The Relations between Residuated Frames and Residuated Connections

Abstract

We introduce the notion of (dual) residuated frames as a viewpoint of relational semantics for a fuzzy logic. We investigate the relations between (dual) residuated frames and (dual) residuated connections as a topological viewpoint of fuzzy rough sets in a complete residuated lattice. As a result, we show that the Alexandrov topology induced by fuzzy posets is a fuzzy complete lattice with residuated connections. From this result, we obtain fuzzy rough sets on the Alexandrov topology. Moreover, as a generalization of the Dedekind–MacNeille completion, we introduce R-R (resp. $DR$-$DR$) embedding maps and R-R (resp. $DR$-$DR$) frame embedding maps.

1. Introduction

Blyth and Janovitz [1] introduced the residuated connection as a pair $(f,g)$ of maps from a partially ordered set $(X,{\le }_X)$ to a partially ordered set $(Y,{\le }_Y)$ such that for all $x\in X,y\in Y$, $f(x){\le }_{Y}y$ if and only if $x{\le }_{X}g(y).$ Examples of maps which form residuated connections play an important role [2,3,4]. Orłowska and Rewitzky [5,6,7] introduced the residuated frame of logical relational systems for residuated connections.

Pawlak [8,9] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. Rough sets form residuated connections in the following sense: let R be an equivalence relation on X. For $A\subset X$ and ${\left[x\right]}_R=\{y\in X|(x,y)\in R\}$,

$$\overline{R}(A)=\{x\in X|{\left[x\right]}_R\cap A\ne \varnothing \},\underline{R}(A)=\{x\in X|{\left[x\right]}_R\subset A\}.$$

(1)

Let $P(X)$ be the class of all subsets of X and $(P(X),\subset )$ be a partially ordered set. A rough set $(\underline{R},\overline{R})$ forms a residuated connection because for all $A,B\subset X$, $\overline{R}(A)\subset B$ if and only if $A\subset \underline{R}(B).$

Ward et al. [10] introduced a complete residuated lattice L as an important algebraic structure for many valued logics [11,12,13,14,15,16]. For an extension of Pawlak’s rough sets, many researchers have developed L-lower and L-upper approximation operators in algebraic structures L [17,18,19,20,21,22,23,24,25]. She and Wang [26] developed an L-fuzzy rough set $(G,H)$ with L-lower approximation operator G and L-upper approximation operator F in complete residuated lattices as follows. Let $(X,e_X)$ be an L-fuzzy partially ordered set. For $A,B\in L^X$,

$$F(A)(y)=\underset{x\in X}{\bigvee }(e_X(x,y)\odot A(x)),G(B)(x)=\underset{y\in X}{\bigwedge }\left(e_X(x,y)\to B(y)\right).$$

(2)

Moreover, fuzzy rough sets form residuated connections in the following sense: for all $A,B\subset X$,

$$\begin{array}{cc}\hfill e_{L^Y}(F(A),B)& =\underset{y\in X}{\bigwedge }(F(A)(y)\to B(y))=\underset{x\in X}{\bigwedge }(A(x)\to G(B)(x))=e_{L^X}(A,G(B)).\hfill \end{array}$$

(3)

Perfilieva [27,28,29,30] introduced the theory of fuzzy transform and inverse fuzzy transform in complete residuated lattices, which is similar to other well-known transform theories such as the Fourier, Laplace, Hilbert and wavelet transforms, as well as fuzzy various concept analysis and fuzzy relation equations [31,32,33]. Oh and Kim [34] interpreted Perfilieva’s fuzzy transform as a residuated connection $(e_{L^X},F,G,e_{L^Y})$ with fuzzy transform and inverse fuzzy transform G. By using the residuated connection, F is a fuzzy join preserving map and G is a fuzzy meet preserving map in a Kim’s fuzzy complete lattice sense [20], as a generalization of a complete lattice [35,36,37,38]. If X and Y are solutions of fuzzy relation equations $F(X)=B$ and $G(Y)=A$, then $G(B)$ and $F(A)$ are solutions, respectively.

Discrete and stone dualities are dualities between algebras and logical relational systems such as Boolean algebras and classical propositional logics; MV-algebra and Lukasiewicz logic; and BL-algebra and basic fuzzy logics [3,4,5,6,39,40,41]. The duality leads in a natural way to relational semantics for a logic [39,40,41].

In this paper, as a duality between algebras and logical relational systems, we introduce the notion of residuated connections and residuated frames in fuzzy logics. In Theorems 3 and 4, we show that (dual) residuated frames induce (dual) residuated connections.

Let $(X,e_X)$ be an L-fuzzy partially ordered set. As a generalization of the classic Tarski’s fixed point theorem [42,43] for isotone maps, we show that ${\tau }_{e_X}=\{A\in L^X|A=F(A)={\bigvee }_{x\in X}(e_X(x,y)\odot A(x))\}$ is an Alexandrov L-topology and $({\tau }_{e_X},\vee ,\wedge ,e_{{\tau }_{e_X}})$ is a fuzzy complete lattice [20].

If $(e_X,R,S,e_Y)$ is a residuated frame, then we show that $F:{\tau }_{e_X}\to {\tau }_{e_Y}$ and $G:{\tau }_{e_Y}\to {\tau }_{e_X}$ are well-defined and $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is a residuated connection; $e_{{\tau }_{e_Y}}(F(A),B)=e_{{\tau }_{e_X}}(A,G(B))$ is defined by

$$F(A)(y)=\underset{x\in X}{\bigvee }(A(x)\odot R(x,y)),G(B)(x)=\underset{y\in Y}{\bigwedge }(S(y,x)\to B(y))$$

(4)

where ${\tau }_{e_X}$ and ${\tau }_{e_Y}$ are Alexandrov L-topologies induced by fuzzy posets $(X,e_X)$ and $(Y,e_Y)$ in Theorem 1. Using this result, one can show that the pair $(F(A),G(A))$ is an fuzzy rough set for A on ${\tau }_{e_X}$ because $(e_X,R=e_X,S=e_X^{-1},e_X)$ is a residuated frame. Moreover, we show the existence of fuzzy rough sets from residuated connections.

Similarly, by Theorem 4, dual residuated frames induce dual residuated connections. In Theorem 5 (resp. 9), (resp. dual) residuated connections induce (resp. dual) residuated frames. Under various relations, we investigate the (dual) residuated connections and frames on Alexandrov L-topologies.

As a generalization of the Dedekind–MacNeille completion [37], we prove the existence of R-R (resp. $DR$-$DR$) embedding maps and R-R (resp. $DR$-$DR$) frame embedding maps.

2. Preliminaries

Definition 1

([10]). An algebra $(L,\wedge ,\vee ,\odot ,\to ,\perp ,\top )$ is called a complete residuated lattice if it satisfies the following conditions:

(L1) 

$(L,\le ,\vee ,\wedge ,\perp ,\top )$ is a complete lattice with the greatest element ⊤ and the least element ⊥;

(L2) 

$(L,\odot ,\top )$ is a commutative monoid;

(L3) 

$x\odot y\le z$ if and only if $x\le y\to z$ for $x,y,z\in L$.

In this paper, we always assume that $(L,\le ,\odot ,\to ,\ast )$ is a complete residuated lattice with $x^{\ast }=x\to \perp$ and ${(x^{\ast })}^{\ast }=x$.

For $\alpha \in L,A\in L^X$, we denote $(\alpha \to A),(\alpha \odot A),{\alpha }_X\in L^X$ by $(\alpha \to A)(x)=\alpha \to A(x),(\alpha \odot A)(x)=\alpha \odot A(x),$${\alpha }_X(x)=\alpha$.

Lemma 1

([2]). Let $x,y,z,x_i,y_i,w\in L$. Then the following hold:

(1)

$\top \to x=x$, $\perp \odot x=\perp ;$

(2)

If $y\le z$, then $x\odot y\le x\odot z$, $x\to y\le x\to z$ and $z\to x\le y\to x$;

(3)

$x\le y$ if and only if $x\to y=\top$;

(4)

$x\to ({\bigwedge }_i{y}_i)={\bigwedge }_i(x\to y_i)$;

(5)

$({\bigvee }_i{x}_i)\to y={\bigwedge }_i(x_i\to y)$;

(6)

$x\odot ({\bigvee }_i{y}_i)={\bigvee }_i(x\odot y_i)$;

(7)

$(x\odot y)\to z=x\to (y\to z)=y\to (x\to z)$;

(8)

$(x\to y)\odot (z\to w)\le (x\odot z)\to (y\odot w)$ and $x\to y\le (x\odot z)\to (y\odot z)$;

(9)

$(x\to y)\odot (y\to z)\le x\to z$;

(10)

${\bigvee }_{i\in \mathsf{\Gamma }}{x}_i\to {\bigvee }_{i\in \mathsf{\Gamma }}{y}_i\ge {\bigwedge }_{i\in \mathsf{\Gamma }}(x_i\to y_i)$ and ${\bigwedge }_{i\in \mathsf{\Gamma }}{x}_i\to {\bigwedge }_{i\in \mathsf{\Gamma }}{y}_i\ge {\bigwedge }_{i\in \mathsf{\Gamma }}(x_i\to y_i)$;

(11)

$x\to y\le (y\to z)\to (x\to z)$ and $x\to y\le (z\to x)\to (z\to y)$;

(12)

${(x\odot y^{\ast })}^{\ast }=x\to y$ and $x\to y=y^{\ast }\to x^{\ast }$.

