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\begin{document}

%********************************************************
\title
    {Aggregating fuzzy QL-subimplications: \\ conjugate and  dual  constructions\thanks{This work is supported by the Brazilian funding agencies CAPES, FAPERGS (Ed. PqG 06/2011, under the process number 11/1520-1); This article extends the work published by IEEE Xplore Pos-Proceedings of WEIT 2013,which was indicated as a select paper to TEMA.}}

%\author
%    {R.~H.~S.~REISER, I.~C.~K.~BENÍTEZ, A.~C.~YAMIN\,
%     \thanks{\{ickbenitez,reiser,yamin\}@inf.ufpel.edu.br}\,,
%     Centro de Desenvolvimento Tecnológico,
%      Universidade Federal de Pelotas,  96010-610 Pelotas, RS, Brasil
%     \\ \\
%      B.~R.~C.~BEDREGAL\,
%     \thanks{bedregal@dimap.ufrn.br}\,
%     Departamento de Inform\'atica e Matem\'atica Aplicada,  Universidade Federal do Rio Grande do Norte,  59078-970 Natal, RN, Brasil
%}


\criartitulo

\runningheads {Reiser et. al.}{Aggregating fuzzy QL-subimplications}

\begin{abstract}
Fuzzy (S,N)- and QL-subimplication classes can be obtained by a distributive   $n$-ary aggregation  performed over the  families  $\mathcal{T}$ of t-subnorms and  $\mathcal{S}$ of  t-subconorms  along with a fuzzy negation.
Since these classes of subimplications are explicitly represented by t-subconorms and t-subnorms verifying the generalized associativity, the corresponding (S,N)- and  QL-subimplications,  referred as $I_{\mathcal{S},N}$ and $I_{\mathcal{S},T,N}$, are  characterized as distributive $n$-ary aggregation together with related generalizations as the exchange and neutrality principles.
Based on these results, the both subclasses $\mathcal{I}_{S,N}$ and $\mathcal{QL}$ of (S,N)- and QL-subimplications which are obtained by the median aggregation operation performed over the standard negation $N_S$ together with  the families  of t-subnorms and  t-subconorms   $\mathcal{S}_P$ and  $\mathcal{T}_P$, respectively. In particular,  the subclass $\mathcal{T}_P$ extends the product t-norm $T_P$ as well as  $\mathcal{S}_P$ extends the algebraic sum $S_P$. As the main results, the family of subimplications $I_{\mathcal{S}_P,N}$ and $I_{\mathcal{S}_P,T_P,N}$ extends the  implication class by preserving the corresponding properties. We also present an extension from (S,N)- and QL-subimplications to  (S,N)- and QL-implications and discuss dual and conjugate constructions.

{\bf Palavras-chave}. median aggregation, fuzzy t-sub(co)norms, fuzzy (sub)implications, QL-implications, (S,N)-implications
\end{abstract}


%********************************************************
\newsec{Introdução}\label{intro}


The study of aggregation operators is a large domain, supported by using aggregation concepts  modelling uncertainty in distinct fields as social, engineering or economical problems which are based on fuzzy logic (FL)~\cite{Mayorand2007,RSK10,IPJ13,GMS13}. Aggregations are important operators in the construction of fuzzy inference  of expert systems. Consequently, they have been applied to many fields of approximate reasoning, e.g.  image processing, data mining, pattern recognition, fuzzy relational equations and fuzzy morphology, see~\cite{KMP00, CKKM02, Torra05,Sola10,BBF11,WLZ12,BGB13}.

 Despite potential distinct areas for applications of aggregation operators, this paper deals with the current status of the theory of aggregation operators in FL and also considers some of their main properties: symmetry, monotonicity, idempotency, homogeneity and distribution. Moreover, many other extensions of fuzzy logic make use of aggregation operators,  e.g. Interval-valued Fuzzy Logic \cite{CDK04, DK05,At08,BDSR10,DBSR11,BBJCF12,RB13b,RBR13}, Intuitionistic Fuzzy Logic \cite{CDK04,LiW10,CBD11,LinZ12,BBP08,REI11} and Hesitant Fuzzy Logic \cite{XiaX11,XXC13,BRBLMT13a}.

Distinguished classes of aggregation operators have been studied in the literature, e.g. the average, the minimum and the maximum, as well as some classical generalizations like the (ordered) weighted mean and the k-order statistics. This work considers the median aggregation, which is applied into a family of  fuzzy connectives  to generate new fuzzy connectives,  preserving the same properties verified by the corresponding family.

Following the studies  presented in~\cite{RBB13} and~\cite{BRYB13}, by relaxing the neutral element property related to triangular (co)norms,   the class $\mathcal{T}$ ($\mathcal{S}$) of
$t$-sub(co)norms is considered. Additionally, the fuzzy (S,N)-subimplication class, explicitly represented by fuzzy negations and  such class $\mathcal{S}$ of  fuzzy t-subconorms,  is also reported. In particular, generalizations of the  product t-norm and probabilistic sum are taken into account and provide interesting examples based on the median aggregation. Since this study considers $n$-ary aggregations,  generalized  associativity, exchange principle and distributivity properties also need to be considered.

In~\cite{BRYB13}, the class $\mathcal{J}$ of fuzzy QL-subimplications is introduced, which is obtained by the median  aggregation performed over a family of t-sub(co)norms $\mathcal{T}(\mathcal{S})$ along with fuzzy negations. These results state the  following constructions as equivalent:
\begin{description}
\item [($i$)] Firstly, we can aggregate all the $t$-sub(co)norms ($T_i$($S_i$)) and then  generate a class $\mathcal{I}_{QL}$ of QL-subimplications;
\item [($ii$)] Secondly, in other order, we can obtain each QL-subimplication which is expressed by composition of a $t$-sub(co)norm and a fuzzy negation and then, by aggregating all the QL-subimplications related to the median  we obtain the same class $\mathcal{I}_{QL}$.
\end{description}
Thus, a new class of  QL-subimplication is obtained. Analogous results in~\cite{RBB13} are reported to (S,N)-implications and R-implications generated by the aggregation of t-sub(co)norms and
fuzzy negations.

As the main contribution, in this work, the converse construction presented in~\cite{BRYB13} is now considered by stating the conditions under which  (S,N)- or QL-subimplications can be extended in order to obtain the corresponding  (S,N)- or  QL-implications. We also discuss the $N$-dual and conjugate constructions of aggregate operators.

The paper is organized as follows.
The preliminaries  in Section~\ref{sec-1} are concerned with fuzzy connectives and their algebraic properties.
Section~\ref{sec-2} reports concepts of aggregation functions together with their main properties and examples.
Focusing  on the median operator and the two classes of $t$-subconorm and $t$-subnorm we analyse the corresponding  properties.
Section~\ref{sec-3} considers both classes, (S,N)-(sub)implications and QL-(sub)implications and their conjugate and dual constructions. 
The main results concerned with aggregating QL-subimplications by applying the median operator are described in Section\ref{sec-4}
Moreover, it is shown that the median operator preserves (S,N)- and QL-implication classes. Lastly, the conclusion and final remarks are presented.


\newsec{Fuzzy Connectives}\label{sec-1}
In the following, basic concepts of an automorphism on the unit interval $U$,  fuzzy negations and  fuzzy subimplications are reported \cite{BDSR10,SHI08}

\begin{defTEMAi}\cite[Def. 0]{BBS03}
A mapping $\rho\colon U\to U$ is an \textbf{order automorphism} if it is continuous, strictly increasing and verifies the boundary conditions $\rho(0)\hspace{-0.1cm}=\hspace{-0.1cm}0$ and $\rho(1)\hspace{-0.1cm}=\hspace{-0.1cm}1$, i.e., if it is an increasing bijection on $U$.
\end{defTEMAi}

Order automorphisms are closed under composition and inverse operators. When $\rho(\vec{x}) =(\rho(x_1),\ldots ,\rho(x_n))\in U^{n}$, the action of an order automorphism $\rho$ on a function $f\colon U^{n}\to U$, refereed as $f^{\rho}$ and named \textbf{$\rho$-conjugate of $f$}, is defined as 
\begin{align}\label{automorphism}
  f^\rho(\vec{x}) &= \rho^{-1}(f (\rho(\vec{x}))), \vec{x} = (x_1, \ldots , x_n) \in U^{n}.
  \end{align}
 The family of all automorphisms is referred as $Aut(U)$. %Moreover, when $f^\rho= f$ we can say that function  $f$ has a $\rho$-invariant operator.

\subsection{Fuzzy negations}

Let $U=[0,1]$ be the unit interval. A \textbf{fuzzy negation} (FN) $N\colon U \to U$ satisfies:
   \begin{description}
       \item [$\mathbf{N1}:$] $N(0) = 1$ and $N(1) = 0$;
         %
       \item [$\mathbf{N2}:$] If $x \geq y$ then $N(x)\leq N(y)$, $\forall$ $x,y \in U$. %\hspace{1.0cm}
   \end{description}
FNs satisfying the involutive property are called \textbf{strong} fuzzy negations (SFNs):
   \begin{description}
       \item [$\mathbf{N3}:$] $N(N(x))=x$, $\forall$ $x \in U$.
   \end{description}
%And, a continuous FN is \textbf{strict}~\cite{KMP00}, when it satisfies
%   \begin{description}
%       \item $\mathbf{N4}:$ if $x > y$ then $N(x)< N(y)$, $\forall$ $x,y \in U$.
%\end{description}
%Strong FNs are also strict negations~\cite{BBS03}.
The standard negation $N_S(x)=1-x$ is a strong fuzzy negation.

  Let $N$ be a FN and $f\colon U^n\to U$ be a real function.  Then, for all \mbox{ $\vec{x}=(x_1,x_2, \ldots, x_n)\in U^n$ },  the \textbf{$N-$dual function} of $f$ is given by the expression:
    \begin{equation}\label{eq func dual}
           f_N(\vec{x})=N(f(N(x_1), N(x_2),\ldots ,N(x_n)))= N(f(N(\vec{x}))).
    \end{equation}

Notice that, when $N$ is involutive, $(f_{N})_N=f$, that is the $N$-dual function of $f_N$ coincides with $f$. In addition, if $f=f_N$ then it is clear that $f$ is a self-dual function. Other properties of fuzzy negations and related main extensions can be founded in \cite{KMP00,KMP04a} and \cite{BJ10}.

\subsection{Fuzzy Subimplications}

A function $I:U^2 \to U$ is a \textbf{fuzzy subimplicator} if it satisfies the  conditions:
  \begin{description}
     \item [$\mathbf{I0}:$] $I(1,1)=I(0,1)=I(0,0)=1;$
  \end{description}
When a fuzzy subimplicator $I:U^2 \to U$ also satisfies this boundary condition:
   \begin{description}
      \item [$\mathbf{I1}:$] $I(1,0)=0$;
   \end{description}
 $I$ is called \textbf{fuzzy implicator}. And, a fuzzy (sub)implicator $I$ satisfying the properties:
     \begin{description}
       \item [$\mathbf{I2}:$] If $x \leq z$ then $I(x,y) \geq I(z,y)$ (left antitonicity);
       \item [$\mathbf{I3}:$] If $y \leq z$ then $I(x,y) \leq I(x,z)$ (right isotonicity);
        \item [$\mathbf{I4}:$]  $I(0,y)=1$ (left boundary property);
     \end{description}
 $I$ is called a \textbf{fuzzy (sub)implication}~\cite[Def. 6]{CDK04}\cite{Kit93}.

\begin{propTEMAi}~\cite[Prop.~4.10]{RBB13}\label{pro-1}
The following statements are equivalent:\\
1. \hspace{-0.1cm} $I\colon \hspace{-0.1cm} U^2  \hspace{-0.1cm}\to U$ is an (S,N)-implication underlying a continuous FN $N$ and a t-subconorm $S$ at point $0$;\\
2. \hspace{-0.1cm} $I$  is continuous at point $x=1$ in the first component, satisfying $\mathbf{I3}$ and the two additional conditions:
 \begin{description}
  \item [$\mathbf{I5}:$] \textbf{Exchange Principle}:
   $I(x,I(y,z))= I(y,I(x,z))$, for all $x,y,z \in U$ ;    %I7 exchange priciple
  \item [$\mathbf{I6}:$] \textbf{Contrapositive Symmetry}: $I(x,y)=I(N(y),N(x))$, for all $x,y \in U$.%I12
  \end{description}
\end{propTEMAi}

