\left({\xi }^{-1}\left(\xi \left(u\right)\circ \xi \left(\alpha \right)\right)\right)$ for all $S\left(\alpha ,u\right)\le e$ and $w\in \left[\alpha ,e\right[$;

$h\left(w\right)\cdot h\left({\xi }^{-1}\left(\xi \left(u\right)\circ \xi \left(w\right)\right)\right)=h\left(\alpha \right)\cdot h\left({\xi }^{-1}\left(\xi \left(u\right)\circ \xi \left(\alpha \right)\right)\right)$ for all $S\left(\alpha ,u\right)>e$ and $w\in \left[\alpha ,e\right[$.

Let $\rho \left(u\right)=h\circ {\xi }^{-1}\left(u\right)$. Then we obtain

$\rho \left(\xi \left(\alpha \right)\right)\cdot \rho \left(\xi \left(u\right)\cdot \xi \left(w\right)\right)=\rho \left(\xi \left(w\right)\right)\cdot \rho \left(\xi \left(\alpha \right)\cdot \xi \left(u\right)\right)$ for all $u\in \left[\beta ,1\right]$, $S\left(\alpha ,u\right)\le e$ and $w\in \left[\alpha ,e\right[$;

$\rho \left(\xi \left(w\right)\right)\cdot \rho \left(\xi \left(u\right)\cdot \xi \left(w\right)\right)=\rho \left(\xi \left(\alpha \right)\right)\cdot \rho \left(\xi \left(\alpha \right)\cdot \xi \left(u\right)\right)$ for all $u\in \left[\beta ,1\right]$, $S\left(\alpha ,u\right)>e$ and $w\in \left[\alpha ,e\right[$.

Case (iii) $\beta =\alpha$.

By Theorem 2.2(v), $S$ is an ordinal sum t-conorm. One has $S_b$ which is strict and Archimedean in a similar way as for Case (ii). By the continuity of $S$ and $S\left(\alpha ,\alpha \right)=\alpha$, we have $S\left(u,\alpha \right)=\text{max}\left(u,\alpha \right)=u$ for all $u\in \left[\alpha ,1\right]$. We have for all $u\in \left[\alpha ,1\right]$ by Theorem 3.2,

$S\left(u,w\right)=\{\begin{array}{ll}{h}^{-1}\left(\frac{h\left(w\right)}{h\left(\alpha \right)}\cdot h\left(S\left(\alpha ,u\right)\right)\right)=h^{-1}\left(\frac{h\left(u\right)\cdot h\left(w\right)}{h\left(\alpha \right)}\right),\hfill & \text{if}S\left(\alpha ,u\right)\le e\text{and}w\in \left[\alpha ,e\right[;\hfill \\ h^{-1}\left(\frac{h\left(\alpha \right)}{h\left(w\right)}\cdot h\left(S\left(\alpha ,u\right)\right)\right)=h^{-1}\left(\frac{h\left(u\right)\cdot h\left(\alpha \right)}{h\left(w\right)}\right),\hfill & \text{if}S\left(\alpha ,u\right)>e\text{and}w\in \left[\alpha ,e\right[.\hfill \end{array}$

Thus we obtain the form of $S$.

Sufficiency. (i) $\beta =0$.

By item (i), we have

$\rho \left(\psi \left(\alpha \right)\right)\cdot \rho \left(\psi \left(u\right)\cdot \psi \left(w\right)\right)=\rho \left(\psi \left(w\right)\right)\cdot \rho \left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)\le e$ and $w\in \left[\alpha ,e\right[$;

$\rho \left(\psi \left(w\right)\right)\cdot \rho \left(\psi \left(u\right)\cdot \psi \left(w\right)\right)=\rho \left(\psi \left(\alpha \right)\right)\cdot \rho \left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)$ for all $S\left(\alpha ,u\right)>e$ and $w\in \left[\alpha ,e\right[$;

where $\rho =h\circ {\psi }^{-1}$. Hence one obtains for all $w\in \left[\alpha ,e\right[$,