Definition 2

([21]). Let X be a set. A function $e_X:X\times X\to L$ is called:

(E1) 

Reflexive if$e_X(x,x)=\top$for all$x\in X$;

(E2) 

Transitive if$e_X(x,y)\odot e_X(y,z)\le e_X(x,z)$, for all$x,y,z\in X$;

(E3) 

If$e_X(x,y)=e_X(y,x)=\top$, then$x=y$. If$e_X$satisfies (E1) and (E2), then$(X,e_X)$is called a fuzzy preorder set. Ifesatisfies (E1), (E2) and (E3), then$(X,e_X)$is called a fuzzy partially order set (simply, fuzzy poset).

Definition 3

([18]). (1) A subset ${\tau }_X\subset L^X$ is called an Alexandrov L-topology on X if it satisfies the following conditions:

(O1) 

${\alpha }_X\in {\tau }_X$;

(O2) 

If$A_i\in {\tau }_X$for all$i\in I$, then${\bigvee }_{i\in I}{A}_i,{\bigwedge }_{i\in I}{A}_i\in {\tau }_X$;

(O3) 

If$A\in {\tau }_X$ and $\alpha \in L$, then $\alpha \odot A,\alpha \to A\in {\tau }_X$. The pair $(X,{\tau }_X)$ is called an Alexandrov L-topological space.

Lemma 2.

Let${\tau }_X\subset L^X$. Define$e_{{\tau }_X}:{\tau }_X\times {\tau }_X\to L$by$e_{{\tau }_X}(A,B)={\bigwedge }_{x\in X}(A(x)\to B(x))$. Then$({\tau }_X,e_{{\tau }_X})$is a fuzzy poset.

Proof. 

(E1) For all $A\in {\tau }_X$, we have $e_{{\tau }_X}(A,A)={\bigwedge }_{x\in X}(A(x)\to A(x))=\top$.

(E2) Let $A,B,C\in {\tau }_X$. Then by Lemma 1(9), we have

$$\begin{array}{cc}{e}_{{\tau }_X}(A,B)\odot e_{{\tau }_X}(B,C)\hfill & =\underset{x\in X}{\bigwedge }(A(x)\to B(x))\odot \underset{x\in X}{\bigwedge }(B(x)\to C(x))\hfill \\ & \le \underset{x\in X}{\bigwedge }((A(x)\to B(x))\odot (B(x)\to C(x)))\hfill \\ & \le e_{{\tau }_X}(A,C).\hfill \end{array}$$

(5)

(E3) Let $e_{{\tau }_X}(A,B)=e_{{\tau }_X}(B,A)=\top$. Then by Lemma 1(3), $A=B$.

Hence $({\tau }_X,e_{{\tau }_X})$ is a fuzzy poset. □

Theorem 1.

([18]) Let $(X,e_X)$ be a fuzzy poset. Define

$${\tau }_{e_X}=\{A\in L^X|A(x)\odot e_X(x,z)\le A(z)\}.$$

(6)

Then ${\tau }_{e_X}$ is an Alexandrov L-topology on X.

Remark 1.

(1) Let$(X,{\top }_{{▵}_X})$be a fuzzy poset where${\top }_{{▵}_X}(x,x)=\top$and${\top }_{{▵}_X}(x,y)=\perp$for$x\ne y\in X$. Then${\tau }_{{\top }_{{▵}_X}}=L^X$and$e_{{\tau }_{{\top }_{{▵}_X}}}=e_{L^X}:L^X\times L^X\to L$as$e_{L^X}(A,B)={\bigwedge }_{x\in X}(A(x)\to B(x))$.

(2) Let$(X,{\top }_{X\times X})$be a fuzzy poset where${\top }_{X\times X}(x,y)=\top$for each$x,y\in X$. Then${\tau }_{{\top }_{X\times X}}=\{{\alpha }_X\in L^X|\alpha \in L\}$and$e_{{\tau }_{{\top }_{X\times X}}}:{\tau }_{{\top }_{X\times X}}\times {\tau }_{{\top }_{X\times X}}\to L$by$e_{{\tau }_{{\top }_{X\times X}}}({\alpha }_X,{\beta }_X)=\alpha \to \beta$.

3. Fuzzy Residuated Frames and Fuzzy Residuated Connections on Alexandrov $\mathit{L}$-topologies

Definition 4.

Let$(X,e_X)$and$(Y,e_Y)$be fuzzy posets. Let$f:X\to Y$ and $g:Y\to X$ be maps.

(1) $(e_X,f,g,e_Y)$ is a residuated connection if $e_Y(f(x),y)=e_X(x,g(y))$ for all $x\in X,y\in Y$;

(2) $(e_X,f,g,e_Y)$ is a dual residuated connection if $e_Y(y,f(x))=e_X(g(y),x)$ for all $x\in X,y\in Y$;

(3) f is an isotone map if $e_Y(f(x_1),f(x_2))\ge e_X(x_1,x_2)$ for all $x_1,x_2\in X$;

(4) f is an antitone map if $e_Y(f(x_1),f(x_2))\ge e_X(x_2,x_1)$ for all $x_1,x_2\in X$;

(5) f is an embedding map if $e_Y(f(x_1),f(x_2))=e_X(x_1,x_2)$ for all $x_1,x_2\in X$.

Theorem 2.

Let $(X,e_X)$ and $(Y,e_Y)$ be fuzzy posets. Let $f:X\to Y$ and $g:Y\to X$ be maps.

(1) $(e_X,f,g,e_Y)$ is a residuated connection if and only if $f,g$ are isotone maps and $e_Y(f(g(y)),y)=e_X(x,g(f(x)))=\top$ for all $x,y\in X$;

(2) $(e_X,f,g,e_Y)$ is a dual residuated connection if and only if $f,g$ are isotone maps and $e_Y(y,f(g(y)))=e_X(g(f(x)),x)=\top$ for all $x,y\in X$.

Proof. 

(1) Let $(f,g)$ be a residuated connection. Since $e_Y(f(x),y)=e_X(x,g(y))$, we have $\top =e_Y(f(x),f(x))=e_X(x,g(f(x)))$ and $e_Y(f(g(y)),y)=e_X(g(y),g(y))=\top$. Furthermore,

$$e_Y(f(x_1),f(x_2))=e_X(x_1,g(f(x_2)))\ge e_X(x_1,x_2)\odot e_X(x_2,g(f(x_2)))=e_X(x_1,x_2).$$

Conversely,

$$e_Y(f(x),y)\ge e_Y(f(g(y)),y)\odot e_Y(f(x),f(g(y)))=e_Y(f(x),f(g(y)))\ge e_X(x,g(y)).$$

Similarly, $e_Y(f(x),y)\le e_X(x,g(y))$.

(2) Since $e_Y(f(x),y)=e_X(g(y),x)$, we have $\top =e_Y(f(x),f(x))=e_X(g(f(x)),x)$ and $e_Y(f(g(y)),y)=e_X(g(y),g(y))=\top$. Furthermore,

$$e_Y(f(x_1),f(x_2))=e_X(g(f(x_2)),x_1)\ge e_X(x_2,x_1)\odot e_X(g(f(x_2)),x_2)=e_X(x_2,x_1).$$

□

For $R_1\in L^{X\times Y}$ and $R_2\in L^{Y\times Z}$, define

$$R_1\circ R_2(x,z)=\underset{y}{\bigvee }(R_1(x,y)\odot R_2(y,z)),R_1^{-1}(y,x)=R_1(x,y).$$

(7)

Lemma 3.

Let $(X,e_X)$ and $(X,e_Y)$ be fuzzy posets. Let $R\in L^{X\times Y}$. Then the following hold:

(1)

${(e_X\circ R)}^{-1}=R^{-1}\circ e_X^{-1}$ and ${(R\circ e_X)}^{-1}=e_X^{-1}\circ R^{-1}$;

(2)

$e_X\circ R\le R$ if and only if $e_X^{-1}\circ R^{\ast }\le R^{\ast }$;

(3)

$R\circ e_X^{-1}\le R$ if and only if $R^{\ast }\circ e_X\le R^{\ast }$;

(4)

$e_X\circ R\circ e_Y\le R$ if and only if $e_X\circ R\le R$ and $R\circ e_Y\le R$;

(5)

$e_X^{-1}\circ R\circ e_Y^{-1}\le R$ if and only if $e_X^{-1}\circ R\le R$ and $R\circ e_Y^{-1}\le R$;

(6)

$e_X^{-1}\circ R\circ e_Y^{-1}\le R$ if and only if $e_X\circ R^{\ast }\circ e_Y\le R^{\ast }$.

Proof. 

(1) ${(e_X\circ R)}^{-1}(y,x)=e_X\circ R(x,y)={\bigvee }_{z\in X}(e_X(x,z)\odot R(z,y))={\bigvee }_{z\in X}(e_X^{-1}(z,x)\odot R^{-1}(y,z))=R^{-1}\circ e_X^{-1}(y,x).$ Similarly, ${(R\circ e_X)}^{-1}=e_X^{-1}\circ R^{-1}$.