\newsec{Aggregation functions}\label{sec-2}

Based on~\cite{Torra05},~\cite{DK05}, and ~\cite{BBJCF12}, the general meaning of an aggregation function in FL is to assign an $n$-tuple of real numbers belonging to $U^n$ to a single real number on $U$, such that it is a non-decreasing and idempotent (i.e., it is the identity when an $n$-tuple is unary) function satisfying boundary conditions. In~\cite[Def.~2]{Sola10}, an $n$-ary \emph{aggregation}  function $A\colon U^n \to U$ demands, for all $\vec{x}=(x_1,x_2, \ldots, x_n)$, $\vec{y}=(y_1,y_2,\ldots, y_n) \in U^n$, the following conditions:
\begin{description}
\item[$\textbf{A1}$:] \textbf{Boundary Conditions}\\
   $A( \vec{0} )= A(0,0, \ldots , 0) = 0$ and \\ $A( \vec{1} )=A(1,1,\ldots, 1) = 1$;
\item[$\textbf{A2}$:] \textbf{Monotonicity}
   If $\vec{x} \leq \vec{y}$ then $A(\vec{x})\leq A(\vec{y})$ where $\vec{x}\leq \vec{y}$ iff $x_i\leq y_i,$ for all $0\leq i\leq n$.
   \end{description}

   Some extra usual properties for aggregation functions are the following:
\begin{description}
\item[$\textbf{A3}$:] \textbf{Symmetry}\\
   $A(\overrightarrow{x_{\sigma}}) = A(x_{\sigma_1},x_{\sigma_2}, \ldots,x_{\sigma_n})=A(\vec{x})$, when $\sigma \colon \mathbb{N}^n \rightarrow \mathbb{N}^n$ is a permutation;
%\item[$\textbf{A4}$ $a$:] \textbf{Divisors of Zero} \\
%There exists $\vec{x}\in \, ]0,1]^n$ such that $A(\vec{x})=0$;
%\item[$\textbf{A4}$ $b$:] \textbf{Divisors of One} \\
%There exists $\vec{y}\in [0,1[^n$ such that  $A(\vec{y})=1$;
\item[$\textbf{A4}$:]\textbf{Idempotency}\\ $A(x,x,\ldots,x) = x$, for all $x \in U$;
\item[$\textbf{A5}$:] \textbf{Continuity }\\ If  for each $i\in\{1,\ldots,n\}$, $ x_1, \ldots,x_{i-1},x_{i+1},\ldots x_n \in U$ and convergent sequence $\{x_{ij}\}_{j \in \mathbb{N}}$ we have that \\
$\lim_{j\to \infty} A(x_1,\ldots,x_{i-1},x_{ij}, x_{i+1}, \ldots , x_n) = A (x_1,\ldots,x_{i-1}, \lim_{j\to \infty} x_{ij}, x_{i+1}, \ldots  x_n)$;
\item[$\textbf{A6}$:]\textbf{$k$-homogeneity} \\ For all $k \in \, ]0,\infty[$ and $\alpha \in [0,\infty[$ such that $\alpha^k\vec{x} = (\alpha^kx_1, \alpha^kx_2, \dots , \alpha^kx_n)\in U^n$,\\
$A(\alpha^k\vec{x} )= \alpha^{k}A(\vec{x})$;
\item [$\textbf{A7}$:] \textbf{Distributivity of an aggregation $A:U^n \rightarrow U$ related to a function $F: U^2 \rightarrow U$} \\$A(F(x,y_1), \ldots, F(x,y_n))=F(x, A(y_1, \ldots , y_n))$, for all $x, y_1,\ldots , y_n\in U$.
\end{description}


\subsection{Median as a self $N_S$-dual operator}

In the following, the median aggregation is a self $N_S$-dual aggregation operator.

\begin{propTEMAi}\cite[Proposition 1]{BRYB13} Let  $\sigma \colon \mathbb{N}^n \rightarrow \mathbb{N}^n$ be a permutation function. For all $\vec{x}  \in U^n$, the $n$-ary aggregation function $M \colon U^n \rightarrow U$ called \textbf{median aggregation} and defined  as follows:
\begin{eqnarray}\label{eq:median}
M(\vec{x} ) = \left\{ \begin{array}{lc}
x_{\sigma (\frac{n+1}{2})}, \hspace{0mm} n \textnormal{~is an odd positive integer number};\\
\frac{1}{2}\left ( x_{\sigma (\frac{n}{2})} + x_{\sigma (\frac{n}{2}+1)}  \right ),\hspace{0mm} \textnormal{otherwise}.
%\textnormal{if } n \textnormal{ is an even positive  integer number}.
\end{array}
\right.
\end{eqnarray}
verifies Property $\mathbf{Ak}$, for $k \in \{4,5,6\} $.
 \end{propTEMAi}\vspace{0.14cm}
% %
% \begin{proof} For all $x,y \in U$, \textbf{A5} follows from  Eq.(\ref{eq:median}). The following also holds.
% \begin{description}
%\item[\textbf{A6}:]  Firstly, when $n$ is an even positive integer number, we have to analyse the following cases:
%
% $(i)$ If $\sigma (\frac{n}{2})=i$ then it holds that
%\begin{equation*}
%\begin{split}
% \lim_{j\to \infty} M(x_1,\ldots,x_{ij},\ldots,x_n) = \lim_{j\to \infty} \frac{1}{2}\left (x_{\sigma (\frac{n}{2})j}+x_{\sigma (\frac{n}{2}+1)} \right ) 
% = \frac{1}{2}\left (\lim_{i\to \infty} x_{\sigma (\frac{n}{2})j}+x_{\sigma (\frac{n}{2}+1)}\right ).
%\end{split}
%\end{equation*}
%\hspace{-0.3cm}Therefore, we have that
%\begin{equation*}
%\begin{split}lim_{j\to \infty} M(x_1,\ldots,x_{ij},\ldots,x_n)= M(x_1,\ldots,\lim_{j\to \infty} x_{ij},\ldots,x_n).
%\end{split}
%\end{equation*}
%
%$(ii)$ If $\sigma (\frac{n}{2}+1)\hspace{-0.05cm}=\hspace{-0.1cm}i$, the result is analogous to item $(i)$.
%
%$(iii)$ Otherwise,  if $\sigma (\frac{n}{2}) \neq i$ and $\sigma (\frac{n}{2}+1) \neq i$, we have that the following holds:
%\begin{equation*}
%\begin{split}
%  \lim_{j\to \infty} M(x_1,\ldots,x_{ij},\ldots,x_n) \hspace{-0.1cm}=\hspace{-0.1cm} \lim_{j\to \infty} \frac{1}{2}(x_{\sigma (\frac{n}{2})} + x_{\sigma (\frac{n}{2}+1)}) =  \frac{1}{2}(x_{\sigma (\frac{n}{2})} + x_{\sigma (\frac{n}{2}+1)}).
%  \end{split}
%\end{equation*}
%\hspace{-0.5cm}Therefore, we can assure that
%$\lim_{j\to \infty} M(x_1,\ldots,x_{ij},\ldots,x_n)\hspace{-0.1cm}=\hspace{-0.1cm} M(x_1,\ldots,\lim_{j\to \infty} x_{ij},\ldots,x_n).
%$
%
% Secondly, when  $n$ is odd, we have to analyse other two cases:
%
%$(i)$ If  $\sigma (\frac{n+1}{2})=i$, then we obtain that
%\begin{equation*}
%\begin{split}
%\lim_{j\to \infty}& M(x_1,\ldots,x_{ij},\ldots,x_n)  =
%\lim_{j\to \infty} x_{\sigma (\frac{n+1}{2})j} = M(x_1,\ldots,\lim_{j\to \infty} x_{\sigma (\frac{n}{2})j},\ldots,x_n).
%\end{split}
%\end{equation*}
%
%$(ii)$ Otherwise, when  $\sigma (\frac{n+1}{2})\neq i$,
%\begin{equation*}
%\begin{split}
%\lim_{j\to \infty} M(x_1,\ldots,x_{ij},\ldots,x_n)= \lim_{j\to \infty} x_{\sigma (\frac{n}{2}+1)} = x_{\sigma (\frac{n}{2}+1)}.
%\end{split}
%\end{equation*}
%Then, we have that $\lim_{j\to \infty} M(x_1,\ldots,x_{ij},\ldots,x_n) \hspace{-0.1cm}=\hspace{-0.1cm}M( x_1,\ldots,\lim_{j\to \infty} x_{ij},\ldots,x_n)$.
%
%Concluding, $M$ verifies \textbf{A6}.
%
%\item[\textbf{A7}:] Firstly, let $M$ be an odd number of arguments. Then, for all $\alpha \in \mathbb{N}$ and $\vec{x}\in U$,
%\begin{equation*}
%\begin{split}
%M(\alpha \vec{x})= \alpha x_{\sigma (\frac{n+1}{2})} = \alpha M(\vec{x}).
%\end{split}
%\end{equation*}
% Otherwise, when $M$ has an even number of arguments, we obtain that
%\begin{equation*}
%\begin{split}
%M(\alpha  \vec{x}) & = \frac{1}{2}(\alpha x_{\sigma (\frac{n}{2})} + \alpha x_{\sigma (\frac{n}{2}+1)})  = \frac{\alpha}{2}(x_{\sigma (\frac{n}{2})} + x_{\sigma (\frac{n}{2}+1)})  = \alpha M(\vec{x}).
%\end{split}
%\end{equation*}
%Thus,  both cases state the median aggregation $M$ as a  $1$-homogeneous function.
%\end{description}
% \end{proof}


\begin{propTEMAi}\label{prop_An} Let $A$ be an aggregation function and $N$ be a SFN such that:
  \begin{description}
       \item [$\mathbf{N5}:$] $N(A(\vec{x}))= A(N(x_1), N(x_2),\ldots ,N(x_n))= A(N(\vec{x}))$.
  \end{description}
 Then we have that
$A_{N}(\vec{x}) = A(\vec{x}), \forall \vec{x}  \in U^n.$
\end{propTEMAi}
\begin{proof}
For all $\vec{x}  \in U^n$, $A_{N}(\vec{x}) = N(A(N(\vec{x}))) = A(N(N(\vec{x}))) = A(\vec{x})$.
\end{proof}

\begin{propTEMAi} The median  aggregation function $M$ verifies:
\begin{eqnarray}\label{eq:median1}
M_{N_S}(\vec{x}) = M(\vec{x}), \forall \vec{x}  \in U^n.
\end{eqnarray}
\end{propTEMAi}
\begin{proof}
Firstly, when $M$ has an odd number of arguments,
$N_S(M(\vec{x}))= 
N_S(x_{\sigma (\frac{n}{2})}) = M(N_S(\vec{x}))$.
 Otherwise, by taking $M$ as an even number of arguments, we obtain that
\begin{equation*}
N_S(M(\vec{x}))= 
N_S(\frac{1}{2}(x_{\sigma (\frac{n}{2})}) + x_{\sigma (\frac{n}{2}+1)})) = 1- \frac{1}{2}(x_{\sigma (\frac{n}{2})} + x_{\sigma (\frac{n}{2}+1)})  = \frac{1}{2}(1-x_{\sigma (\frac{n}{2})}+ 1-x_{\sigma (\frac{n}{2}+1)}).
\end{equation*}  
Thus, for all $\vec{x}\in U$, both cases state that $M$ verifies $N_S(M(\vec{x}))=M(N_S(\vec{x}))$. Therefore, by Proposition~\ref{prop_An}, $M_{N_S}(\vec{x})= M(\vec{x})$ and  Eq.~(\ref{eq:median1}) is verified.
\end{proof}

\begin{propTEMAi} Let $\phi(x,y)=x^r \in Aut(U)$. The median  aggregation function $M$ verifies:
\begin{equation}
M^{\phi}(\vec{x}) =
\left\{ \begin{array}{l}
M(\vec{x}) ,  \mbox{if $n$ is odd,} \\
\sqrt[n]{\frac{1}{2} \left(x^{r}_{\sigma{(\frac{n}{2})}}+ x^{r}_{\sigma{(\frac{n+1}{2})}} \right)} ,\mbox{otherwise;}
\end{array}\right.
\end{equation}
\end{propTEMAi}
\begin{proof} Straightforward.
\end{proof}