$h\left({\psi }^{-1}\left(\psi \left(\alpha \right)\right)\right)\cdot h\left({\psi }^{-1}\left(\psi \left(u\right)\cdot \psi \left(w\right)\right)\right)=h\left({\psi }^{-1}\left(\psi \left(w\right)\right)\right)\cdot h\left({\psi }^{-1}\left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)\right)$ for all $S\left(\alpha ,u\right)\le e$;

$h\left({\psi }^{-1}\left(\psi \left(w\right)\right)\right)\cdot h\left({\psi }^{-1}\left(\psi \left(u\right)\cdot \psi \left(w\right)\right)\right)=h\left({\psi }^{-1}\left(\psi \left(\alpha \right)\right)\right)\cdot h\left({\psi }^{-1}\left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)\right)$ for all $S\left(\alpha ,u\right)>e$.

That is,

$h\left(\alpha \right)\cdot h\circ S\left(u,w\right)=h\left(w\right)\cdot h\circ S\left(\alpha ,u\right)$ for all $S\left(\alpha ,u\right)\le e$ and $w\in \left[\alpha ,e\right[$;

$h\left(w\right)\cdot h\circ S\left(u,w\right)=h\left(\alpha \right)\cdot h\circ S\left(\alpha ,u\right)$ for all $S\left(\alpha ,u\right)>e$ and $w\in \left[\alpha ,e\right[$.

Therefore, $\left(S,I^h\right)$ is $\alpha$-cross-migrative by Theorem 3.2.

In the sequel, we only verify the sufficiency for item (ii), i.e., $\beta \in \left]0,\alpha \right[$. The item (iii) can be proven in a similar way. We prove Equation (2) in the following cases.

Case (a) $u\in \left[0,\beta \right]$ and $w\in \left[\alpha ,e\right[$.

By the continuity of $S$ and $S\left(\alpha ,\alpha \right)=\alpha$, we have $S\left(u,w\right)=\text{max}\left(u,w\right)=w$ and $S\left(\alpha ,u\right)=\text{max}\left(u,\alpha \right)=\alpha <e$. Hence we obtain

$h\left(\alpha \right)\cdot h\left(w\right)=h\left(\alpha \right)\cdot h\left(S\left(u,w\right)\right)=h\left(w\right)\cdot h\left(S\left(\alpha ,u\right)\right)=h\left(\alpha \right)\cdot h\left(w\right)$ for all $\left(u,w\right)\in \left[0,\beta \right]\times \left[\alpha ,e\right[$.

Case (b) $u\in \left[\beta ,1\right]$ and $w\in \left[\alpha ,e\right[$.

The following can be proven in a similar way as for the sufficiency of item (i),

$h\left(\alpha \right)\cdot h\left(S\left(u,w\right)\right)=h\left(w\right)\cdot h\left(S\left(\alpha ,u\right)\right)$ for all $u\in \left[\beta ,1\right]$, $S\left(\alpha ,u\right)\le e$ and $w\in \left[\alpha ,e\right[$;

$h\left(w\right)\cdot h\left(S\left(u,w\right)\right)=h\left(\alpha \right)\cdot h\left(S\left(\alpha ,u\right)\right)$ for all $u\in \left[\beta ,1\right]$, $S\left(\alpha ,u\right)>e$ and $w\in \left[\alpha ,e\right[$.

Therefore, $\left(S,I^h\right)$ is $\alpha$-cross-migrative by Theorem 3.2. □

In the equal, we characterize $\alpha =e$.

Theorem 3.6. Let $S$ be a t-conorm and $I^h$ be an $h$-generated implication. Then $\left(S,I^h\right)$ is not $e$-cross-migrative.

Proof. Suppose $\left(S,I^h\right)$ is $e$-cross-migrative, we have for all $u\in \left]0,1\right[$ and $S\left(\alpha ,w\right)\le e$,

$h^{-1}\left(u\cdot h\left(S\left(e,w\right)\right)\right)=S\left(w,h^{-1}\left(u\cdot h\left(e\right)\right)\right)=S\left(w,h^{-1}\left(0\right)\right)=S\left(w,e\right).$

Hence one obtains for all $u\in \left]0,1\right[$ and $S\left(\alpha ,w\right)\le e$,

$h\left(S\left(w,e\right)\right)=u\cdot h\left(S\left(e,w\right)\right).$

Then we obtain $u=1$, which is a contradiction. □

Finally, we discuss $\alpha \in \left]e,1\right[$.