(2) $e_X(x,z)\odot R(z,y)\le R(x,y)$ if and only if $R(z,y)\le e_X(x,z)\to R(x,y)$ if and only if $e_X(x,z)\odot R^{\ast }(x,y)\le R^{\ast }(z,y)$ if and only if $e_X^{-1}(z,x)\odot R^{\ast }(x,y)\le R^{\ast }(z,y)$.

(3) $R(w,y)\odot e_X^{-1}(y,x)\le R(w,x)$ if and only if $R(w,y)\odot e_X(x,y)\le R(w,x)$ if and only if $e_X(x,y)\to R^{\ast }(w,y)\ge R^{\ast }(w,x)$ if and only if $e_X(x,y)\odot R^{\ast }(w,x)\le R^{\ast }(w,y)$.

(4) $e_X\circ R\circ e_Y(x,y)={\bigvee }_{y_1\in Y}(e_X\circ R)(x,y_1)\odot e_Y(y_1,y)\ge (e_X\circ R)(x,y)\odot e_Y(y,y)=(e_X\circ R)(x,y)$. Similarly, $R\circ e_Y\le R$. The converse part can be proved easily.

(5) and (6) can be proved easily by using (2)–(4). □

Definition 5.

Let$(X,e_X)$and$(Y,e_Y)$be fuzzy posets. Let$R\in L^{X\times Y}$and$S\in L^{Y\times X}$. A structure$(e_X,R,S,e_Y)$is called:

(1) A residuated frame if $S=R^{-1}$ and $e_X\circ R\circ e_Y\le R$;

(2) A dual residuated frame if $S=R^{-1}$ and $e_X^{-1}\circ R\circ e_Y^{-1}\le R$.

Lemma 4.

Let $(X,e_X)$ and $(Y,e_Y)$ be fuzzy posets. Then the following hold:

(1)

Let $(e_X,f,g,e_Y)$ be a residuated connection. Define maps $R:X\times Y\to L$ and $S:Y\times X\to L$ by

$$R(x,y)=e_X(x,g(y))=e_Y(f(x),y),S(y,x)=R(x,y).$$

(8)

Then $(e_X,R,S,e_Y)$ is a residuated frame;

(2)

Let $(e_X,f,g,e_Y)$ be a dual residuated connection. Define maps $R:X\times Y\to L$ and $S:Y\times X\to L$ by

$$R(x,y)=e_X(g(y),x)=e_Y(y,f(x)),S(y,x)=R(x,y).$$

(9)

Then $(e_X,R,S,e_Y)$ is a dual residuated frame;

(3)

If g is isotone and $R_1(x,y)=e_X(x,g(y))$ (resp. $R_2(x,y)=e_X(g(y),x)$), then $e_X\circ R_1\circ e_Y\le R_1$ (resp. $e_X^{-1}\circ R_2\circ e_Y^{-1}\le R_2$);

(4)

If f is isotone and $R_1(x,y)=e_Y(y,f(x))$ (resp. $R_2(x,y)=e_Y(f(x),y)$), then $e_X^{-1}\circ R_1\circ e_Y^{-1}\le R_1$ (resp. $e_X\circ R_2\circ e_Y\le R_2$).

Proof. 

(1) For all $x,x_1\in X$ and $y,y_1\in Y$,

$$\begin{array}{cc}{e}_X(x,x_1)\odot R(x_1,y_1)\odot e_Y(y_1,y)\hfill & =e_X(x,x_1)\odot e_X(x_1,g(y_1))\odot e_Y(y_1,y)\hfill \\ & \le e_X(x,g(y_1))\odot e_X(y_1,y)\hfill \\ & =e_Y(f(x),y_1)\odot e_Y(y_1,y)\hfill \\ & \le e_Y(f(x),y)=R(x,y).\hfill \end{array}$$

(10)

Hence $e_X\circ R\circ e_Y\le R$.

(3) For all $x,x_1\in X$ and $y,y_1\in Y$,

$$\begin{array}{cc}{e}_X(x,x_1)\odot R_1(x_1,y_1)\odot e_Y(y_1,y)\hfill & =e_X(x,x_1)\odot e_X(x_1,g(y_1))\odot e_Y(y_1,y)\hfill \\ & \le e_X(x,x_1)\odot e_X(x_1,g(y_1))\odot e_X(g(y_1),g(y))\hfill \\ & \le e_X(x,x_1)\odot e_X(x_1,g(y))\hfill \\ & \le e_X(x,g(y))=R(x,y).\hfill \end{array}$$

(11)

Hence $e_X\circ R_1\circ e_Y\le R_1$.

(2) and (4) can be proved similarly. □

Theorem 3.

Let $(e_X,R,S,e_Y)$ be a residuated frame. Let ${\tau }_{e_X}$ and ${\tau }_{e_Y}$ be Alexandrov L-topologies. Then the following hold:

(1)

$(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is a residuated connection where

$$F(A)(y)=\underset{x\in X}{\bigvee }(A(x)\odot R(x,y)),G(B)(x)=\underset{y\in Y}{\bigwedge }(S(y,x)\to B(y));$$

(12)

(2)

$(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is an dual residuated connection where

$$F(A)(y)=\underset{x\in X}{\bigwedge }(R^{\ast }(x,y)\to A(x)),G(B)(x)=\underset{y\in Y}{\bigvee }(R^{\ast }(x,y)\odot B(y)).$$

(13)

Proof. 

(1) Since $R\circ e_Y\le R$ and $e_X\circ R\le R$ by Lemma 3(4), we have $F(A)\in {\tau }_{e_Y}$ and $G(B)\in {\tau }_{e_X}$ from:

$$\begin{array}{cc}F(A)(y)\odot e_Y(y,w)\hfill & =\underset{x\in X}{\bigvee }(A(x)\odot R(x,y)\odot e_Y(y,w))\le \underset{x\in X}{\bigvee }(A(x)\odot R(x,w))=F(A)(w),\hfill \end{array}$$

(14)

and

$$\begin{array}{cc}G(B)(x)\odot e_X(x,z)\odot R(z,y)\hfill & \le \underset{y\in Y}{\bigwedge }((R(x,y)\to B(y))\odot R(x,y))\le B(y)\hfill \\ & \iff G(B)(x)\odot e_X(x,z)\le G(B)(z).\hfill \end{array}$$

(15)

Moreover, for all $A\in {\tau }_{e_X}$ and $B\in {\tau }_{e_Y}$,

$$\begin{array}{cc}{e}_{{\tau }_{e_Y}}(F(A),B)\hfill & =\underset{y\in Y}{\bigwedge }(F(A)(y)\to B(y))=\underset{y\in Y}{\bigwedge }\left(\underset{x\in Y}{\bigvee }(R(x,y)\odot A(x))\to B(y)\right)\hfill \\ & =\underset{x\in X}{\bigwedge }\underset{y\in Y}{\bigwedge }\left(A(x)\to (R(x,y)\to B(y))\right)=\underset{x\in X}{\bigwedge }\left(A(x)\to \underset{x\in X}{\bigwedge }(R(x,y)\to B(y))\right)\hfill \\ & =\underset{x\in X}{\bigwedge }\left(A(x)\to G(B)(x)\right)=e_{{\tau }_{e_X}}(A,G(B)).\hfill \end{array}$$

(16)

(2) Since $R^{\ast }\circ e_Y^{-1}\le R^{\ast }$ and $e_X^{-1}\circ R^{\ast }\le R^{\ast }$ by Lemma 3 (5)–(6), we have

$$\begin{array}{cc}\hfill F(A)(y)\odot e_Y(y,w)\odot R^{\ast }(x,w)& =(\underset{x\in X}{\bigwedge }(R^{\ast }(x,y)\to A(x)))\odot e_Y(y,w)\odot R^{\ast }(x,w)\hfill \\ & \le \underset{x\in X}{\bigwedge }(R^{\ast }(x,y)\to A(x)\odot R^{\ast }(x,y))\le A(x),\hfill \\ \hfill G(B)(x)\odot e_X(x,z)& \le \underset{y\in Y}{\bigvee }((R^{\ast }(x,y)\odot B(y))\odot e_X(x,z))\hfill \\ & \le \underset{y\in Y}{\bigvee }(R^{\ast }(z,y)\odot B(y))=G(B)(z).\hfill \end{array}$$

(17)

Thus $F(A)\in {\tau }_{e_Y}$ and $G(B)\in {\tau }_{e_X}$.

Moreover, for all $A\in {\tau }_{e_X}$ and $B\in {\tau }_{e_Y}$,

$$\begin{array}{cc}{e}_{{\tau }_{e_X}}(G(B),A)\hfill & =\underset{x\in X}{\bigwedge }(G(B)(x)\to A(x))=\underset{x\in X}{\bigwedge }\left(\underset{y\in Y}{\bigvee }(R^{\ast }(x,y)\odot B(y))\to A(x)\right)\hfill \\ & =\underset{x\in X}{\bigwedge }\underset{y\in Y}{\bigwedge }\left(B(y)\to (R^{\ast }(x,y)\to A(x))\right)=\underset{y\in Y}{\bigwedge }\left(B(y)\to \underset{x\in X}{\bigwedge }(R^{\ast }(x,y)\to A(x))\right)\hfill \\ & =\underset{y\in Y}{\bigwedge }\left(B(y)\to F(A)(y)\right)=e_{{\tau }_{e_Y}}(B,F(A)).\hfill \end{array}$$

(18)

□

Remark 2.