\subsection{Triangular sub(co)norms}
\label{sec-tconorm}

% A \textbf{triangular sub(co)norm} (t-sub(co)norm) is a binary aggregation function $(S)T \colon U^2 \rightarrow U$ satisfying the commutativity, associativity and
% monotonicity properties~\cite{KMP00} which are, respectively, given by the  expressions:
% \begin{center}
% \begin{tabular}{ll}
% %  \textbf{T{1}}: $T(x,1) = x$; & \textbf{S{1}}: $S(x,0)=x$; \\
%   \hspace{-3.8cm}\textbf{T{1}}: $T(x,y)=T(y,x)$; & \textbf{S{1}}: $S(x,y)=S(y,x)$; \\
%    \hspace{-3.8cm}\textbf{T{2}}: $T(x,T(y,z))=T(T(x,y),z)$; & \textbf{S{2}}: $S(x,S(y,z))=S(S(x,y),z)$. \\
%   \hspace{-3.9cm} \textbf{T{3}}: $T(x,z) \leq T(y,z)$, if $x\leq y$; & \textbf{S{3}}: $S(x,z) \leq S(y,z)$, if $x\leq y$. \\
%   \hspace{-3.9cm} \textbf{T{4}}: $T(x,y) \leq \min\{x,y\}$; & \textbf{S{4}}: $S(x,y) \geq \max\{x,y\}$.
% \end{tabular}
% \end{center}
% A t-(co)norm is a t-sub(co)norm satisfying the following boundary condition:
% \begin{description}
% \item [T5:] $T(x,1) = x$; \hspace{2.8cm} \textbf{S{5}}: $S(x,0)=x$.
% \end{description}
%  \vspace{0.2cm}

According with~\cite{KMP00}, a \textbf{triangular sub(co)norm} (t-sub(co)norm) is a binary aggregation function $(S)T \colon U^2 \rightarrow U$ such that,  for all $x,y \in U$, the following holds:
  \begin{description}
       \item [$\mathbf{T0}:$]  $T(x,y) \leq \min(x,y)$  \hspace{1.0cm} \textbf{S{0}}:
$S(x,y) \geq \max(x,y)$
  \end{description}
and also verifying the commutativity, associativity and monotonicity properties which are, respectively, given by the next four expressions:

\begin{description}
       \item [$\mathbf{T1}:$]  $T(x,y)=T(y,x)$; \hspace{1.5cm}\textbf{S{1}}: $S(x,y)=S(y,x)$; 
       
       \item [$\mathbf{T2}:$] $T(x,T(y,z))\hspace{-0.1cm}=\hspace{-0.1cm}T(T(x,y),z)$; \hspace{0.15cm}\textbf{S{2}}: $S(x,S(y,z))\hspace{-0.1cm}=\hspace{-0.1cm}S(S(x,y),z)$; 
       
       \item [$\mathbf{T3}:$] $T(x,z) \leq T(y,z)$, if $x\leq y$; \hspace{0.3cm}\textbf{S{3}}: $S(x,z) \leq S(y,z)$, if $x\leq y$. 
%  \hspace{-3.9cm} \textbf{T{4}}: $T(x,y) \leq \min\{x,y\}$; & \textbf{S{4}}: $S(x,y) \geq \max\{x,y\}$.
\end{description}

A \textbf{t-(co)norm} is a t-sub(co)norm satisfying the following boundary condition:
\begin{description}
\item [$\mathbf{T4}:$] $T(x,1) = x$; \hspace{2.2cm} \textbf{S{4}}: $S(x,0)=x$.
\end{description}

% \begin{description}
% \item [\textbf{T4}:] \hspace{-0.48cm} $T(x,1) = x$; \hspace{2.8cm} \textbf{S{4}}: $S(x,0)=x$.
% \end{description}\vspace{0.2cm}

\begin{remark}\label{rem-1}
Based on Properties \textbf{S0} and \textbf{T0}, we have that:
\begin{center}
\begin{tabular}{llll}
$S(0,0) \geq 0$; & $S(0,1)=1$; & $S(1,0)=1$; & $S(1,1)=1$. \\
$T(1,1) \leq 1$; & $T(1,0)=0$; & $T(0,1)=0$; & $T(0,0)=0$. \\
\end{tabular}
\end{center}
\end{remark}


\subsubsection{Triangular sub(co)norms and $N_S$-dual constructions}

In the following, we discuss properties of t-subnorms and t-subconorms as extension of product and probabilistic sum in $\mathcal{T}_P$ and $\mathcal{S}_P$.

\begin{propTEMAi}\label{pro-fam-Ti(Si)}
For $i \geq 1$ and $x,y \in U$, $T_i \, (S_i): U^2\rightarrow U$ is a t-sub(co)norm  given by
\begin{eqnarray} \label{eq_T_S}
 T_i (x,y)=\frac{1}{i}xy, && S_i(x,y)= 1-\frac{1}{i}(1-x-y+xy).
 \end{eqnarray}
\end{propTEMAi}
\begin{proof} Straightforward.
\end{proof}

The families of all t-sub(co)norms $T_i$  ($S_i$) is referred as $\mathcal{T}$ ($\mathcal{S}$).

\begin{remark}
Observe that, for $i=1$, Eqs.(\ref{eq_T_S})a and (\ref{eq_T_S})b are named as the product t-norm and the the algebraic sum, respectively, and corresponding expression can be given as
\begin{eqnarray}
\begin{split}
T_P(x,y)=xy, && S_P(x,y)=x+y-xy.
\end{split}
\end{eqnarray}
Additionally, each pair  $(T_i,S_i) \in \mathcal{T} \times \mathcal{S}$ defines a pair of $N_S$-mutual dual functions. That means, 
$T_P$ and its $N_S$-dual construction $S_P$ is a pair of $N_S$-mutual dual functions.
\end{remark}


\begin{propTEMAi}\label{pro-fam-Ti(Si)1}
For $i \geq 1$ and $x,y \in U$, $T_i \, (S_i): U^2\rightarrow U$ is a t-sub(co)norm  verifying 
%\begin{eqnarray} \label{eq_T_S34}
% T_i (x,N(x))= 0 \mbox{  iff  } x=0 \mbox{  or  } x=1; && S_i(x,N(x))= 1 \mbox{  iff  } x=0 \mbox{  or  } x=1.
% \end{eqnarray}
\begin{description}
       \item [$\mathbf{T5}:$]  $T_i (x,N(x))= 1$ iff  $x=0$ or $x=1$; \hspace{1.0cm}$\mathbf{S{5}}$: $S_i (x,N(x))= 0$ iff  $x=0$ or $x=1$; 
 \end{description}
\end{propTEMAi}
\begin{proof} For all $x,y \in U$, we have that
\begin{eqnarray*}
T_i(x,N(x)) = 1 & \Leftrightarrow &   1-\frac{1}{i}(x-x^r)=1 \Leftrightarrow x-x^2=0 \Leftrightarrow   x=0 \mbox{  or  } x=1;\\
S_i(x,N(x)) = 1 & \Leftrightarrow &    1-\frac{1}{i}(x(1-x))=1 
 \Leftrightarrow  x-x^2=0 \Leftrightarrow  x=0 \mbox{  or  } x=1.
\end{eqnarray*}
Therefore, $T_i$ and $(S_i)$ verify Properties $\mathbf{T{5}}$ and $\mathbf{S{5}}$, respectively. 
\end{proof}


\subsubsection{Extending triangular sub(co)norms to triangular (co)norms}

In the following proposition, we can obtain t-(co)norms from t-sub(co)norms.

\begin{propTEMAi}\label{pro:Si1}
 Let $i$ be an index such that $i\geq1$. 
 %$U_i=[0,i]$ be an $U_i$-extension of unit interval $U$ and $T_{U_i}(S_{U_i}):U_i\times U \rightarrow U$ be the corresponding $U_i$-extension of a t-sub(co)norm $T_i$ ($S_i$), which means $T_{U_i} (x,y)=T_i(x,y)$ ($S_{U_i}(x,y)=S_i(x,y)$). 
 
 For all $ x,y \in U$, we have that
\begin{eqnarray*} \label{eq-extSiTi}
T_P(x,y)= i T_{i} (x,y),  && S_{P} (x,y) = 1-(i-i(S_{i}(x,y))),
\end{eqnarray*} 
\end{propTEMAi}
 \begin{proof}
 Straightforward.
%For all $x,y \in U_i$,
%$T_{U_i}(ix,y) = 1-\frac{1}{i}(ixy)= xy$ and $S_{U_i}(1-i(1-x), 1-y)= 1-\frac{1}{i}(1-1+i(1-x)-1+y+1-i+ix+y-yi-ixy = x+y-xy=S_{P}(x,y)$.
 \end{proof}
 
 
 \subsubsection{Conjugate Triangular Sub(co)norms}
 
 \begin{propTEMAi}\label{prop-aut-subt} Consider $\phi(x)=x^r,\psi(x)=1-(1-x)^r$  in $Aut(U)$ whose reverse operators are $\phi^{-1}(x)=\sqrt[r]{x}$ and $\psi^{-1}(x)=1-\sqrt[r]{1-x}$, respectively. Then, the following holds:
 \begin{eqnarray} \label{eq_T_Saut}
 T_{i}^{\phi}(x,y)=T_{\sqrt[r]{i}}(x,y), & \mbox{and} & S_{i}^{\psi}(x,y)= S_{\sqrt[r]{i}}(x,y), \,\, \forall x,y \in U.
 \end{eqnarray}
 \end{propTEMAi}
 \begin{proof} For all $i\geq1$ and $x,y \in U$, we have that:
 \begin{eqnarray*}
  T_{i}^{\phi}(x,y)&=& \phi^{-1}(\frac{1}{i}(\phi(x)\phi(y)) =  \phi^{-1}(\frac{1}{i}x^ry^r) =  \frac{1}{\sqrt[r]{i}}xy; =T_{\sqrt[r]{i}}(x,y);\\
 S_{i}^{\psi}(x,y)&=& \psi^{-1}(1-\frac{1}{i}(1-\psi(x))(1-\psi(y))= \psi^{-1}(1-\frac{1}{i}(1-x)^r(1-y^r) \\ &=& 1- \sqrt[r]{\frac{1}{i}(1-x)^2(1-y^2)} = 1-\frac{1}{\sqrt[r]{i}}(1-x)(1-y)=S_{\sqrt[r]{i}}(xy).
\end{eqnarray*} 
Therefore, $T_{i}^{\phi} \in \mathcal{T}$ and  $S_{i}^{\phi} \in \mathcal{S}$.
 \end{proof}
 
 \begin{remark} Based on Proposition~\ref{prop-aut-subt}, if $\phi(x)=x^2$  and $\psi(x)=1-(1-x)^2$ we have that
 \begin{center}
  $T_i^{\psi}(x,y)=1-\sqrt[r]{(1-y)^r+y^r(1-x)^r}$  \,\,and \,\, $S_{i}^{\phi}(x,y)=\sqrt[r]{1-\frac{1}{i}(1-x^r)(1-y^r)}$.
  \end{center}
   Therefore, $T_i^{\psi}$ and  $S_{i}^{\phi}$ can not be expressed as members of $\mathcal{T}$ and $\mathcal{S}$, respectively. 
 \end{remark}

\newsec{(S,N)- and QL-(sub)implication classes}\label{sec-3}

The main results considered in this section are reported from \cite{BBS03,RBB13,Bac20081836} and \cite{BBS03}.