Theorem 3.7. Take $\alpha \in \left]e,1\right[$. Then $S$ be $\alpha$-cross-migrative over $I^h$ iff

$S\left(u,w\right)=h^{-1}\left(\frac{h\left(w\right)}{h\left(\alpha \right)}\cdot h\left(S\left(\alpha ,u\right)\right)\right)$ for all $\left(u,w\right)\in \left[0,1\right]\times \left[\alpha ,1\right]$.(4)

Proof. ( $\Rightarrow$). By definition 2.4, we have for all $u,v\in \left[0,1\right]$,

$I\left(v,S\left(\alpha ,u\right)\right)=S\left(u,I\left(v,\alpha \right)\right).$

Consider $v=0$. Then it is obvious. By the property of $S$, one obtains $S\left(\alpha ,u\right)\ge \text{max}\left(\alpha ,u\right)\ge \alpha >e$ for all $u\in \left[0,1\right]$. One has for all $\left(v,u\right)\in \left]0,1\right]\times \left[0,1\right]$

$h^{-1}\left(\frac{1}{v}\cdot h\left(S\left(\alpha ,u\right)\right)\right)=S\left(u,h^{-1}\left(\frac{1}{v}\cdot h\left(\alpha \right)\right)\right).$

Denote $w=h^{-1}\left(\frac{1}{v}\cdot h\left(\alpha \right)\right)$. Then one obtains

$\alpha =h^{-1}\left(h\left(\alpha \right)\right)\le w=h^{-1}\left(\frac{1}{v}\cdot h\left(\alpha \right)\right)=I^h\left(v,\alpha \right)<I^h\left(0,\alpha \right)=1.$

It is $w\in \left[\alpha ,1\right[$. Hence we obtain for all $\left(u,w\right)\in \left[0,1\right[\times \left[\alpha ,1\right]$

$S\left(u,w\right)=h^{-1}\left(\frac{h\left(w\right)}{h\left(\alpha \right)}\cdot h\left(S\left(\alpha ,u\right)\right)\right).$

Consider $w=1$. Then we have $1=S\left(w,1\right)=h^{-1}\left(\frac{h\left(1\right)}{h\left(\alpha \right)}\cdot h\left(S\left(\alpha ,u\right)\right)\right)=h^{-1}\left(+\infty \right)=1$. Thus we obtain the conclusion.

( $\Leftarrow$). It is obvious. □

Proposition 3.8. Take $\alpha \in \left]e,1\right[$. If $S$ be $\alpha$-cross-migrative over $I^h$, then $S\left(u,u\right)>u$ for all $u\in \left]\alpha ,1\right[$.

Proof. By Theorem 3.7, we have for all $u\in \left[0,1\right]$ and $w\in \left[\alpha ,1\right]$,

$h\left(\alpha \right)\cdot h\left(S\left(u,w\right)\right)=h\left(w\right)\cdot h\left(S\left(\alpha ,u\right)\right).$(5)

Suppose that there exists some $u_0\in \left]\alpha ,1\right[$ such that $S\left(u_0,u_0\right)=u_0$. Then we have for all $u\in \left[\alpha ,u_0\right]$,

$u_0=\text{max}\left(u,u_0\right)\le S\left(u,u_0\right)\le S\left(u_0,u_0\right)=u_0.$

Let $u=u_0$ and $w\in \left]\alpha ,u_0\right[$ in Equation (5). Then one has

$h\left(u_0\right)\cdot h\left(\alpha \right)=h\left(\alpha \right)\cdot h\left(S\left(u_0,w\right)\right)=h\left(w\right)\cdot h\left(S\left(\alpha ,u_0\right)\right)=h\left(w\right)\cdot h\left(u_0\right).$

It implies $h\left(\alpha \right)=h\left(w\right)$, i.e., $\alpha =w$, which is a contradiction. □

Lemma 3.9. Take $\alpha \in \left]e,1\right[$. If $S$ be $\alpha$-cross-migrative over $I^h$, then $S$ is not nilpotent.