Since$({\top }_{{▵}_X},e_X,e_X^{-1},{\top }_{{▵}_X})$is a residuated frame where$e_X$is a fuzzy poset and${\tau }_{{\top }_{{▵}_X}}=L^X$by Remark 1(1),$(e_{L^X},F,G,e_{L^X})$is a residuated connection where

$$F(A)(y)=\underset{x\in X}{\bigvee }(A(x)\odot e_X(x,y)),G(B)(x)=\underset{y\in X}{\bigwedge }(e_X(x,y)\to B(y)).$$

(19)

The pair $(G,F)$ is a fuzzy rough set ([26]).

Theorem 4.

Let $(e_X,R,S,e_Y)$ be a dual residuated frame. Let ${\tau }_{e_X}$ and ${\tau }_{e_Y}$ be Alexandrov L-topologies. Then the following hold:

(1) $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is a dual residuated connection where

$$F(A)(y)=\underset{x\in X}{\bigwedge }(R(x,y)\to A(x)),G(B)(x)=\underset{y\in Y}{\bigvee }(R(x,y)\odot B(y));$$

(20)

(2) $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is a residuated connection where

$$F(A)(y)=\underset{x\in X}{\bigvee }(A(x)\odot R^{\ast }(x,y)),G(B)(x)=\underset{y\in Y}{\bigwedge }(R^{\ast }(x,y)\to B(y)).$$

(21)

Proof. 

(1) Since $R\circ e_Y^{-1}\le R$ and $e_X^{-1}\circ R\le R$ by Lemma 3(5), we have

$$\begin{array}{cc}F(A)(y)\odot e_Y(y,w)\odot R(x,w)\hfill & =(\underset{x\in X}{\bigwedge }(R(x,y)\to A(x)))\odot e_Y^{-1}(w,y)\odot R(x,w)\hfill \\ & \le \underset{x\in X}{\bigwedge }(R(x,y)\to A(x)\odot R(x,y))\le A(x),\hfill \\ \hfill G(B)(x)\odot e_X(x,z)& \le \underset{y\in Y}{\bigvee }(R(x,y)\odot B(y)\odot \odot e_X(x,z))\le G(B)(z).\hfill \end{array}$$

(22)

Moreover, for all $A\in {\tau }_{e_X}$ and $B\in {\tau }_{e_Y}$,

$$\begin{array}{cc}{e}_{{\tau }_{e_X}}(G(B),A)\hfill & =\underset{x\in X}{\bigwedge }(G(B)(x)\to A(x))=\underset{x\in X}{\bigwedge }\left(\underset{y\in Y}{\bigvee }(R(x,y)\odot B(y))\to A(x)\right)\hfill \\ & =\underset{x\in X}{\bigwedge }\underset{y\in Y}{\bigwedge }\left(B(y)\to (R(x,y)\to A(x))\right)=\underset{y\in Y}{\bigwedge }\left(B(y)\to \underset{x\in X}{\bigwedge }(R(x,y)\to A(x))\right)\hfill \\ & =\underset{y\in Y}{\bigwedge }\left(B(y)\to F(A)(y)\right)=e_{{\tau }_{e_Y}}(B,F(A)).\hfill \end{array}$$

(23)

Thus $F(A)\in {\tau }_{e_Y}$ and $G(B)\in {\tau }_{e_X}$.

(2) Since $R^{\ast }\circ e_Y\le R^{\ast }$ and $e_X\circ R^{\ast }\le R^{\ast }$ by Lemma 3(2–3), we have

$$\begin{array}{cc}\hfill F(A)(y)\odot e_Y(y,w)& =\underset{x\in X}{\bigvee }(A(x)\odot R^{\ast }(x,y)\odot e_Y(y,w))\hfill \\ & \le \underset{x\in X}{\bigvee }(A(x)\odot R^{\ast }(x,w))=F(A)(w),\hfill \\ \hfill G(B)(x)\odot e_X(x,z)\odot R^{\ast }(z,y)& \le \underset{y\in Y}{\bigwedge }((R^{\ast }(x,y)\to B(y))\odot R^{\ast }(x,y))\le B(y).\hfill \end{array}$$

(24)

Thus $F(A)\in {\tau }_{e_Y}$ and $G(B)\in {\tau }_{e_X}$.

Moreover, for all $A\in {\tau }_{e_X}$, and $B\in {\tau }_{e_Y}$,

$$\begin{array}{cc}{e}_{{\tau }_{e_Y}}(F(A),B)\hfill & =\underset{y\in Y}{\bigwedge }(F(A)(y)\to B(y))=\underset{y\in Y}{\bigwedge }\left(\underset{x\in Y}{\bigvee }(R^{\ast }(x,y)\odot A(x))\to B(y)\right)\hfill \\ & =\underset{x\in X}{\bigwedge }\underset{y\in Y}{\bigwedge }\left(A(x)\to (R^{\ast }(x,y)\to B(y))\right)=\underset{x\in X}{\bigwedge }\left(A(x)\to \underset{x\in X}{\bigwedge }(R^{\ast }(x,y)\to B(y))\right)\hfill \\ & =\underset{x\in X}{\bigwedge }\left(A(x)\to G(B)(x)\right)=e_{{\tau }_{e_X}}(A,G(B)).\hfill \end{array}$$

(25)

□

Remark 3.

Since$({\top }_{{▵}_X},e_X,e_X^{-1},{\top }_{{▵}_X})$is a dual residuated frame where$e_X$is a fuzzy poset and${\tau }_{{\top }_{{▵}_X}}=L^X$by Remark 1(1),$(e_{L^X},F,G,e_{L^X})$is a dual residuated connection where

$$F(A)(y)=\underset{x\in X}{\bigwedge }(e_X(x,y)\to A(x)),G(B)(x)=\underset{y\in X}{\bigvee }(e_X(x,y)\odot B(y)).$$

(26)

Example 1.

Let$(X,e_X)$and$(Y,e_Y)$be fuzzy posets. Let$f:X\to Y$ and $g:Y\to X$ be maps. Let ${\tau }_{e_X}$ and ${\tau }_{e_Y}$ be Alexandrov L-topologies.

(1) Let g be isotone and $R(x,y)=e_X(x,g(y))$. By Lemma 4(3), $(e_X,R,S=R^{-1},e_Y)$ is a residuated frame. By Theorem 3(1), $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is a residuated connection with

$$F(A)(y)=\underset{x\in X}{\bigvee }(A(x)\odot e_X(x,g(y))),G(B)(x)=\underset{y\in Y}{\bigwedge }(e_X(x,g(y))\to B(y)).$$

(27)

(2) Let g be isotone and $R(x,y)=e_X(g(y),x)$. By Lemma 4(3), $(e_X,R,S=R^{-1},e_Y)$ is a dual residuated frame. By Theorem 4(1), $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is a dual residuated connection where

$$F(A)(y)=\underset{y\in Y}{\bigwedge }(e_X(g(y),x)\to A(x)),G(B)(x)=\underset{y\in Y}{\bigvee }(B(y)\odot e_X(g(y),x)).$$

(28)

(3) Let f be isotone and $R(x,y)=e_Y(y,f(x))$. By Lemma 4(4), $(e_X,R,S=R^{-1},e_Y)$ is a dual residuated frame. By Theorem 4(1), $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_X}})$ is a dual residuated connection where

$$F(A)(y)=\underset{x\in X}{\bigwedge }(e_Y(y,f(x))\to A(x)),G(B)(y)=\underset{y\in Y}{\bigvee }(B(y)\odot e_Y(y,f(x))).$$

(29)

(4) Let f be isotone and $R(x,y)=e_Y(f(x),y)$. By Lemma 4(4), $(e_X,R,S=R^{-1},e_Y)$ is a residuated frame. By Theorem 3(1), $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is a residuated connection where

$$F(A)(y)=\underset{x\in X}{\bigvee }(e_Y(f(x),y)\odot A(x)),G(B)(y)=\underset{y\in Y}{\bigwedge }(e_Y(f(x),y)\to B(y)).$$

(30)

Theorem 5.

Let $(X,e_X)$ and $(Y,e_Y)$ be fuzzy posets. Let ${\tau }_{e_X}$ and ${\tau }_{e_Y}$ be Alexandrov L-topologies. Then the following hold:

(1) $(e_X,f,g,e_Y)$ is a residuated connection. That is, $e_Y(f(x),y)=e_X(x,g(y))$ for all $x,y\in X$ if and only if there exist relations $R:{\tau }_{e_X}\times {\tau }_{e_Y}\to L$ and $S:{\tau }_{e_Y}\times {\tau }_{e_X}\to L$ by

$$R(A,B)=\underset{x\in X}{\bigwedge }(A(x)\to B(f(x))),S(B,A)=\underset{y\in Y}{\bigwedge }(A(g(y))\to B(y))$$

(31)

with isotone maps $f:X\to Y$, $g:Y\to X$ such that $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_Y}})$ is a residuated frame.