\subsection{Fuzzy (S,N)-subimplications and dual construction}
A function $I_{S,N} (J_{S,N}) \colon U^2 \to U$ is called an  \textbf{(S,N)-subimpli\-cation} (\textbf{(T,N)-subcoimpli\-cation}) if there exists a t-subconorm  $S$ (t-norm $T$) and a fuzzy negation $N$ such that
\begin{eqnarray*}
I_{S,N}(x,y)= S(N(x),y), &&  J_{S,N}(x,y)= T(N(x),y), \forall x,y \in U \label{eq-sn-imp}
\end{eqnarray*}
for all $x,y\in U$. 
If $N$ is a strong FN, then $I_S(J_T)$ is called an \textbf{S-subimplication} (\textbf{T-subcoimpli\-cation}). 


\begin{propTEMAi} An (S,N)-sub(co)implicator is a sub(co)implicator.
\end{propTEMAi}
 \begin{proof} 
By Property $\mathbf{I0}$, it holds that $I_{S,N}(0,0)= S(N(0),0)=S(1,0)=1$; $I_{S,N}(1,1)= S(N(1),1)=S(0,1)=1$; and $I_{S,N}(0,1)= S(N(0),1)=S(0,1)=1$.
 \end{proof}

Clearly, a fuzzy (co)implication $I_{S,N}$ ($J_{S,N}$) is also a fuzzy sub(co)implication.
The family of all (S,N)-subimplications ((T,N)-subcoimplications) is referred as $\mathcal{I}_S$ ($\mathcal{J}_T$). Additionally, if $S=T_N$, the  $N$-dual function of an subimplication $I_{S,N}$ is a subcoimplication ${J}_{S_N,N}$, meaning that
${J}_{S_N,N}= (I_{S,N})_{N}$.



Since the dual construction of Proposition \ref{pro-1} is also verified, the following holds:

\begin{propTEMAi}\label{pro:Ii1}
For all $x,y \in U$, the binary function $I_i,({J}_{i}) :U \rightarrow U$, defined as
\begin{eqnarray}\label{eq_snimp}
I_i (x,y)=1-\frac{1}{i}(x-xy) && (J_{i} (x,y)=\frac{1}{i}(1-x)(1-y))
\end{eqnarray}
 is a fuzzy (S,N)-sub(co)implicator.
 \end{propTEMAi}
 \begin{proof}
$\mathbf{I0}$  is immediate. Additionally, for all $x,y \in U$, by taking $S_i(x,y)=1-\frac{1}{i}(1-x-y+xy)$, for $i\leq1$, we have that
 \begin{equation*}
 \begin{split}
 S_i(N_S(x),y) = 1-\frac{1}{i}(1-(1-x)-y+(1-x)y)  = 1-\frac{1}{i}(x-xy).
 \end{split}
 \end{equation*}
Consequently,  
$I_i(x,y)=S_i(N_S(x),y)$.
Therefore $I_i$ is an ($S_i$,$N_S$)-implicator. Analogously, it is proved related subcoimplication.
 \end{proof}
 



 
\begin{propTEMAi} \cite[Proposition~5]{BRYB13}\label{pro:Ii2}
An (S,N)-subimplication verifies Property $\mathbf{I_k}$, for $\mathbf{k} \in\{0,2,3,4,5,6\}$.
 \end{propTEMAi}
%\begin{proof}
%The following holds.
%\begin{description}
%\item [$\mathbf{I0}$:] Straightforward from definition of $I_i$ by Eq. (\ref{eq_snimp}) in Proposition \ref{pro:Ii1}.
%
%\item [$\mathbf{I2}$:] If $x_1\leq x_2$, for all $x_1,y,x_2\in U$, it holds that \\ $I_i(x_1,y) =1-\frac{1}{i}(x_1-x_1y) \geq x_1-\frac{1}{i}(x_1-x_1y)= I_i(x_2,y)$.
%
%\item [$\mathbf{I3}$:] If $y_1\leq y_2$,  for all $y_1,y_2,x\in U$ it holds that
%\begin{equation*}I_i(x,y_1) = 1-\frac{1}{i}(x-xy_1) \leq x-\frac{1}{i}(x-xy_2)= I_i(x,y_2).
%\end{equation*}
%
%\item [$\mathbf{I4}$:] $I_i(0,y)= 1-\frac{1}{i}\cdot{0}=1$, for all $y\in U$.\vspace{0.1cm}
%
%\item [$\mathbf{I5}$:] For all $x,y,z \in U$, it holds that
%\begin{equation*}I_i(x,I_i(y,z))= 1-\frac{1}{i} x-x(1-\frac{1}{i}(y-yz))=   1-\frac{x}{i^2}(y-yz).
%\end{equation*}
%Therefore, we obtain the following:
%\begin{equation*}
%\begin{split}
%I_i(x,I_i(y,z)) & = 1-\frac{y}{i^2}(x-xz) =  1-\frac{1}{i} y-y(1-\frac{1}{i}(x-xz)) = I_i(y,I_i(x,z)).
%\end{split}
%\end{equation*}
%
%\item [$\mathbf{I6}$:]  For all $x,y\in U$, the following is held:
%\begin{equation*}
%\begin{split}
%I_i(N_S(y),N_S(x)) &= 1\frac{1}{i}(1-y-(1-y-x+xy))  =1-\frac{1}{i}(x-xy) = I_i(x,y).
%\end{split}
%\end{equation*}
%\end{description}
%Concluding, Proposition \ref{pro:Ii2} is verified.
%\end{proof}
%
%\begin{thmTEMA}
%The operator $I_i: U^2\rightarrow U$ given by Eq. (\ref{eq_snimp})  is an ($S_i$,$N_S$)-implication underlying the continuous negation $N_S$ and the continuous t-subconorm $S_i$ at point $0$.
%\end{thmTEMA}
% \begin{proof} Straightforward from Propositions~\ref{pro-1},~\ref{pro:Ii1} and~\ref{pro:Ii2}.
% \end{proof}


\subsubsection{Extending (S,N)-subimplications to an (S,N)-implications}


See in Fig.~\ref{fig:Ii} graphical representations for three  examples of subimplications,  $I_1,I_2,I_3 \in \mathcal{I}_{S}$. In particular,  $I_1$ is referred as the Reichenbach's implication and indicted as $I_{RH}$.

\begin{figure}[htp]
  \centering
  \subfigure{\includegraphics[width=2.9cm]{Imagens/I1.png}}\quad
  \subfigure{\includegraphics[width=2.9cm]{Imagens/I2.png}}\quad
  \subfigure{\includegraphics[width=2.9cm]{Imagens/I3.png}}
  \caption{Fuzzy (S,N)-subimplications of family $\mathcal{I}=\{I_{1},I_2,I_3\}$.}
  \label{fig:Ii}
\end{figure}
% \begin{figure}[h] \label{f800} % Duas Fig.s lado a lado
%   \centering
%   \subfigure[{\footnotesize Fuzzy implication $I_1$}]{\includegraphics[width=4.0cm]{Imagens/I1.png}}
%   \hspace{1cm}
%   \subfigure[{\footnotesize Fuzzy subimplication $I_2$}]{\includegraphics[width=4.0cm]{Imagens/I2.png}}
%   \hspace{1cm}
%     \subfigure[{\footnotesize Fuzzy subimplication $I_3$}]{\includegraphics[width=4.0cm]{Imagens/I3.png}}
%     \caption{Fuzzy subimplications of family $\mathcal{I}$.}
% \end{figure}

In the following, we discuss the extension of an $(S_i,N_S)$-subimplication  $I_i$  to an $(S_i,N_S)$-implication. Based on the duality stated by $N_S$ in the class $\mathcal{I}_T$, the extension of an $(T_i,N_S)$-subcoimplication to an $(T_i,N_S)$-implication can also be obtained, analogously. 

\begin{propTEMAi}\label{pro:Ii1}
 Let $i$ be an index such that $i \geq 1$. 
 %Consider  $I_{U_i}:[0,i] \times U \rightarrow U$, given by $I_{U_i} (x,y)=1-\frac{1}{i}(x-xy)$, as the $U_i$-extension of an  (S,N)-implication $I_i$. 
 Then, for all $x,y, \in U$,  it holds that
\begin{eqnarray*}\label{eq_snimp}
I_{RH}(x,y) = i(1-I_{i} (x,y)), &&  {J_{RH}}(x,y) = i({J_i} (x,y)).
\end{eqnarray*}
\end{propTEMAi}
 \begin{proof}
 Straightforward.
%For all $x,y \in U$,
%$i(1-I_{i}(x,y)) = i(1-(1-\frac{1}{i}(x+xy)) = I_{1}(x,y)$.
 \end{proof}
 
As a consequence derived from Proposition~\ref{pro:Ii1}, we can obtain the Reichenbach's implication from each $U_I$-extended member $I_i$ of family $\mathcal{I}$.

\subsubsection{Conjugate (S,N)-subimplications}

In the following, the action of an automorphism on $U$ is discussed.
 
 \begin{propTEMAi}\label{prop-aut-subt} Consider $\phi(x)=x^r, \psi(x)=1-(1-x)^r$ in $Aut(U)$ whose reverse operators are $\phi^{-1}(x)=\sqrt[r]{x}$ and $\psi^{-1}(x)=1-\sqrt[r]{1-x}$, respectively. Then, the following holds:
 \begin{eqnarray} \label{eq_b}
\left(I_{S_{i},N_S}\right)^{\psi}(x,y)=I_{S^{\psi},N_S}(x,y) & \mbox{ and } & \left(J_{T_{i},N_S}\right)^{\psi}(x,y)=J_{T^{\psi},N_S}(x,y), \,\,\,\, \forall x,y \in U.
 \end{eqnarray}
 \end{propTEMAi}
 \begin{proof} For all $i\geq1$ and $x,y \in U$, we have that:
 \begin{eqnarray*}
 \left(I_{S_{i},N_S}\right)^{\psi}(x,y)&=& 
\psi^{-1} (S_i(N_S(\psi(x)), \psi(y))) 
 \\
 &=& \psi^{-1}(1-\frac{1}{i}(\psi(x)(1-\psi(y))= \psi^{-1}(1-\frac{1}{i}(1-x)^r(1-y^r)) 
 \\
  &=& 1- \sqrt[r]{\frac{1}{i}(1-x)^r(1-y^r)} = 1-\frac{1}{\sqrt[r]{i}}(1-x)(1-y)
\\ 
&=& I_{S_{\sqrt[r]{i}},N_S}(x,y) = I_{\mathcal{S}_{i}^{\psi},N_S}(x,y) .
\end{eqnarray*} 
Therefore, $I_{S_{i},N_S}^{\phi} \in \mathcal{I}_{S,N}$. Dually, it can also be proved that $J_{i}^{\phi} \in \mathcal{J}_{T,N}$.
 \end{proof}
 
 \subsection{Fuzzy QL-(sub)implication class}

 This section reviews the main properties of fuzzy QL-(sub)implication class. See \cite{BJ10,MMT07} and  \cite{SHI08} for additional Inf..

\begin{defTEMAi}\label{def-ql-sub-imp}
Let $N$ be a fuzzy negation.
A function $I_{S,T,N} \colon U^2 \rightarrow U$ is called a \textbf{QL-subimplicator} if, for all $x,y \in U$, there exist a t-subconorm $S$ and a t-norm $T$ such that:
\begin{eqnarray}\label{eq-ql-imp}
I_{S,N,T}(x,y) &= & S(N(x),T(x,y)).
\end{eqnarray}
Dually, $J_{T,S,N}\colon U^2 \rightarrow U$is called a \textbf{QL-subcoimplicator} if, for all $x,y \in U$, there exist a t-subnorm $T$ and a t-conorm $S$ such that:
\begin{eqnarray}\label{eq-ql-imp}
 I_{T,N,S}(x,y) &=& T(N(x),S(x,y)).
\end{eqnarray}
\end{defTEMAi}

\begin{propTEMAi} A QL-subimplicator is a subimplicator.
\end{propTEMAi}
 \begin{proof} 
By Property $\mathbf{I0}$, it holds that $I_{S,N,T}(0,0)= S(N(0),T(0,0))=S(1,0)=1$; $I_{S,N,T}(1,1)= S(N(1),T(1,1))=S(0,1)=1$; and $I_{S,N,T}(0,1)= S(N(0),T(0,1))=S(1,0)=1$.
 \end{proof}

%\begin{propTEMAi} When a \textbf{QL-(sub)implicator} $I_{QL}\colon U^2 \rightarrow U$ verifies $\mathbf{Ik}$, for $k \in \{2,3,4\}$, then $I_{QL}$ is a QL-(sub)implication.
%\end{propTEMAi}

Thus, a QL-subimplication $I_{S,N,T}$ indicates the underlying t-subconorm, negation and t-norm as $S$, $N$ and $T$, respectively. Analogously, it can be indicated by a QL-subcoimplication $I_{T,N,S}$.  The family of all fuzzy QL-sub(co)implicators is referred as $\mathcal{I}_{QL}$ ($\mathcal{J}_{QL}$).