Proof. Assume that $S$ is nilpotent, there exists a strictly increasing bijection $\psi :\left[0,1\right]\to \left[0,1\right]$ such that $S\left(u,w\right)={\psi }^{-1}\left(\text{min}\left\{\psi \left(u\right)+\psi \left(w\right),1\right\}\right)$ for all $u,w\in \left[0,1\right]$. Hence there must exist $u_{\alpha }$ such that $\psi \left(\alpha \right)+\psi \left(u_{\alpha }\right)=1$. Choose $w_0\in \left]\alpha ,1\right[$ such that $u_0={\psi }^{-1}\left(1-\psi \left(w_0\right)\right)$. Then one obtains $\psi \left(u_0\right)+\psi \left(w_0\right)=1$ and

$u_0={\psi }^{-1}\left(1-\psi \left(w_0\right)\right)<{\psi }^{-1}\left(1-\psi \left(\alpha \right)\right)=u_{\alpha }.$

That is $u_0\in \left]0,u_{\alpha }\right[$. Hence we have $S\left(\alpha ,u_0\right)<1$. Let $u_0\in \left]0,u_{\alpha }\right[$ and $w_0\in \left]\alpha ,1\right[$ in Equation (4). Then one obtains

$1=S\left(u_0,w_0\right)=h^{-1}\left(\frac{h\left(w_0\right)}{h\left(\alpha \right)}\cdot h\left(S\left(\alpha ,u_0\right)\right)\right)<h^{-1}\left(\frac{h\left(w_0\right)}{h\left(\alpha \right)}\cdot h\left(1\right)\right)=h^{-1}\left(+\infty \right)=1,$

which is a contradiction.

Theorem 3.10. Take $\alpha \in \left]e,1\right[$, $S$ be a continuous t-conorm. Then $S$ be $\alpha$-cross-migrative over $I^h$ iff one of the items holds. [(i)]

i) $S$ is strict. There exists a strictly decreasing bijection $\psi :\left[0,1\right]\to \left[0,1\right]$ such that

$\rho \left(\psi \left(\alpha \right)\right)\cdot \rho \left(\psi \left(u\right)\cdot \psi \left(w\right)\right)=\rho \left(\psi \left(w\right)\right)\cdot \rho \left(\psi \left(\alpha \right)\cdot \psi \left(u\right)\right)$ for all $\left(u,w\right)\in \left[0,1\right]\times \left[\alpha ,1\right]$.

where $\rho :\left[0,\psi \left(\alpha \right)\right]\to \left[h\left(\alpha \right),+\infty \right]$ and $\rho \left(u\right)=h\circ {\psi }^{-1}\left(u\right)$.

ii) $\beta \in \left]0,\alpha \right[$ and $S=\left(\langle 0,\beta ,S_a\rangle ,\langle \beta ,1,S_b\rangle \right)$, where $\beta =\text{sup}\left\{u\in \left[0,1\right[|S\left(u,u\right)=u\right\}$. There exists a strictly decreasing bijection $\xi :\left[\beta ,1\right]\to \left[0,1\right]$ such that

$\rho \left(\xi \left(\alpha \right)\right)\cdot \rho \left(\xi \left(u\right)\cdot \xi \left(w\right)\right)=\rho \left(\xi \left(w\right)\right)\cdot \rho \left(\xi \left(\alpha \right)\cdot \xi \left(u\right)\right)$ for all $\left(u,w\right)\in \left[\beta ,1\right]\times \left[\alpha ,1\right]$,

where $\rho :\left[0,\xi \left(\alpha \right)\right]\to \left[h\left(\alpha \right),+\infty \right]$ and $\rho \left(u\right)=h\circ {\xi }^{-1}\left(u\right)$.