(2) In $(1)$,

$$R(A,B)=e_{{\tau }_{e_X}}(A,f^{\leftarrow }(B))=e_{{\tau }_{e_Y}}(F(A),B)=e_{{\tau }_{e_X}}(A,G(B))$$

(32)

where $F(A)(y)={\bigvee }_{z\in X}(e_Y(f(z),y)\odot A(z))$ and $G(B)={\bigwedge }_{y\in Y}(e_Y(f(z),y)\to B(y)))$.

$$S(B,A)=e_{{\tau }_{e_Y}}(g^{\leftarrow }(A),B)=e_{{\tau }_{e_Y}}(F_1(A),B)=e_{{\tau }_{e_X}}(A,G_1(B))$$

(33)

where $F_1(A)(w)={\bigvee }_{z\in X}(e_Y(z,g(w))\odot A(z))$ and $G_1(B)(z)={\bigwedge }_{w\in Y}(e_Y(z,g(w))\to B(w)).$

Proof. 

(1) $(\Rightarrow )$ Let $A\in {\tau }_{e_X}$ and $B\in {\tau }_{e_Y}$. Since $B(f(g(y)))\odot e_Y(f(g(y)),y)\le B(y)$, $e_Y(f(g(y)),y)=\top$, $A(x)\odot e_X(x,g(f(x)))\le A(g(f(x)))$ and $e_X(x,g(f(x)))=\top$,

$$\begin{array}{cc}R(A,B)\hfill & =\underset{x\in X}{\bigwedge }(A(x)\to B(f(x)))\le \underset{y\in Y}{\bigwedge }(A(g(y))\to B(f(g(y)))\odot e_Y(f(g(y)),y))\hfill \\ & \le \underset{y\in Y}{\bigwedge }(A(g(y))\to B(y))=S(B,A)\hfill \end{array}$$

(34)

and

$$\begin{array}{cc}S(B,A)\hfill & =\underset{x\in X}{\bigwedge }(A(g(y))\to B(y))\le \underset{x\in X}{\bigwedge }(A(g(f(x)))\to B(f(x)))\hfill \\ & \le \underset{y\in X}{\bigwedge }(A(x)\odot e_X(x,g(f(x)))\to B(f(x)))=R(A,B).\hfill \end{array}$$

(35)

Thus we have $R(A,B)=S(B,A)$. For all $A,A_1\in {\tau }_{e_X},B,B_1\in {\tau }_{e_Y}$, we have

$$\begin{array}{cc}\hfill & e_{{\tau }_{e_X}}(A,A_1)\odot R(A_1,B_1)\odot e_{{\tau }_{e_Y}}(B_1,B)\hfill \\ & =e_{{\tau }_{e_X}}(A,A_1)\odot \underset{x\in X}{\bigwedge }(A_1(x)\to B_1(f(x)))\odot \underset{x\in X}{\bigwedge }(B_1(f(x))\to B(f(x)))\hfill \\ & \le \underset{x\in X}{\bigwedge }(A(x)\to B(f(x)))=R(A,B).\hfill \end{array}$$

(36)

Thus $e_{{\tau }_{e_X}}\circ R\circ e_{{\tau }_{e_Y}}\le R$.

$(\Leftarrow )$ Since $e_Y(z,w)\odot e_Y(w,y)\le e_Y(z,y)$ if and only if ${(e_Y)}_y^{-1\ast }(z)\odot e_Y(z,w)\le {(e_Y)}_y^{-1\ast }(w)$, we have ${(e_Y)}_y^{-1\ast }\in {\tau }_{e_Y}$. For all ${(e_X)}_x\in {\tau }_{e_X}$ and ${(e_Y)}_y^{-1\ast }\in {\tau }_{e_Y}$,

$$\begin{array}{cc}R({(e_X)}_x,{(e_Y)}_y^{-1\ast })\hfill & =\underset{z\in X}{\bigwedge }({(e_X)}_x(z)\to {(e_Y)}_y^{-1\ast }(f(z)))\hfill \\ & \le {(e_X)}_x(x)\to {(e_Y)}_y^{-1\ast }(f(x))=e_Y{(f(x),y)}^{\ast }.\hfill \end{array}$$

(37)

Since $e_X(x,z)\odot e_Y(f(z),y)\le e_Y(f(x),f(z))\odot e_Y(f(z),y)\le e_Y(f(x),y)$, we have $e_X(x,z)\to e_Y^{\ast }(f(z),y)\ge e_Y^{\ast }(f(x),y)$. Hence $R({(e_X)}_x,{(e_Y)}_y^{-1\ast })=e_Y^{\ast }(f(x),y)$. Moreover,

$$\begin{array}{cc}S({(e_Y)}_y^{-1\ast },{(e_X)}_x)\hfill & =\underset{z\in X}{\bigwedge }({(e_X)}_x(g(z))\to {(e_Y)}_y^{-1\ast }(z))\hfill \\ & \le {(e_X)}_x(g(y))\to {(e_Y)}_y^{-1\ast }(y)=e_X{(x,g(y))}^{\ast }.\hfill \end{array}$$

(38)

Since $e_X(x,g(z))\odot e_Y(z,y)\le e_X(x,g(z))\odot e_X(g(z),g(y))\le e_X(x,g(y))$, we have $e_X(x,g(z))\to e_Y^{\ast }(z,y)\ge e_X^{\ast }(x,g(y))$. Hence $S({(e_Y)}_y^{-1\ast },{(e_X)}_x)=e_X^{\ast }(x,g(y))$. Now, from

$$R({(e_X)}_x,{(e_Y)}_y^{-1\ast })=e_Y^{\ast }(f(x),y)=S({(e_Y)}_y^{-1\ast },{(e_X)}_x)=e_X^{\ast }(x,g(y)),$$

(39)

we have $e_Y(f(x),y)=e_X(x,g(y))$ for all $x,y\in X$.

(2) Let $A\in {\tau }_{e_X}$ and $B\in {\tau }_{e_Y}$. Since $A={\bigvee }_{z\in X}(A(z)\odot e_X(z,-))$ and $B={\bigwedge }_{y\in Y}(B^{\ast }(y)\to e_Y^{\ast }(-,y))$, we have

$$\begin{array}{cc}R(A,B)\hfill & =\underset{x\in X}{\bigwedge }(A(x)\to B(f(x)))=\underset{x\in X}{\bigwedge }(\underset{z\in X}{\bigvee }(A(z)\odot e_X(z,x))\to \underset{y\in Y}{\bigwedge }(B^{\ast }(y)\to e_Y^{\ast }(f(x),y)))\hfill \\ & =\underset{x,z\in X}{\bigwedge }\underset{y\in Y}{\bigvee }(A(z)\odot B^{\ast }(y)\to (e_X(z,x)\to e_Y^{\ast }(f(x),y)))\hfill \\ & =\underset{z\in X}{\bigwedge }\underset{y\in Y}{\bigvee }(A(z)\odot B^{\ast }(y)\to \underset{x\in X}{\bigwedge }(e_X(z,x)\to e_Y^{\ast }(f(x),y)))\hfill \\ & =\underset{z\in X}{\bigwedge }\underset{y\in Y}{\bigvee }(A(z)\odot B^{\ast }(y)\to e_Y^{\ast }(f(z),y))\hfill \\ & =\underset{y\in Y}{\bigwedge }(\underset{z\in X}{\bigvee }(e_Y(f(z),y)\odot A(z))\to B(y))=e_{{\tau }_{e_Y}}(F(A),B)\hfill \\ & =\underset{z\in X}{\bigwedge }(A(z)\to \underset{y\in Y}{\bigwedge }(e_Y(f(z),y)\to B(y)))=e_{{\tau }_{e_X}}(A,G(B))\hfill \end{array}$$

(40)

and

$$\begin{array}{cc}S(B,A)\hfill & =\underset{y\in Y}{\bigwedge }(A(g(y))\to B(y))=\underset{y\in Y}{\bigwedge }(\underset{z\in X}{\bigvee }(A(z)\odot e_X(z,g(y)))\to \underset{w\in Y}{\bigwedge }(B^{\ast }(w)\to e_Y^{\ast }(y,w)))\hfill \\ & =\underset{y,w\in Y}{\bigwedge }\underset{z\in X}{\bigvee }(A(z)\odot B^{\ast }(w)\to (e_X(z,g(y))\to e_Y^{\ast }(y,w)))\hfill \\ & =\underset{w\in Y}{\bigwedge }\underset{z\in X}{\bigvee }(A(z)\odot B^{\ast }(w)\to \underset{y\in Y}{\bigwedge }(e_X(z,g(y))\to e_Y^{\ast }(y,w)))\hfill \\ & =\underset{w\in Y}{\bigwedge }\underset{z\in X}{\bigvee }(A(z)\odot B^{\ast }(w)\to e_Y^{\ast }(z,g(w)))\hfill \\ & =\underset{w\in Y}{\bigwedge }(\underset{z\in X}{\bigvee }(e_Y(z,g(w))\odot A(z))\to B(w))=e_{{\tau }_{e_Y}}(F_1(A),B)\hfill \\ & =\underset{z\in X}{\bigwedge }(A(z)\to \underset{w\in Y}{\bigwedge }(e_Y(z,g(w))\to B(w)))=e_{{\tau }_{e_X}}(A,G_1(B)).\hfill \end{array}$$

(41)

□

Example 2.