  \begin{propTEMAi}\label{pro-ji-qlimp} For all $x,y\in U$, the function $I_i (J_i) : U^2\rightarrow U$, given by the expression
  \begin{eqnarray}\label{eq-ql-imp-Ji}
 I_{{S_i},N_S,T_P}(x,y) = 1-\frac{1}{i}(x-x^2y) && (J_{T_i,N_S,S_P}(x,y) = \frac{1}{i}(1-x)(1-y+xy)),
  \end{eqnarray}
is a fuzzy QL-sub(co)implicator.
 \end{propTEMAi}
 \begin{proof} 
 For all $x,y\in U$, we have that 
 \begin{equation*}
 I_{{S_i},N_S,T_P}(x,y) = S_i(N_S(x),T(x,y)) 
 =  1-\frac{1}{i}(1-(1-x)-xy+(1-x)xy)  = 1-\frac{1}{i}(x-x^2y).
 \end{equation*}
 Concluding, $ I_{{S_i},N_S,T_P}\in \mathcal{I}_{QL}$. Analogously, we can prove that $J_{T_i,N_S,S_P}\in \mathcal{J}_{QL}$.
\end{proof}


%\begin{propTEMAi}\cite[Proposition~4.2]{BJ10} \label{pro-I1}
%A QL-implicator $I_{S,T,N}$ verifies $\textbf{I0}, \textbf{I2}$ and $\textbf{I4}$ together with  the properties:
% \begin{description}
%        \item [$\mathbf{I7}:$]  \textbf{Left Neutrality}:\\ $I(1,y)= y$, for all $y \in U$;
%        \item [$\mathbf{I8}:$] \textbf{Natural Negation}: \\$N_I(x)=I(x,0)$ is a fuzzy negation, for all $x \in U$,
%     \end{description}
%when each member-function $J_i \in \mathcal{J}$ also verifies the corresponding properties.
%\end{propTEMAi}




The following proposition is an extension of Prop.~4.2 in \cite{BJ10} by considering the main algebraic properties which  characterize  the fuzzy QL-subimplication class.

\begin{propTEMAi}\label{pro-7-8} A QL-subimplicator $I_{S,T,N} \in \mathcal{I}_{QL}$ verifies   $\textbf{Ik}$ for $\mathbf{k} \in \{0,2,4\}$ together with the additional two properties:
 \begin{description}
        \item [$\mathbf{I9}:$]   $I(1,y)\geq y$, for all $y \in U$;
        \item [$\mathbf{I10}:$] if $x_1\geq x_2$ then $I(x_1,0) \leq I(x_2,0)$, for all $x_1,x_2 \in U$.
     \end{description}
\end{propTEMAi}
\begin{proof} For $x_1,x_2,x,y_1,y_2,y \in U$, the following is verified.
\begin{description}
\item[\textbf{I0}] By results  in Remark \ref{rem-1}, it follows that

\begin{tabular}{l}
$I_{S,T,N}(0,0) = S(1,T(0,0))=S(1,0)=1$;
$I_{S,T,N}(0,1) = S(1,T(0,1))=S(1,0)=1$; \\ $I_{S,T,N}(1,0) = S(0,T(1,0))=S(0,0)\geq 0$; 
$I_{S,T,N}(1,1) = S(0,T(1,1))=S(1,1)=1$.
\end{tabular}

\item [\textbf{I2}] Since $S,T$ are monotonic functions, if $y_1 \leq y_2$ then $T(x,y_1) \leq T(x,y_2)$ and consequently, $I_{S,N,T}(x,y_1)= S(N(x),T(x,y_1)) \leq  S(N(x), T(x,y_2)) = I_{S,N,T}(x,y_2)$.
\item [\textbf{I4}] $I_{S,N,T}(0,y) = S(1,T(0,y)) = S(1,0) = 1$.
\item [\textbf{I9}] $I_{S,N,T}(1,y)= S(0,T(1,y))= S(0,y)\geq y$.
\item [\textbf{I10}] When $x \geq y$ then $N(x) \leq N(y)$. Then, $I_{S,N,T}(x,0)= S(N(x),T(x,0)) = S(N(x),0) \leq  S(N(y),0)= S(N(y),T(y,0))= I_{S,N,T}(y,0)$.
\end{description}
Therefore, Proposition \ref{pro-7-8} is verified.
\end{proof}





\begin{coroTEMAi}\label{pro-ik-ql}
The operator $I_{{S_P}_I,N_S,T_P}\in \mathcal {J}$ verifies $\textbf{Ik}$ for $\mathbf{k} \in \{0,2,4,9,10\}$.
\end{coroTEMAi}
 \begin{proof}
Straightforward from Proposition \ref{pro-7-8}.
 \end{proof}

 
% \begin{propTEMAi}\label{pro-7-8} Let $I_{S_i,N,T_P}$ be a QL-subimplicator in $\mathcal{I}_{QL}$. The  verifying    the property:
% \begin{description}
%        \item [$\mathbf{I11}:$]   $T_P(1-x, I_{S_i,N,T_P}(x,y)) \leq 1-x$, for all $y \in U$;
%        \item [$\mathbf{I12}:$] if $x\geq x_k$ then $I_{S_i,N,T_P}(x,0) \leq I_{S_i,N,T_P}(x_k,0)$, for all $x_1,x_2 \in U$.
%     \end{description}
%\end{propTEMAi}
%
%\begin{proof}
%For all $x,y\in U$, $T_P(1-x, I_{S_i,N,T_P}(x,y)) = (1-x)(1-\frac{1}{i}(x-x^2y)) \leq (1-x)$. Moreover, if $x\geq x_k$ then
%$I_{S_i,N,T_P}(x,0) = \frac{1}{i}(1-x)x \leq \frac{1}{i}(1-x_k)x_k = I_{S_i,N,T_P}(x_k,0)$.
%\end{proof}
 



 \begin{remark} Let $I:U^2\rightarrow U$ be a function given by Eq.(\ref{eq-ql-imp}). By taking a t-subconorm $S$, a fuzzy negation $N$ and a t-subnorm $T$, the function $I$
does not verify neither $\mathbf{I0}$ nor $\mathbf{I1}$:
\begin{description}
\item [(i)] $I(1,1)=S(N(1), T(1,1))=S(0,T(1,1))\geq T(1,1)$;
\item [(ii)] $I(1,0)=S(N(1), T(1,0))=S(0,0)\geq 0$.
\end{description}
Therefore, $I$ is not necessarily a subimplicator.
\end{remark}

 \subsubsection{Extending QL-subimplications to QL-implications}

Clearly, a QL-implication is always a QL-subimplication. This section discusses a converse construction in the class $\mathcal{J}$, by considering the main conditions under which a QL-subimplication can be extended to a QL-implication. 

 In Figure~\ref{fig:Ji},  instances $J_1,J_2$ and $J_3$ of such class $\mathcal{J}$ is graphically presented. In particular, $J_1 \in \mathcal{J}$ is a QL-implication~\cite{Bac20081836}.


\begin{figure}[htp]
  \centering
  \subfigure{\includegraphics[width=2.9cm]{Imagens/J1.png}}\quad
  \subfigure{\includegraphics[width=2.9cm]{Imagens/J2.png}}\quad
  \subfigure{\includegraphics[width=2.9cm]{Imagens/J3.png}}
  \caption{Fuzzy QL-subimplications of family $\mathcal{I}_{QL}=\{I_{QL1},I_{QL2},I_{QL3}\}$.}
  \label{fig:Ji}
\end{figure}

\begin{propTEMAi}\label{pro:Ii1}
 Let $i$ be an index such that $i \geq 1$. For all $x,y \in U$, the following holds:
 %Consider  $J_{U_i}:[0,i] \times U \rightarrow U$, given by $J_{U_i} (x,y)=1-\frac{1}{i}(x-xy)$, as the $U_i$-extension of an  (S,N)-implication $J_i$. Then, it holds that
%\begin{equation}\label{eq_snimp}
%J_1(x,y) = J_{U_i} (ix,y), \forall x,y \in U.
%\end{equation}
 \begin{eqnarray}\label{eq-ql-imp-Ji}
 I_{QL1}(x,y) = i-iI_{QLi}(x,y) & \mbox{ and } & J_{QL1}(x,y) = i{J}_{QLi}(x,y)).
  \end{eqnarray}
\end{propTEMAi}
 \begin{proof} Straightforward.
%For all $x,y \in U$,
%$ I_{U_i}(ix,y) = 1-\frac{1}{i}(ix+ixy)  = 1-x+xy$. Therefore,  $I_{U_i}(ix,y)= I_{1}(x,y)$.
 \end{proof}
 

\subsubsection{Conjugate QL-subimplications}
In the following, an automorphism $\psi(\phi) \in Aut(U)$  is considered in order to obtain conjugate functions in the class of QL-sub(co)implications $\mathcal{I}_{QL}$($\mathcal{J}_{QL}$).
 
 \begin{propTEMAi}\label{prop-aut-subt} Consider $\phi(x)=x^r,\psi(x)=1-(1-x)^r$  in $Aut(U)$ whose reverse operators are $\phi^{-1}(x)=\sqrt[r]{x}$ and $\psi^{-1}(x)=1-\sqrt[r]{1-x}$, respectively. Then, the following holds:
 \begin{eqnarray} \label{eq_T_Saut200}
\left(I_{S_{i},T_P,N_S}\right)^{\psi}(x,y)=I_{S_i^{\psi},T_P,N_S}(x,y) & \mbox{and} &
\left(J_{S_P,T_{i},N_S}\right)^{\psi}(x,y) =  J_{S_P,T_i^{\psi},N_S}(x,y).
 \end{eqnarray}
 \end{propTEMAi}
 \begin{proof} For all $i\geq1$ and $x,y \in U$, we have that:
 \begin{eqnarray*}\label{eq-car}
 \left(I_{S_{i},T_P,N_S}\right)^{\psi}(x,y)&=& 
\psi^{-1} (S_i(N_S(\psi(x)), T_P(\psi(x),\psi(y)))= \psi^{-1} (S_i(1-\psi(x)),\psi(x)\psi(y))
 \\
 &=& \psi^{-1}(1-\frac{1}{i}(\psi(x)(1-\psi(y))= \psi^{-1}(1-\frac{1}{i}(1-x)^r(1-y^r)) 
 \\
  &=& 1-\sqrt[r]{\frac{1}{i}(1-x)^r(1-y)^r}=  1-\frac{1}{\sqrt[r]{i}}(1-x)(1-y)=  \left(I_{S_{\sqrt{i}},T_P,N_S}\right)^{\psi}(x,y).
\end{eqnarray*} 
Therefore, $I_{i}^{\phi} \in \mathcal{I}_{QL}$. Analogously, it can be proved that  $J_{i}^{\phi} \in \mathcal{J}_{QL}$.
 \end{proof}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \newsec{Aggregating connectives from the median  operator}\label{sec-4}
\label{sec-4}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Consider $A\colon U^n \to U$ as an $n$-ary aggregation function and \mbox{$\mathcal{F}= \{F_i \colon U^k \to U\}$}, with $i\in \{1,2, \dots , n\}$ as a family of binary functions in the following results of this section.
\vspace{1mm}

\begin{defTEMAi}\cite[Prop. 5.1]{RBB13}
An $k$-ary function $\mathcal{F}_{A} \colon U^k \to U$ is called  as  \textbf{$(A,\mathcal{F})$-operator on $U$} and  given by:
\begin{eqnarray}
\label{eq:agrega_function}
   \mathcal{F}_{A} (x_1, \ldots , x_k) = A(F_1(x_1, \ldots , x_k),  \ldots ,  F_n(x_1, \ldots , x_k)).
\end{eqnarray}
\end{defTEMAi}

\subsection{Aggregating fuzzy t-sub(co)norms}

In this section, the conditions under which a class of t-sub(co)norms is preserved by the median aggregation operator are discussed. Additionally,  the conjugate and dual constructions related to the family of t-sub(co)norms $(\mathcal{S}) \mathcal{T} = \{(S_i) T_i \colon U^2 \to U\}$, with $i\in \{1,2, \dots , n\}$ are also analysed.