iii) $\beta =\alpha$ and $S=\left(\langle 0,\beta ,S_a\rangle ,\langle \beta ,1,S_b\rangle \right)$, where $\beta =\text{sup}\left\{u\in \left[0,1\right[|S\left(u,u\right)=u\right\}$. Then

$S\left(u,w\right)=\{\begin{array}{ll}\alpha S_a\left(\frac{u}{\alpha },\frac{w}{\alpha }\right),\hfill & \text{if}\left(u,w\right)\in {\left[0,\alpha \right]}^2;\hfill \\ h^{-1}\left(\frac{h\left(u\right)\cdot h\left(w\right)}{h\left(\alpha \right)}\right),\hfill & \text{if}\left(u,w\right)\in {\left[\alpha ,1\right]}^2;\hfill \\ \text{max}\left(u,w\right),\hfill & \text{otherwise}.\hfill \end{array}$

Proof. It can be proven in a similar way as for Theorem 3.5.

4. Conclusion

In this paper, full characterizations for the $\alpha$-cross-migrativity of continuous t-conorms over $I^h$ implications are obtained. Moreover, we investigate all solutions of the cross-migrativity equation for all possible combinations of continuous t-conorms and $I^h$ implications.

References

1. Fodor, J., Klement, E.P. and Mesiar, R. (2012) Cross-Migrative Triangular Norms. International Journal of Intelligent Systems, 27, 411-428. >https://doi.org/10.1002/int.21526

2. Li, S., Qin, F. and Fodor, J. (2015) On the Cross-Migrativity with Respect to Continuous T-Norms. International Journal of Intelligent Systems, 30, 550-562. >https://doi.org/10.1002/int.21708

3. Su, Y., Zong, W., Qin, F. and Zhao, B. (2019) The Cross-Migrativity with Respect to Continuous Triangular Norms Revisited. Information Sciences, 486, 114-118. >https://doi.org/10.1016/j.ins.2019.02.029

4. Qiao, J. and Zhao, B. (2022) On Α-Cross-Migrativity of Overlap (0-Overlap) Functions. IEEE Transactions on Fuzzy Systems, 30, 448-461. >https://doi.org/10.1109/tfuzz.2020.3040038

5. Zhu, K., Zeng, X. and Qiao, J. (2022) On the Cross-Migrativity between Uninorms and Overlap (Grouping) Functions. Fuzzy Sets and Systems, 451, 113-129. >https://doi.org/10.1016/j.fss.2022.10.009

6. Li, W. and Qin, F. (2021) On the Cross-Migrativity of Uninorms Revisited. International Journal of Approximate Reasoning, 130, 246-258. >https://doi.org/10.1016/j.ijar.2020.12.012

7. Zhan, H. and Liu, H. (2016) The Cross-Migrative Property for Uninorms. Aequationes Mathematicae, 90, 1219-1239. >https://doi.org/10.1007/s00010-016-0437-8

8. Zhan, H. and Liu, H. (2017) A Corrigendum to “Cross-Migrative Uninorms with Different Neutral Elements”. Journal of Intelligent&Fuzzy Systems, 33, 1303-1311. >https://doi.org/10.3233/jifs-17351

9. Zhan, H. and Liu, H. (2017) Cross-Migrative Uninorms with Different Neutral Elements. Journal of Intelligent&Fuzzy Systems, 32, 1877-1889. >https://doi.org/10.3233/jifs-161206

10. Fang, B.W. (2023) On Alpha-Cross-Migrativity of T-Conorms over Fuzzy Implications. Fuzzy Sets and Systems, 466, Article 108463. >https://doi.org/10.1016/j.fss.2022.12.019

11. He, F.M. and Fang, B.W. (2024) Characterizations for the Alpha-Cross-Migrativity of Continuous T-Conorms over Generated Implications. University of Sistan and Baluchestan.

12. Massanet, S. and Torrens, J. (2011) On a New Class of Fuzzy Implications: H-Implications and Generalizations. Information Sciences, 181, 2111-2127. >https://doi.org/10.1016/j.ins.2011.01.030

13. Klement, E.P., Mesiar, R. and Pap, E. (2000) Triangular Norms. Kluwer Academic Publishers.