Let $(L^X,F,G,L^Y)$ be a residuated connection where for $R\in L^{X\times Y}$,

$$F(A)(y)=\underset{x\in X}{\bigvee }(R(x,y)\odot A(x)),G(B)(x)=\underset{y\in Y}{\bigwedge }(R(x,y)\to B(y)).$$

(42)

Let ${\tau }_{e_{L^X}}=\{\alpha \in L^{L^X}|\alpha (A)\odot e_{L^X}(A,B)\le \alpha (B)\}$ and ${\tau }_{e_{L^Y}}=\{\beta \in L^{L^Y}|\beta (A)\odot e_{L^Y}(A,B)\le \beta (B)\}$. Define two maps $T_1,S_1^{-1}:{\tau }_{e_{L^X}}\times {\tau }_{e_{L^Y}}\to L$ by

$$T_1(\alpha ,\beta )=\underset{A\in L^X}{\bigwedge }(\alpha (A)\to \beta (F(A))),S_1(\beta ,\alpha )=\underset{B\in L^X}{\bigwedge }(\alpha (G(B))\to \beta (B)).$$

(43)

Then $(e_{{\tau }_{e_{L^X}}},T_1,S_1,e_{{\tau }_{e_{L^Y}}})$ is a residuated frame.

Theorem 6.

Let $(X,e_X)$ be a fuzzy poset. Let ${\tau }_{e_X}$ be an Alexandrov L-topology. Let ${\tau }_{e_{{\tau }_{e_X}}}=\{\alpha \in L^{{\tau }_{e_X}}|\alpha (A)\odot e_{{\tau }_{e_X}}(A,B)\le \alpha (B)\}$. Define a map $h:X\to {\tau }_{e_{{\tau }_{e_X}}}$ by $h(x)(A)=\widehat{x}(A)=A(x).$ Then $h:(X,e_X)\to ({\tau }_{e_{{\tau }_{e_X}}},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is an embedding map.

Proof. 

Assume that $h(x)(A)=h(y)(A)$ for all $A\in {\tau }_{e_X}$. Then $h(x)({(e_X)}_x)=h(y)({(e_X)}_x)=e_X(x,y)=\top$ for ${(e_X)}_x\in {\tau }_{e_X}$, and $h(x)({(e_X)}_y)=h(y)({(e_X)}_y)=e_X(y,x)=\top$ for ${(e_X)}_y\in {\tau }_{e_X}$. Thus $x=y$. Hence h is injective.

Since

$$\begin{array}{cc}\hfill \widehat{x}(A)\odot e_{{\tau }_{e_X}}(A,B)& =\widehat{x}(A)\odot \underset{y\in X}{\bigwedge }(A(y)\to B(y))\le A(x)\odot (A(x)\to B(x))\le B(x)=\widehat{x}(B),\hfill \end{array}$$

(44)

we have $h(x)=\widehat{x}\in {\tau }_{e_{{\tau }_{e_X}}}$. Let $A\in {\tau }_{e_X}$. Since $A(x)={\bigwedge }_{y\in Y}(e_X(x,y)\to A(y))$, we have

$$\begin{array}{c}\hfill e_X(x,y)\le \underset{A\in {\tau }_{e_X}}{\bigwedge }(A(x)\to A(y))=\underset{A\in {\tau }_{e_X}}{\bigwedge }(\widehat{x}(A)\to \widehat{y}(A))=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},\widehat{y}).\end{array}$$

(45)

Let ${(e_X)}_z(x)=e_X(z,x)$. Since ${(e_X)}_z(x)\odot e_X(x,y)\le {(e_X)}_z(y)$, we have ${(e_X)}_z\in {\tau }_{e_X}$ for all $z\in X$. Note that

$$\begin{array}{cc}{e}_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},\widehat{y})\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(A(x)\to A(y))\le \underset{{(e_X)}_z\in {\tau }_{e_X}}{\bigwedge }({(e_X)}_z(x)\to {(e_X)}_z(y))\hfill \\ & =\underset{z\in X}{\bigwedge }(e_X(z,x)\to e_X(z,y))=e_X(x,y).\hfill \end{array}$$

(46)

Hence $e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},\widehat{y})=e_X(x,y).$ □

Definition 6.

Let$(e_X,f,g,e_X)$ and $(e_Z,\tilde{f},\tilde{g},e_Z)$ be residuated connections. An injective function $k:(e_X,f,g,e_X)\to (e_Z,\tilde{f},\tilde{g},e_Z)$ is an R-R embedding if

$$e_X(x,y)=e_Z(k(x),k(y)),e_X(f(x),y)=e_Z(\tilde{f}(k(x)),k(y)),e_X(x,g(y))=e_Z(k(x),\tilde{g}(k(y))).$$

(47)

If k is a bijective R-R embedding map, then k is called an R-R isomorphism.

Theorem 7.

Let $(e_X,f,g,e_X)$ be a residuated connection, ${\tau }_{e_X}$ be an Alexandrov L-topology and ${\tau }_{e_{{\tau }_{e_X}}}=\{\alpha \in L^{{\tau }_{e_X}}|\alpha (A)\odot e_{{\tau }_{e_X}}(A,B)\le \alpha (B)\}$. Define a map $h:X\to {\tau }_{e_{{\tau }_{e_X}}}$ by $h(x)(A)=\widehat{x}(A)=A(x).$ Then the map $h:(e_X,f,g,e_X)\to (e_{{\tau }_{e_{{\tau }_{e_X}}}},F,G,e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is an R-R embedding map with

$$e_X(x,y)=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},\widehat{y}),F(h(x))(B)=F(\widehat{x})(B)=\widehat{f(x)}(B)$$

(48)

for all $B\in {\tau }_{e_X}$ and $G(h(y))(A)=G(\widehat{y})(A)=\widehat{g(y)}(A)$ for all $A\in {\tau }_{e_X}$ where

$$R(A,B)=\underset{x\in X}{\bigwedge }(A(x)\to B(f(x))),S(B,A)=\underset{y\in X}{\bigwedge }(A(g(y))\to B(y)),$$

(49)

$$F(\widehat{x})(B)=\underset{A\in {\tau }_{e_X}}{\bigvee }(R(A,B)\odot \widehat{x}(A)),G(\widehat{y})(A)=\underset{B\in {\tau }_{e_X}}{\bigwedge }(S(B,A)\to \widehat{y}(B)).$$

(50)

Moreover, $e_{{\tau }_{e_{{\tau }_{e_X}}}}(F(\widehat{x}),\widehat{y})=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},G(\widehat{y})).$

Proof. 

By Theorem 6, $h:(X,e_X)\to ({\tau }_{e_{{\tau }_{e_X}}},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is an embedding map. By Theorem 5(1), $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_X}})$ is a residuated frame where

$$R(A,B)=\underset{x\in X}{\bigwedge }(A(x)\to B(f(x))),S(B,A)=\underset{y\in X}{\bigwedge }(A(g(y))\to B(y)).$$

(51)

By Theorem 3(1), $(e_{{\tau }_{e_{{\tau }_{e_X}}}},F,G,e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a residuated connection where

$$F(\alpha )(B)=\underset{A\in {\tau }_{e_X}}{\bigvee }(R(A,B)\odot \alpha (A))=\underset{A\in {\tau }_{e_X}}{\bigvee }\left(\underset{z\in X}{\bigwedge }(A(z)\to B(f(z)))\odot \alpha (A)\right),$$

(52)

$$G(\alpha )(A)=\underset{B\in {\tau }_{e_X}}{\bigwedge }(S(B,A)\to \alpha (B))=\underset{B\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(A(g(z))\to B(z))\to \alpha (B)\right).$$

(53)

Moreover,

$$\begin{array}{cc}F(\widehat{x})(B)=\underset{A\in {\tau }_{e_X}}{\bigvee }(R(A,B)\odot \widehat{x}(A))\hfill & =\underset{A\in {\tau }_{e_X}}{\bigvee }\left(\underset{z\in X}{\bigwedge }(A(z)\to B(f(z)))\odot A(x)\right)\hfill \\ & \le B(f(x))=\widehat{f(x)}(B).\hfill \end{array}$$

(54)

Since f is isotone and $B\in {\tau }_{e_X}$, we have $B(f(x))\odot e_X(x,y)\le B(f(x))\odot e_X(f(x),f(y))\le B(f(y))$. Hence $f^{\leftarrow }(B)\in {\tau }_{e_X}.$

Let $A=f^{\leftarrow }(B)$. Note that

$$\begin{array}{cc}F(\widehat{x})(B)=\underset{A\in {\tau }_{e_X}}{\bigvee }(R(A,B)\odot \widehat{x}(A))\hfill & \ge \left(\underset{z\in X}{\bigwedge }(f^{\leftarrow }(B)(z)\to B(f(z)))\odot f^{\leftarrow }(B)(x)\right)\hfill \\ & =B(f(x))=\widehat{f(x)}(B).\hfill \end{array}$$

(55)

Hence $F(\widehat{x})=\widehat{f(x)}$. Note that

$$\begin{array}{cc}\hfill G(\widehat{y})(A)& =\underset{B\in {\tau }_{e_X}}{\bigwedge }(S(B,A)\to \widehat{y}(B))=\underset{B\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(A(g(z))\to B(z))\to B(y)\right)\hfill \\ & \ge \underset{B\in {\tau }_{e_X}}{\bigwedge }\left(A(g(y))\to B(y))\to B(y)\right)\ge A(g(y))=\widehat{g(y)}(A).\hfill \end{array}$$

(56)

Since g is isotone, we have $g^{\leftarrow }(A)\in {\tau }_{e_X}.$ Thus $G(\widehat{y})\le \widehat{g(y)}$. Moreover,

$$\begin{array}{cc}\hfill e_{{\tau }_{e_{{\tau }_{e_X}}}}(F(\widehat{x}),\widehat{y})& =e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{f(x)},\widehat{y})=e_X(f(x),y)=e_X(x,g(y))=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},\widehat{g(y)})=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},G(\widehat{y})).\hfill \end{array}$$

(57)

□

Definition 7.