\begin{propTEMAi}~\cite[Proposition 6.1]{RBB13}\label{pro-101}
\label{pro:agrega-connective}
Let $A\colon U^n \to U$ be an  aggregation function and $(\mathcal{S}) \mathcal{T} = \{(S_i) T_i \colon U^2 \to U\}$, with $i\in \{1,2, \dots , n\}$ be a family of t-sub(co)norms. Then the function
 ($\mathcal{S}_A: U^2 \rightarrow U$) $\mathcal{T}_A:U^2 \rightarrow U$, called (($A, \mathcal{S}$)-operator) ($A, \mathcal{T}$)-operator, is a t-sub(co)norm  whenever the following two conditions are verified:
 \begin{description}
 \item [(i)] $A$ satisfies property $\mathbf{A7}$; and
 \item [(ii)] each t-sub(co)norm ($S_i$) $T_i$ satisfies the \textbf{generalized associativity}\footnote{Eq.~\eqref{eq:general-associa-S} are particular cases of Eq. (GA) in \cite{Maksa05}.}:
\begin{eqnarray}
 S_i(x, S_j(y,z))  =  S_i(S_j(x,y),z); &&  T_i(x, T_j(y,z))  =  T_i(T_j(x,y),z),\label{eq:general-associa-S}
\end{eqnarray}
for all $i,j$ such that $0\leq i,j\leq n \mbox{ and } x,y,z \in U$.
\end{description}
\end{propTEMAi}

\begin{propTEMAi}\label{Pro10}
 Let $\mathcal{T}$ and $\mathcal{S}$ be the families  of t-subnorms and t-subconorms described in Proposition \ref{pro-fam-Ti(Si)}. For all $i,j\geq 1$, each pair $T_i,T_j \in \mathcal{T}$ and  $S_i,S_j \in \mathcal{S}$ verifies  Eqs.~(\ref{eq:general-associa-S})a and~(\ref{eq:general-associa-S})b, respectively.
\end{propTEMAi}
\begin{proof} For all $x$, $y$, $z \in U$,
$ T_i(x, T_j(y,z))  = T_i(x,\frac{1}{j}yz)
 = \frac{1}{ij} (xyz)
= \frac{1}{i} (T_j(x,y) \cdot z)
 = T_i(T_j(x,y),z)$
Then, $\mathcal{T}$ satisfies the Eq.(\ref{eq:general-associa-S})a. The proof for $\mathcal{S}$ and related to
Eq.(\ref{eq:general-associa-S}) can be analogously obtained.
\end{proof}

\begin{propTEMAi}~\cite[Proposition 10]{RBB13}\label{Pro11}
 Let $\mathcal{T}_{P_i}$ and $\mathcal{S}_P$ be the  corresponding families:
 \begin{eqnarray}
 \mathcal{T}_{P_i} =\{T_i =\frac{1}{i}xy  \colon i \geq 1\} & \mbox{  and  } &
 \mathcal{S}_{P_i} =\{S_i(x)= 1-\frac{1}{i}(1-x-y+xy)\colon i \geq 1\}.
 \end{eqnarray}
According to Eq. (\ref{eq:agrega_function}), the operators $\mathcal{S}_{M}, \mathcal{T}_{M} \colon U^2 \to U$ given as 
%%

\begin{equation}
(\mathcal{T}_{P_i})_M(x,y) =
\left\{ \begin{array}{l}
\frac{1}{\sigma{(\frac{n+1}{2})}} xy ,  \mbox{if $n$ is odd,} \\
 \left( \frac{1}{2\sigma{(\frac{n}{2})}}+ \frac{1}{2\sigma{(\frac{n+1}{2})}} \right)  xy ,\mbox{otherwise;}
\end{array}\right.\hspace{2.0cm}
\end{equation}
%%
and
%%
\begin{equation}
(\mathcal{S}_{P_i})_M(x,y) \hspace{-0.1cm}=\hspace{-0.1cm}
\left\{ \begin{array}{l}
 \hspace{-0.1cm}1\hspace{-0.1cm}-\hspace{-0.1cm}\frac{1}{2\sigma{(\frac{n+1}{2})}} (1-x)(1-y),  \mbox{if $n$ is odd,} \\
1- \left(\frac{1}{2 \sigma{(\frac{n}{2})}}+ \frac{1}{2\sigma{(\frac{n+1}{2})}} \right) (1-x)(1-y),  \mbox{otherwise;}
\end{array}\right.
\end{equation}
%%
respectively, verify Property $\mathbf{A7}$.
\end{propTEMAi}
\begin{proof} For all $x,y_1, \ldots, y_n \in U$, consider the following two distinct cases.
\begin{description}
\item [(i)] Firstly, if $n$ is odd, we obtain the following:
\begin{align*}
(\mathcal{T}_{P_i})_M(x,y) &=
M(T_i(x,y_1), \ldots, T_i(x,y_n)) = \frac{1}{\sigma(\frac{n+1}{2})}xy_{\sigma(\frac{n+1}{2})} = T_i(x,M(y_1, \ldots, y_n)).
\end{align*}

\item [(ii)] Otherwise, when $n$ is even, it holds that:
\begin{align*} (\mathcal{T}_{P_i})_M(x,y) &= M(T_i(x,y_1),  \ldots , T_i(x,y_n)) 
=  \left( \frac{1}{2\sigma{(\frac{n}{2})}}+ \frac{1}{2\sigma{(\frac{n+1}{2})}} \right)  x  \left(y_{\sigma(\frac{n}{2})}+ y_{\sigma(\frac{n+1}{2})} \right)\\
& =  \left( \frac{1}{\sigma{(\frac{n}{2})}}+ \frac{1}{\sigma{(\frac{n+1}{2})}} \right)  x  \frac{1}{2}\left(y_{\sigma(\frac{n}{2})}+ y_{\sigma(\frac{n+1}{2})} \right) = T_i(x,M(y_1, \ldots, y_n)).
\end{align*}
\end{description}
 Therefore $\mathcal{T}_M$satisfies $\mathbf{A7}$. The proof related to $\mathcal{S}_M$ can be analogously obtained.
\end{proof}


\begin{coroTEMAi}
The operator (($\mathcal{S}_{P_i})_{M}$) $(\mathcal{T}_{P_i})_{M}$ is  a t-sub(co)norm.
\end{coroTEMAi}
\begin{proof} Straightforward from Propositions~\ref{pro:agrega-connective},~\ref{Pro10} and~\ref{Pro11}.
\end{proof}

The following proposition, reported from \cite{RBB13}, states the conditions under which a fuzzy subimplication $I_M$ verifies the generalized exchange principle.

\begin{propTEMAi}\cite[Proposition 5.5]{RBB13}
 \label{pro:dist-general-excahnge} Let $A\colon U^n \to U$ be an $n$-ary aggregation function and $\mathcal{I}= \{I_i \colon U^k \to U\}$, for $i\in \{1,2, \dots , n\}$ be a family of fuzzy subimplication functions.
$\mathcal{I}_A$ verifies $\mathbf{I5}$ when the aggregation $A$ verifies $\mathbf{A7}$ and the following properties are verified:
\begin{description}
\item [$\mathbf{I10}$:] \textbf{Generalized Exchange Principle}: $\forall x,y,z \in U$  and  $I_i, I_j \in \mathcal{I}$, such that $0\leq i,j \leq n,$\footnote{This property  also can be considered as a generalization of the extended migrative property, see ~\cite[Definition 2]{FR11}.}
   \begin{equation}
   \label{eq:general-exchange}
       \hspace{-1.5cm}I_i(x,I_j(y,z))=I_i(y,I_j(x,z)).
   \end{equation}
   \end{description}
\end{propTEMAi}

\begin{propTEMAi}\label{pro-fam-Ti(Si)1}
 $\mathcal{T}_M \, (\mathcal{S}_M): U^2\rightarrow U$ is a t-sub(co)norm  verifying $\mathbf{T5}$ ($\mathbf{S5}$).
\end{propTEMAi}
\begin{proof} For all $x,y \in U$, we have that
\begin{equation*}
(\mathcal{T}_P)_M(x,N(x)) = 1 \Leftrightarrow
\left\{ \begin{array}{l}
\frac{1}{\sigma{(\frac{n+1}{2})}} x(1-x) = 1  \Leftrightarrow  x=0 \mbox{ or } x=1 ,  \mbox{if $n$ is odd,} \\
 \left( \frac{1}{2\sigma{(\frac{n}{2})}}+ \frac{1}{2\sigma{(\frac{n+1}{2})}} \right)  x(1-x) =1   \Leftrightarrow  x=0 \mbox{ or } x=1 ,\mbox{otherwise;}
\end{array}\right.\hspace{2.0cm}
\end{equation*}
Therefore, Property $\mathbf{T{5}}$ is verified by $\mathcal{T}_M$. Analogously, its dual construction can be proven. 
\end{proof}

\subsubsection{Conjugate and dual t-subnorms obtained by  median aggregation}

Our perspectives in this section are to study in more detail the interrelations between these classes of aggregated t-(co)norms  and their possible conjugate functions.
Another interesting issue is to study how the method can take into account their dual constructions, the classes of (T,N)-subimplications \cite{Bed07,MMT06}.

\begin{propTEMAi} \label{prop_F}
Let $A$ be an aggregation function and $N$ be a SFN verifying $\mathbf{N5}$. When $(T_i,S_i) \in \mathcal{S} \times \mathcal{S}$ is a pair of mutual $N_S$-dual functions on $U$, the following holds:
\begin{eqnarray}\label{eq34}
(\mathcal{S}_{M})_{N_S} (x,y) = (\mathcal{T}_P)_M (x,y) & \mbox{  and  } & (\mathcal{T}_{M})_{N_S} (x,y) = (\mathcal{S}_P)_M (x,y), \forall x,y \in U. 
\end{eqnarray}
\end{propTEMAi} 
\begin{proof} For all $x,y \in U$, we have that
\begin{eqnarray*}
(\mathcal{S}_{M})_{N_S} (x,y) &=& N_S (\mathcal{S}_{M}(N_S(x),N_S(y))) \mbox{  by Eq.~(\ref{eq func dual})}
\\
&=& N_S(M(S_1(N_S(x),N_S(y))), \ldots , (S_n(N_S(x),N_S(y)))) \mbox{  by Eq.~(\ref{eq:median1})} \\
&=& M(N_S(S_1(N_S(x),N_S(y))), \ldots ,N_S (S_n(N_S(x),N_S(y)))))   \mbox{  by Eq.~(\ref{eq func dual})}
\\
&=& M(T_1(x,y),  \ldots , T_n(x,y)) = \mathcal{T}_M(x,y).
\end{eqnarray*}
Therefore, Eq~(\ref{eq34}) is held.
\end{proof}

\begin{coroTEMAi}
$({\mathcal{S}_{P_i}}_{M}, {\mathcal{T}_{P_i}}_{M})$ is a pair of mutual $N_S$-dual functions.
\end{coroTEMAi}
\begin{proof}  Straightforward from  Proposition \ref{prop_F}. 
\end{proof}