Let $(e_X,R,S,e_X)$ and $(e_Z,\tilde{R},\tilde{S},e_Z)$ be residuated frames. An injective map $k:(e_X,R,S,e_X)\to (e_Z,\tilde{R},\tilde{S},e_Y)$ is an R-R frame embedding if

$$e_X(x,y)=e_Z(k(x),k(y)),R(x,y)=\tilde{R}(k(x),k(y)),S(x,y)=\tilde{S}(k(x),k(y)).$$

(58)

If k is a bijective R-R embedding map, then k is called an R-R frame isomorphism.

Theorem 8.

Let $(e_X,R,S,e_X)$ be a residual frame, ${\tau }_{e_X}$ be an Alexandrov L-topology and ${\tau }_{e_{{\tau }_{e_X}}}=\{\alpha \in L^{{\tau }_{e_X}}|\alpha (A)\odot e_{{\tau }_{e_X}}(A,B)\le \alpha (B)\}$. Define a map $k:X\to {\tau }_{e_{{\tau }_{e_X}}}$ by $k(x)(A)=\widehat{x}(A)=A(x).$ Then the map $k:(e_X,R,S,e_X)\to (e_{{\tau }_{e_{{\tau }_{e_X}}}},\widehat{R},\widehat{S},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is an R-R frame embedding map with $e(x,y)=e_{{\tau }_{e_{{\tau }_{e_X}}}}(k(x),k(y))$, $R(x,y)=\widehat{R}(k(x),k(y))=\widehat{R}(\widehat{x},\widehat{y})$ and $S(x,y)=\widehat{S}(k(x),k(y))=\widehat{S}(\widehat{x},\widehat{y})$ where

$$F(A)(y)=\underset{x\in X}{\bigvee }(R(x,y)\odot A(x)),G(B)(x)=\underset{y\in X}{\bigwedge }(R(x,y)\to B(y)),$$

(59)

$$\widehat{R}(\alpha ,\beta )=\underset{A\in {\tau }_{e_X}}{\bigwedge }(\alpha (A)\to \beta (F(A))),\widehat{S}(\beta ,\alpha )=\underset{B\in {\tau }_{e_X}}{\bigwedge }(\alpha (G(B))\to \beta (B)).$$

(60)

Proof. 

By Theorem 6, $k:(X,e_X)\to ({\tau }_{e_{{\tau }_{e_X}}},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is an embedding map. Hence $e_X(x,y)=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},\widehat{y})$. By Theorem 3(1), $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_X}})$ is a residuated connection where

$$F(A)(y)=\underset{x\in X}{\bigvee }(R(x,y)\odot A(x)),G(B)(x)=\underset{y\in X}{\bigwedge }(R(x,y)\to B(y)).$$

(61)

By Theorem 5(1), $(e_{{\tau }_{e_{{\tau }_{e_X}}}},\widehat{R},\widehat{S},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a residuated frame where

$$\widehat{R}(\alpha ,\beta )=\underset{A\in {\tau }_{e_X}}{\bigwedge }(\alpha (A)\to \beta (F(A))),\widehat{S}(\beta ,\alpha )=\underset{B\in {\tau }_{e_X}}{\bigwedge }(\alpha (G(B))\to \beta (B)).$$

(62)

Note that for all $\widehat{x},\widehat{y}\in {\tau }_{e_{{\tau }_{e_X}}}$,

$$\begin{array}{cc}\widehat{R}(\widehat{x},\widehat{y})\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(\widehat{x}(A)\to \widehat{y}(F(A)))=\underset{A\in {\tau }_{e_X}}{\bigwedge }(A(x)\to F(A)(y))\hfill \\ & =\underset{A\in {\tau }_{e_X}}{\bigwedge }\left(A(x)\to \underset{z\in X}{\bigvee }(R(z,y)\odot A(z))\right)\ge \underset{A\in {\tau }_{e_X}}{\bigwedge }\left(A(x)\to (R(x,y)\odot A(x))\right)\hfill \\ & \ge R(x,y).\hfill \end{array}$$

(63)

Let ${(e_X)}_x(z)=e_X(x,z)$. Then ${(e_X)}_x\in {\tau }_{e_X}$. Since $e_X\circ R\circ e_X\le R$, we have $e_X\circ R\le R$. Thus

$$\begin{array}{cc}\hfill \widehat{R}(\widehat{x},\widehat{y})& =\underset{A\in {\tau }_{e_X}}{\bigwedge }\left(A(x)\to \underset{z\in X}{\bigvee }(R(z,y)\odot A(z))\right)\le \left({(e_X)}_x(x)\to \underset{z\in X}{\bigvee }(R(z,y)\odot {(e_X)}_x(z))\right)\hfill \\ \hfill & =R(x,y)\hfill \end{array}$$

(64)

and

$$\begin{array}{cc}\widehat{S}(\widehat{y},\widehat{x})\hfill & =\underset{B\in {\tau }_{e_X}}{\bigwedge }(\widehat{x}(G(B))\to \widehat{y}(B))=\underset{B\in {\tau }_{e_X}}{\bigwedge }(G(B)(x)\to B(y))\hfill \\ & =\underset{B\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(R(x,z)\to B(z))\to B(y)\right)\ge \underset{B\in {\tau }_{e_X}}{\bigwedge }\left((R(x,y)\to B(y))\to B(y)\right)\hfill \\ & \ge R(x,y)=S(y,x).\hfill \end{array}$$

(65)

Since $R\circ e_X\le e_X\circ R\circ e_X\le R$, we have $R(x,y)\odot e_X(y,w)\le R(x,w)$. Thus $R_x=R(x,-)\in {\tau }_{e_X}$. Hence

$$\begin{array}{cc}\widehat{S}(\widehat{y},\widehat{x})\hfill & =\underset{B\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(R(x,z)\to B(z))\to B(y)\right)\le \left(\underset{z\in X}{\bigwedge }(R(x,z)\to R_x(z))\to R_x(y)\right)\hfill \\ & =R(x,y)=S(y,x).\hfill \end{array}$$

(66)

□

Corollary 1.

Let $(e_X,R=e_X,S=e_X^{-1},e_X)$ be a residual frame and ${\tau }_{e_{{\tau }_{e_X}}}=\{\alpha \in L^{{\tau }_{e_X}}|\alpha (A)\odot e_{{\tau }_{e_X}}(A,B)\le \alpha (B)\}$. Define a map $k:X\to {\tau }_{e_{{\tau }_{e_X}}}$ by $k(x)(A)=\widehat{x}(A)=A(x).$ Then the map

$$k:(e_X,R=e_X,S=e_X^{-1},e_X)\to (e_{{\tau }_{e_{{\tau }_{e_X}}}},\widehat{R}=\widehat{e_X},\widehat{S}=\widehat{e_X^{-1}},e_{{\tau }_{e_{{\tau }_{e_X}}}})$$

(67)

is an embedding map with $e_X(x,y)=e_{{\tau }_{e_{{\tau }_{e_X}}}}(k(x),k(y))$, $e_X(x,y)=\widehat{e_X}(\widehat{x},\widehat{y})$ and $e_X^{-1}(x,y)=\widehat{e_X^{-1}}(\widehat{x},\widehat{y})$ where

$$\begin{array}{cc}\widehat{e_X}(\widehat{x},\widehat{y})\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(\widehat{x}(A)\to \widehat{y}(F(A)))=\underset{A\in {\tau }_{e_X}}{\bigwedge }(A(x)\to \underset{z\in X}{\bigvee }(e_X(z,y)\odot A(z)))=e_X(x,y),\hfill \\ \widehat{e_X^{-1}}(\widehat{y},\widehat{x})\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(\widehat{x}(G(B))\to \widehat{y}(B))=\underset{A\in {\tau }_{e_X}}{\bigwedge }(\underset{z\in X}{\bigwedge }(e_X(x,z)\to B(z))\to B(y))=e_X^{-1}(y,x).\hfill \end{array}$$

(68)

Example 3.

Let $X=\{a,b,c\}$ be a set. Let $f:X\to X$ be a map by $f(a)=b,f(b)=a,f(c)=c$ and $f=f^{-1}$. Define a binary operation ⊙ on $L=[0,1]$ by

$$x\odot y=max\{0,x+y-1\},x\to y=min\{1-x+y,1\}.$$

(69)

(1) Let $(X=\{a,b,c\},e_X)$ be a fuzzy poset where

$$e_X=\left(\begin{array}{ccc}1& 0.6& 0.5\\ 0.6& 1& 0.5\\ 0.7& 0.7& 1\end{array}\right).$$

(70)

Since $e_X(x,y)=e_X(f(x),f(y)),e_X(x,f(f(x)))=e_X(f(f(x)),x)=1$, we have that $(e_X,f,f,e_X)$ are both residuated and dual residuated connections. Since $(e_X,f,f,e_X)$ is a residuated connection, we have that $e_X(f(x),y)=e_X(x,f(y))$ for all $x,y\in X$ if and only if there the exist relations $R:{\tau }_{e_X}\times {\tau }_{e_X}\to L$ and $S:{\tau }_{e_X}\times {\tau }_{e_X}\to L$ by

$$R(A,B)=\underset{x\in X}{\bigwedge }(A(x)\to B(f(x))),S(B,A)=\underset{y\in Y}{\bigwedge }(A(f(y))\to B(y))$$

(71)

with an isotone map $f:X\to Y$ such that $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_X}})$ is a residuated frame.