\begin{propTEMAi} \label{prop_F1}
Let $A$ be the median aggregation  and $N$ be a SFN. Additionally, let  $\phi, \psi:U \rightarrow U$ be functions in $Aut(U)$ given as $\phi(x)=x^r$ and $\psi(x)=1-(1-x)^r$, respectively. Then, for all $x,y \in U$, the following holds:
\begin{eqnarray}\label{eq34}
\left({\mathcal{T}_{P_{i}}}_{M}\right)^{\phi} (x,y) = \left(\mathcal{T}_{{P_{i}}^\phi}\right)_{M^{\phi}} (x,y) & \mbox {and } & 
\left({\mathcal{S}_{P_{i}}}_{M}\right)^{\psi} (x,y) = 
\left(\mathcal{S}_{P_{i}^{\psi}}\right)_{M^{\psi}} (x,y). 
\end{eqnarray}
\end{propTEMAi} 
\begin{proof} For all $x,y \in U$, based on results presented in Proposition~\ref{prop-aut-subt}, we have that
\begin{eqnarray*}
\left({\mathcal{T}_{P_{i}}}_{M}\right)^{\phi} (x,y)  &=& \phi^{-1}(\mathcal{T}_{M}(\phi(x),\phi(y))) \mbox{  by Eq.~(\ref{eq func dual})}
\\
&=& \phi^{-1}(M(T_1(\phi(x),\phi(y))),T_2(\phi(x),\phi(y))), \ldots , T_n(\phi(x),\phi(y))) \mbox{  by Eq.~(\ref{eq:median1})} 
\\
&=& \phi^{-1} (M((\phi \circ \phi^{-1})(T_1^{\phi}(x,y)),(\phi \circ \phi^{-1})(T_2^{\phi}(x,y)),  \ldots , (\phi \circ \phi^{-1})(T_n^{\phi}(x,y)))   \mbox{  by Eq.~(\ref{eq func dual})}
\\
&=& \phi^{-1} (M(\phi(T_1^{\phi}(x,y)), \phi(T_2^{\phi}(x,y)) ,\ldots , \phi(T_n^{\phi}(x,y)))   \mbox{  by Eq.~(\ref{eq func dual})}
\\
&=& \phi^{-1} (M(\phi(T_1(x,y)), \phi(T_{\sqrt[r]{2}}(x,y)) ,\ldots , \phi(T_{\sqrt[r]{n}}(x,y)))   \mbox{  by Eq.~(\ref{eq_T_Saut})}
\\
&=& M^{\phi}(T_1(x,y),  T_{\sqrt[r]{2}}(x,y), \ldots , T_{\sqrt[r]{n}}(x,y))  \mbox{  by Eq.~(\ref{eq:median1})}
\\
&=&  {\left(\mathcal{T}_{P_{\sqrt[r]{i}}}\right)}_{M^{\phi}} (x,y) =  \left(\mathcal{T}_{{P_{i}}^\phi}\right)_{M^{\phi}} (x,y).
\end{eqnarray*}
Therefore, Eq~(\ref{eq34})a is held. The dual construction can also be proven, analogously.
\end{proof}



\subsection{Aggregating fuzzy (S,N)-subimplications}
This section describes the class of aggreggating fuzzy (S,N)-subimplications obtained by considering the median operator.

\begin{propTEMAi}~\cite[Proposition~12]{BRYB13} \label{pro-2} Let $\mathcal{I}_{M}$ be the $(M, \mathcal{I})$-operator  defined by the median aggregation $M$ and the family $\mathcal{I}_{S,N}$ of (S,N))-subimplications, which is previously defined in Eq~(\ref{eq_snimp}). Then, $\mathcal{I}_{M}$ verifies  $\mathbf{I0}, \mathbf{I2},\mathbf{I3},\mathbf{I4}$,$\mathbf{I6},\mathbf{I7} and \mathbf{I8}$ when all the member-function of  $I_i \in \mathcal{I}$ verifies  $\mathbf{I0}, \mathbf{I2},\mathbf{I3},\mathbf{I4}$,$\mathbf{I6},\mathbf{I7}and\mathbf{I8}$, respectively.
\end{propTEMAi}
%\begin{proof} In~\cite[Proposition~12]{BRYB13}, it is proved that $I_i \in \mathcal{I}$ verifies  $\mathbf{I0}, \mathbf{I2},\mathbf{I3},\mathbf{I4}$ and $\mathbf{I6}$.
%From Proposition~\ref{pro:Ii2} we have that each $I_{ij}$ with $1\leq j\leq n$ satisfies $\mathbf{I7}$ and $\mathbf{I8}$. So, for all $x,y,z \in U$,  the following holds.
% \begin{description}
%%
%\item [$\textbf{I7}$:] When $M$ is an idempotent function, it holds that
%$\mathcal{I}_{M}(1,y) = M(I_1(1,y) \ldots , I_n(1,y))= M(y,y, \ldots, y) = y$.
%\item [$\textbf{I8}$:] If $x\leq y$, 
%$N_{\mathcal{I}_{M}}(x)=\mathcal{I}_{M}(x,0) = M(I_1(x,0), \ldots , I_n(x,0)) \geq M(I_1(y,0), \ldots , I_n(y,0)) = N_{\mathcal{I}_{M}}(y)$.
%%
%%\item[$\textbf{I6}$:] $\mathcal{I}_{M}(N_S(y),N_S(x))
%% = 1- \frac{1}{i}((1-y)-(1-y)(1-x)) = 1-\frac{1}{i}(x-xy) = \mathcal{I}%_{M}(x,y)$, concluding that $\mathcal{I}_{M}$ verifies the contrapositive %symmetry.
% \end{description}
% Therefore, Proposition \ref{pro-2} is verified.
%\end{proof}

\begin{propTEMAi}\label{pro-20}~\cite[Proposition~13]{BRYB13} The $(A, \mathcal{I})$-operator  defined by the median aggregation $M$ and the family of (S,N))-subimplications $\mathcal{I}$, which is previously defined in Eq~(\ref{eq_snimp}), verifies $\mathbf{I5}$.
\end{propTEMAi}
%\begin{proof} According to~Proposition \ref{pro:dist-general-excahnge}, it is enough to prove the next two conditions: that  and for all $x,y_1, \ldots, y_n \in U$, consider the two distinct cases.
%\begin{description}
%\item
% [(i)] Firstly, if $n$ is odd, we obtain that
% \begin{equation*}
% \begin{split}
%& M(I_i(x,y_1), \ldots, I_i(x,y_n)) = M(1-\frac{1}{i}(x-xy_1), \ldots, 1-\frac{1}{i}(x-xy_n)) = 1-\frac{1}{i}(x-xy_{\sigma(\frac{n+1}{2})}).
% \end{split}
% \end{equation*}
%So, $ M(I_i(x,y_1), \ldots, I_i(x,y_n)) =I_i(x,M(y_1, \ldots, y_n))$. Otherwise, if $n$ is even then
%\begin{equation*}
%\begin{split}
%& M(I_i(x,y_1), \ldots, I_i(x,y_n)) = \\ & = M(1-\frac{1}{i}(x-xy_1), \ldots, 1-\frac{1}{i}(x-xy_n)) = 1-\frac{1}{i}(2x\frac{1}{2}-x\frac{1}{2}\left (y_{\sigma(\frac{n}{2})}+y_{\sigma(\frac{n}{2}+1)})\right ) \\
%&  =  1-\frac{1}{i}\left (x-xM(y_1, \ldots, y_n)\right )= I_i(x, M(y_1, \ldots, y_n)).
%\end{split}
%\end{equation*}
%So, we can conclude that the median aggregation $M$ verifies $\mathbf{A7}$.
% %
% \item [(ii)]Now, for $I_{i_1}, I_{i_2} \in \mathcal{I}$, we obtain the following:
%\begin{equation*}
%\begin{split}
%I_{i_1}(x,I_{i_2}(y,z)) & = I_{i_1}(x,1-\frac{1}{i_2}(y-yz)) = 1-\frac{1}{i_1}(x-x(1-\frac{1}{i_2}(y-yz)))\\
%   & = 1-\frac{xy}{i_1i_2}(1-z) = 1-\frac{1}{i_1}(y-y(1-\frac{1}{i_2}(x-xz))) = I_{i_1}(y,I_{i_2}(x,z)).
%   \end{split}
%\end{equation*}
%Additionally, $\mathcal{I}_{M}$ verifies the generalized exchange principle.
% \end{description}
% Concluding, $\mathcal{I}_{M}$ verifies $\mathbf{I5}$.
%\end{proof}



\begin{propTEMAi}~\cite[Corollary~3]{BRYB13}\label{pro-100}
%Let $M\colon U^n \to U$ be the median aggregation and $\mathcal{I}= \{I_i \colon U^k \to U\}$, for $i\in \{1,2, \dots , n\}$ be a family of fuzzy  subimplications
%$I_i = 1-\frac{1}{i}(x-xy)$.
Let $(A, \mathcal{I})$-operator be the median aggregation $M$ and $\mathcal{I}$ ($\mathcal{J}$) be the family of (S,N)-sub(co)implications previously defined in Eq.(\ref{eq_snimp}). The operators
$\mathcal{I}_{M}$  and $\mathcal{J}_{M}$   given by
\begin{equation}\label{eq-finalS}
\mathcal{I}_{M} (x,y)=\mathcal{S}_{M}(N(x),y) \mbox{  and } \mathcal{J}_{M} (x,y)=\mathcal{T}_{M}(N(x),y),
\end{equation}
are a $(\mathcal{S}_{M},N)$-subimplication   and a $(\mathcal{T}_{M},N)$-subcoimplication, respectively. 
\end{propTEMAi}
%\begin{proof} Straightforward from Propositions~\ref{pro-101},~\ref{pro-2} and~\ref{pro-20}.
%\end{proof}

\begin{coroTEMAi} For all $x,y \in U$, the following functions 
\begin{eqnarray}\label{eq340}
\hspace{-0.8cm}{\mathcal{I}_{\mathcal{S}_{P_{i}},N_S}}_{M} (x,y) = \left({\mathcal{I}_{{\mathcal{S}_{P_{i}}},N_S}}\right)_{M} (x,y) \hspace{-0.2cm}&\mbox{and}& \hspace{-0.2cm}
\left({\mathcal{J}_{\mathcal{T}_{P_{i}},N_S}}_{M}\right) (x,y) = \left({\mathcal{J}_{{\mathcal{T}_{P_{i}}},N_S}}\right)_{M} (x,y), 
\end{eqnarray}
are a $({\mathcal{S}_{P_i}}_{M},N_S)$-subimplication   and a $({\mathcal{T}_{P_i}}_{M},N_S)$-subcoimplication, respectively.
\end{coroTEMAi}



Summarizing the main result in Proposition \ref{pro-109}, the diagram presented in Figure~\ref{fig-sql-im-dual3} is showing that  the median aggregation $M$ preserves the (S,N)-subimplication class defined in Proposition~\ref{pro-2}, which means, ${\mathcal{I}_{\mathcal{S}_M,N_S}}$ is also an (S,N)-subimplication.
Analogously, it can be obtained by aggregating subimplications in the S-subimplication class.


\begin{figure}[htp]
\centerline{
\begin{diagram}[tight]
 C(N) \times \mathcal{S} \times M & &
\rTo^{Eq.(\ref{eq:agrega_function})} & &  C(N) \times \mathcal{S}_{M} \\
%%%%
\dTo^{Eqs.(\ref{eq-sn-imp})}
&  &     & & \dTo_{Eq.(\ref{eq-finalS})} \\
%%%
\mathcal{I}_{\mathcal{S},N} \times M& & \rTo^{Eq.(\ref{eq:agrega_function})} & &
{(\mathcal{I}_{\mathcal{S},N}})_{M}  \\
\end{diagram}
} \caption{$(\mathcal{S}_M,N_S)$-implication class.}
\label{fig-sql-im-dual3}
\end{figure}




\subsubsection{Conjugate and dual (S,N)-sub(co)implications obtained by  the median}

Let $(A, \mathcal{I})$-operator be the median aggregation $M$ and $\mathcal{J}$ be the family of (S,N)-subcoimpli\-cations obtained by the dual construction. The operators
$\mathcal{I}_{M}$ is a $(\mathcal{T}_{M},N_S)$-subcoimplication, respectively. Their corresponding expression are
$\mathcal{J}_{M} (x,y)=\mathcal{T}_{M}(N_S(x),y)$.

In the following, we consider an automorphism $\phi:U \rightarrow U$ together with the subclass $\mathcal{S}_{P_{i}}$ of  t-subconorms obtained by the median aggregation in order to present   conjugate functions which are preserved by (S,N)-subimplications also obtained by the median aggregation. The corresponding dual construction is also discussed.


\begin{propTEMAi} \label{prop_F}
Let  $\phi, \psi:U \rightarrow U$ be functions in $Aut(U)$ given as $\phi(x)=x^r$ and $\psi(x)=1-(1-x)^r$, respectively. Then, for all $x,y \in U$, the following holds:
\begin{eqnarray}\label{eq340}
\hspace{-0.8cm}\left({\mathcal{I}_{\mathcal{S}_{P_{i}},N_S}}_{M}\right)^{\psi} (x,y) = \left({\mathcal{I}_{{\mathcal{S}_{P_{i}}}^{\psi},N_S}}\right)_{M^{\psi}} (x,y) \hspace{-0.2cm}&\mbox{and}& \hspace{-0.2cm}
\left({\mathcal{J}_{\mathcal{T}_{P_{i}},N_S}}_{M}\right)^{\phi} (x,y) = \left({\mathcal{J}_{{\mathcal{T}_{P_{i}}}^{\phi},N_S}}\right)_{M^{\phi}} (x,y). 
\end{eqnarray}
\end{propTEMAi} 
\begin{proof} For all $x,y \in U$, by Propositions~\ref{prop-aut-subt} and~\ref{prop_F1} and  Eq.(\ref{eq34}), we have that
\begin{eqnarray*}
\left({\mathcal{I}_{\mathcal{S}_{P_{i}},N_S}}_{M}\right)^{\psi} (x,y)  &=&  \left( {\mathcal{S}_{P_{i},N_S}}_M \right)^{\psi} (x,y)  {\mathcal{S}_{P_I^{\psi}}}_{M^{\psi}} \left(N_S(x),T_P(x,y)\right) 
=\left({\mathcal{I}_{{\mathcal{S}_{P_{i}}}^{\psi},N_S}}\right)_{M^{\psi}} (x,y). 
\end{eqnarray*}
Therefore, Eq~(\ref{eq34})a is held. The dual construction can also be proven, analogously.
\end{proof}



\subsection{Aggregating fuzzy QL-subimplications}

This section analyses under which conditions the class of fuzzy QL-subimplications are preserved by the median aggregation operator, investigating properties.


Additionally, we present the subclass of fuzzy QL-subimplication represented by a t-norm $T_P$,  the standard negation $N_S$ together with a t-subconorm $\mathcal{S}_P$, which is obtained by aggregating $n$ fuzzy t-subconorms of the family $\mathcal{S}_P$.



\begin{propTEMAi}\cite[Proposition 14]{BRYB13} \label{pro-109}Let $N$ be a fuzzy negation and $M\colon U^n \to U$ be the median aggregation operator. Then $\mathcal{I}_{M}(\mathcal{J}_{M}) \colon U^2 \to U$ given by
\begin{eqnarray}\label{eq-finalQL}
\mathcal{I}_{M}(x,y) = {\mathcal{I}}_{\mathcal{S}_{M},T,N} (x,y) & \mbox{  and  }  & \mathcal{J}_{M}(x,y) = {\mathcal{J}}_{\mathcal{T}_{M},S,N} (x,y),
\end{eqnarray}
is a $QL$-sub(co)implicator in $\mathcal{I}_{QL}$ ($\mathcal{J}_{QL}$).
\end{propTEMAi}
%\begin{proof} $\mathbf{I0}$ is immediate. Additionally, according to Proposition~\ref{pro-1}, the following  holds:
%\begin{description}
%\item [(i)] Firstly, if $n$ is odd, we obtain the following:
%\begin{align*}
%\mathcal{J}_{M}(x,y) &= M(J_1(x,y), \ldots , J_n(x,y))= M(S_1(N(x),T_1(x,y), \ldots , S_n(N(x),T_n(x,y))\\
%&= \frac{1}{\sigma(\frac{n+1}{2})} \left( S_{\sigma(\frac{n+1}{2})} (N(x),  T(x,y) \right)= \mathcal{S}_{M}(N(x),T(x,y)).
%\end{align*}
%
%\item [(ii)] Otherwise, when $n$ is even, it holds that:
%\begin{align*}
%\mathcal{J}_{M}(x,y)
%&= M(J_1(x,y), \ldots , J_n(x,y))= \frac{1}{2}(J_{\sigma(\frac{n}{2})})+J_{\sigma(\frac{n+1}{2})}) \\
%&= \frac{1}{2}(S_{\sigma(\frac{n}{2})}(N(x),T(x,y))+S_{\sigma(\frac{n+1}{2})}(N(x),T(x,y))) \\
%&= M(S_1(N(x),T(x,y), \ldots, S_n(N(x),T(x,y))= \mathcal{S}_M(N(x),T(x,y))
%\end{align*}
%\end{description}
%Therefore, Eq.(\ref{eq-finalQL})a is verified. Analogously, Eq.(\ref{eq-finalQL})b can be verified.
%\end{proof}
%
%
%\begin{propTEMAi} \label{pro-200} Let $M\colon U^n \to U$ be the median aggregation operator and $\mathcal{I}_{M}$ be a fuzzy QL-sub(co)implicator  given by Eq.(\ref{eq-finalQL})a. Then  $(\mathcal{I},M)$-operator  verifies the properties  $\mathbf{I0}, \mathbf{I2}$, $\mathbf{I4}$, $\mathbf{I9}$ and $\mathbf{I10}$.
%\end{propTEMAi}
%\begin{proof}
% Straightforward from Proposition \ref{pro-7-8}.
%\end{proof}


Analogously, the following results can be stated by an $(\mathcal{J},M)$-operator obtained by the median aggregation operator performed over a set of  fuzzy QL-subcoimplicators. 

\begin{coroTEMAi}Let $M\colon U^n \to U$ be the median aggregator and $\{J_i \colon U^k \to U\}$  be a family of QL-subimplications given by Eq. (\ref{eq-ql-imp}).
 Then $\mathcal{I}_{M}$ verifies  $\mathbf{I0}, \mathbf{I3}$, $\mathbf{I4}$, $\mathbf{I7}$ and $\mathbf{I8}$.
\end{coroTEMAi}
\begin{proof}
Straightforward from Propositions~\ref{pro-ji-qlimp} and \ref{pro-109}.
\end{proof}

\begin{coroTEMAi} For all $x,y \in U$ and  the  following holds: 
\begin{eqnarray}\label{eq340}
\mathcal{I}_{{\mathcal{S}_{P_{i}}}_{M},T_P,N_S}(x,y) &=& \left({\mathcal{I}_{{\mathcal{S}_{P_{i}}},T_P,N_S}}\right)_{M} (x,y)
 \\
\mathcal{J}_{{\mathcal{T}_{P_{i}}}_{M},S_P,N_S} (x,y) &=& \left({\mathcal{J}_{{\mathcal{T}_{P_{i}}},S_P,N_S}}\right)_{M} (x,y). 
\end{eqnarray}
Additionally,   ${I}_{{\mathcal{S}_{P_{i}}}_M,T_P,N_S}  \in \mathcal{I}_{QL}$  and ${J}_{{\mathcal{T}_{P_{i}}}_M,S_P,N_S} \in \mathcal{J}_{QL}$, respectively.
\end{coroTEMAi}
\begin{proof} Straightforward.
\end{proof}


Summarizing, in Figure~\ref{fig-sql-im-dual2}, a diagrammatic representation of the result stated in Proposition~\ref{pro-109} is presented. In such graphical description the median aggregation $M$ preserves the fuzzy QL-subimplication class, meaning that the statements are both equivalent:

\begin{enumerate}
\item   Firstly, we obtain $\mathcal{S}_P$ by the median aggregation operator performed over $n$ t-subconorms in the family $\mathcal{S}_P$. And after that, we are able to define an $(\mathcal{J},A)$-operator as a fuzzy QL-subimplication represented by a t-norm $T_P$,  the standard negation $N_S$ together with a t-subconorm $\mathcal{S}_P$.

\item For each t-subconorm $S_i$,  the family $\mathcal{J}$ of QL-implications whose explicitly representable  member-functions are given by  $I_{{S_I}_P, T_P, N_S}$, are  constructed. And after that, as a consequence, by aggregating $n$ member-functions of $\mathcal{J}$, we obtain an $(\mathcal{J},A)$-operator.
\end{enumerate}


\begin{figure}[htp]
\centerline{
\begin{diagram}[tight]
 C(N) \times T_P \times \mathcal{S} \times M & &
\rTo^{Eq.(\ref{eq:agrega_function})} & &  C(N) \times T_P \times \mathcal{S}_{M} \\
%%%%
\dTo^{Eqs.(\ref{eq-ql-imp})}
&  &     & & \dTo_{Eq.(\ref{eq-finalQL})} \\
%%%
\mathcal{I}_{\mathcal{S},\mathcal{T},N} \times M& & \rTo^{Eq.(\ref{eq:agrega_function})} & &
({\mathcal{I}_{\mathcal{S},T_P,N}})_{M}\\
\end{diagram}
} \caption{$(\mathcal{S}_{\mathcal{A}},\mathcal{T}_{\mathcal{A}},N)$-implication class obtained by the median aggregation operator.}
\label{fig-sql-im-dual2}
\end{figure}

\subsubsection{Conjugate and dual QL-subimplications obtained by the median}

In this section, conjugate and dual QL-sub(co)implications obtained by the median aggregation are discussed.

\begin{propTEMAi} \label{prop_F}
Let  $\phi, \psi:U \rightarrow U$ be functions in $Aut(U)$ given as $\phi(x)=x^r$ and $\psi(x)=1-(1-x)^r$, respectively. Then, for all $x,y \in U$, the following holds:
\begin{eqnarray}
\left({\mathcal{I}_{\mathcal{S}_{P_{i}},T_P,N_S}}_{M}\right)^{\psi} (x,y) &=& \left({\mathcal{I}_{{\mathcal{S}_{P_{i}}}^{\psi},T_P,N_S}}\right)_{M^{\psi}} (x,y) \label{eq340}\\
\left({\mathcal{J}_{\mathcal{T}_{P_{i}},S_P,N_S}}_{M}\right)^{\phi} (x,y) &=& \left({\mathcal{J}_{{\mathcal{T}_{P_{i}}}^{\phi},T_P,N_S}}\right)_{M^{\phi}} (x,y).\label{eq3400} 
\end{eqnarray}
\end{propTEMAi} 
\begin{proof} For all $x,y \in U$, based on results in Propositions~\ref{prop-aut-subt},~\ref{eq34} and~\ref{pro-109}, we have that
\begin{eqnarray*}
\left({\mathcal{I}_{\mathcal{S}_{P_{i}},T_P,N_S}}_{M}\right)^{\psi} (x,y)  &=&  \left( {\mathcal{S}_{P_{i},T_P,N_S}}_M \right)^{\psi} (x,y)  \mbox{  by Eq.(\ref{eq-finalQL})}
\\ 
&=& {\mathcal{S}_{P_i^{\psi}}}_{M^{\psi}} \left(N_S(x),T_P(x,y)\right) 
=\left({\mathcal{I}_{{\mathcal{S}_{P_{i}}}^{\psi},T_P,N_S}}\right)_{M^{\psi}} (x,y). 
\end{eqnarray*}
So, Eq~(\ref{eq340}) is held.  Eq~(\ref{eq3400}) can also be proven, analogously.
\end{proof}


\begin{propTEMAi}
For all $x,y \in U$, the following holds:
\begin{eqnarray}
{\mathcal{I}_{\mathcal{S}_{P_{i}},T_P,N_S}}_{N_S} (x,y) &=& \mathcal{J}_{\mathcal{S}_{P_{i}},T_P,N_S}; \\
{\mathcal{J}_{\mathcal{T}_{P_{i}},S_P,N_S}}_{N_S} (x,y) &=& \mathcal{I}_{\mathcal{T}_{P_{i}},S_P,N_S}.
\end{eqnarray}
\end{propTEMAi}


\newsec{Conclusion and Final Remarks}\label{sec-4}

In this paper we characterize both (S,N)- and  QL-subimplications with respect to the median aggregation operator.
In particular, the underlying principle of the proof related to properties preserved by the new (S,N)- and  QL-subimplications, which are obtained by the median aggregation, is similar to the (S,N)- and  QL-implications.

Just to be able to compare the two cases, since such classes of subimplication are represented by t-subconorms and t-subnorms which are characterized by generalized associativity, the corresponding (S,N)- and  QL-subimplications are  characterized by distributive $n$-ary aggregation together with related generalizations, as the exchange and neutrality principles.


Ongoing work on application of Atanassov's intuitionistic extension of fuzzy connectives provides relevant methods to obtain other operators by distributive $n$-ary aggregation, preserving their main properties and discussion the conditions under which conjugate and dual  fuzzy intuitionistic connectives can be also preserved.

%\newsec*{Acknowledgment}
%
%This work is supported by the Brazilian funding agencies CAPES, FAPERGS (Ed. PqG 06/2011, under the process number 11/1520-1) CNPqCNPq(309533/2013-9).





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\end{document}