Let ${(e_X)}_z(x)=e(z,x)$ for all $z\in X$. Then ${(e_X)}_z\in {\tau }_{e_X}$. Now, we have

$$\begin{array}{cc}\hfill R({(e_X)}_a,{(e_X)}_b)& =\underset{x\in X}{\bigwedge }(e_X(a,x)\to e_X(b,f(x)))=1,\hfill \\ \hfill R({(e_X)}_b,{(e_X)}_a)& =1,R({(e_X)}_a,{(e_X)}_a)=R({(e_X)}_b,{(e_X)}_b)=0.6,R({(e_X)}_c,{(e_X)}_c)=1,\hfill \\ \hfill R({(e_X)}_a,{(e_X)}_c)& =0.7,R({(e_X)}_c,{(e_X)}_a)=0.5,R({(e_X)}_b,{(e_X)}_c)=0.7,R({(e_X)}_c,{(e_X)}_b)=0.5.\hfill \\ \hfill S({(e_X)}_x,{(e_X)}_y)& =R({(e_X)}_y,{(e_X)}_x)\mathrm{for}\mathrm{all}x,y\in X.\hfill \end{array}$$

(72)

Moreover,

$$R({(e_X)}_a,{(e_X)}_b^{-1\ast })=\underset{x\in X}{\bigwedge }(e_X(a,x)\to e_X^{\ast }(f(x),b))=e_X^{\ast }(f(a),b).$$

(73)

Since f is isotone and $R(x,y)=e_X(x,f(y))=e_X(f(x),y)$, we have by Example 1(4) that $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is a residuated connection with

$$F(A)(y)=\underset{x\in X}{\bigvee }(A(x)\odot e_X(x,f(y)),G(B)(x)=\underset{y\in X}{\bigwedge }(e_X(x,f(y))\to B(y)).$$

(74)

Since f is isotone and $R(x,y)=e_X(f(y),x)=e_X(y,f(x))$, we have by Example 1(3) that $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is a dual residuated connection with

$$F(A)(y)=\underset{x\in X}{\bigwedge }(e_X(f(y),x)\to A(x)),G(B)(x)=\underset{y\in X}{\bigvee }(B(y)\odot e_X(f(y),x)).$$

(75)

Since $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_X}})$ is a residuated frame, we have by Theorem 7 that $(e_{{\tau }_{e_{{\tau }_{e_X}}}},F,G,e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a residuated connection where

$$\begin{array}{c}F(\alpha )(B)=\underset{A\in {\tau }_{e_X}}{\bigvee }\left(\underset{z\in X}{\bigwedge }(A(z)\to B(f(z)))\odot \alpha (A)\right),G(\alpha )(B)\hfill \\ \hfill =\underset{C\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(B(f(z))\to C(z))\to \alpha (C)\right).\end{array}$$

(76)

Since

$$(A(z)\to B(f(z)))\odot (B(f(z))\to A(z))\odot \alpha (A)\le \alpha (A),$$

(77)

we have

$$(A(z)\to B(f(z)))\odot \alpha (A)\le (B(f(z))\to A(z))\to \alpha (A).$$

(78)

Hence $F(\alpha )(B)\le G(\alpha )(B)$.Since f is isotone, we have that $f^{\leftarrow }(B)\in {\tau }_{e_X}$ for all $B\in {\tau }_{e_X}$, and so

$$\begin{array}{cc}\hfill G(\alpha )(B)\le (B(f(z))\to B(f(z)))\to \alpha (f^{\leftarrow }(B))& \\ \hfill =& (B(f(z))\to B(f(z)))\odot \alpha (f^{\leftarrow }(B))\le F(\alpha )(B).\hfill \end{array}$$

(79)

Hence the map$h:(e_X,f,f,e_X)\to (e_{{\tau }_{e_{{\tau }_{e_X}}}},F,F,e_{{\tau }_{e_{{\tau }_{e_X}}}})$is an R-R embedding map.

(2) Let $(X=\{a,b,c\},e_X)$ be a fuzzy poset where

$$e_X=\left(\begin{array}{ccc}1& 0.6& 0.5\\ 0.6& 1& 0.7\\ 0.7& 0.5& 1\end{array}\right).$$

(80)

Since

$$0.7=e_X(c,a)\nleqq e_X(f(c),f(a))=e_X(c,b)=0.5,$$

f is not an isotone map. Hence $(e_X,f,f,e_X)$ are neither residuated nor dual residuated connections.Let $R(x,y)=e_X(x,f(y))$. Then $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is not a residuated connection with

$$F(A)(y)=\underset{x\in X}{\bigvee }(A(x)\odot e_X(x,f(y)),G(B)(x)=\underset{y\in X}{\bigwedge }(e_X(x,f(y))\to B(y)),$$

(81)

because $F({(e_X)}_c)\notin {\tau }_{e_X}$ for ${(e_X)}_c\in {\tau }_{e_X}$ from $F({(e_X)}_c)(c)\odot e_X(c,a)=0.7\nleqq F({(e_X)}_c)(a)=0.5$ where

$$\begin{array}{cc}F({(e_X)}_c)(c)\hfill & =\underset{x\in X}{\bigvee }({(e_X)}_c(x)\odot e_X(x,f(c))=e_X(c,c)=1,\hfill \\ F({(e_X)}_c)(a)\hfill & =\underset{x\in X}{\bigvee }({(e_X)}_c(x)\odot e_X(x,f(a))=e_X(c,b)=0.5.\hfill \end{array}$$

(82)

Let $R(x,y)=e_X(f(y),x)$. Then $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_Y}})$ is not a dual residuated connection with

$$F(A)(y)=\underset{y\in X}{\bigwedge }(e_X(f(y),x)\to A(x)),G(B)(x)=\underset{y\in X}{\bigvee }(B(y)\odot e_X(f(y),x)),$$

(83)

because$F({(e_X^{-1\ast })}_c)\notin {\tau }_{e_X}$ for ${(e_X^{-1\ast })}_c\in {\tau }_{e_X}$ from $F({(e_X^{-1\ast })}_c)(b)\odot e_X(b,c)=0.2\nleqq F({(e_X^{-1\ast })}_c)(c)=0$ where

$$\begin{array}{cc}F({(e_X^{-1\ast })}_c)(b)\hfill & =\underset{y\in X}{\bigwedge }(e_X(f(b),x)\to {(e_X^{-1\ast })}_c(x))=e_X^{\ast }(f(b),c)=0.5,\hfill \\ F({(e_X^{-1\ast })}_c)(c)\hfill & =\underset{y\in X}{\bigwedge }(e_X(f(c),x)\to {(e_X^{-1\ast })}_c(x))=e_X^{\ast }(f(c),c)=0.\hfill \end{array}$$

(84)

(3) Let $(X=\{a,b,c\},e_X)$ be a fuzzy poset where

$$e_X=\left(\begin{array}{ccc}1& 1& 0.7\\ 0.6& 1& 0.7\\ 0.7& 0.7& 1\end{array}\right).$$

(85)

Let $g,h:X\to X$ be maps by

$$g(a)=g(b)=a,g(c)=c\mathrm{and}h(a)=h(b)=b,h(c)=c.$$

(86)

Since

$$\begin{array}{cc}\hfill e_X(x,y)& \le e_X(g(x),g(y)),e_X(x,y)\le e_X(h(x),h(y)),g(h(a))=g(h(b))=a,\hfill \\ \hfill g(h(c))& =c,h(g(a))=h(g(b))=b,g(h(c))=c,\hfill \end{array}$$

(87)

we have

$$e_X(g(h(x)),x)=e_X(x,h(g(x)))=1,e_X(h(g(a)),a)=e_X(b,g(h(b)))=0.6.$$

(88)

Hence $(e_X,g,h,e_X)$ is a residuated connection, but not a dual residuated connection.Since $(e_X,g,h,e_X)$ is a residuated connection, we have by Theorem 5 that $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_X}})$ is a residuated frame where

$$\begin{array}{cc}\hfill R(A,B)& =\underset{x\in X}{\bigwedge }(A(x)\to B(g(x))),S(B,A)=\underset{y\in Y}{\bigwedge }(A(h(y))\to B(y)).\hfill \end{array}$$

(89)

Since $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_X}})$ is a residuated frame, we have by Theorem 7 that $(e_{{\tau }_{e_{{\tau }_{e_X}}}},F,G,e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a residuated connection where

$$F(\alpha )(B)=\underset{A\in {\tau }_{e_X}}{\bigvee }(R(A,B)\odot \alpha (A))=\underset{A\in {\tau }_{e_X}}{\bigvee }\left(\underset{z\in X}{\bigwedge }(A(z)\to B(g(z)))\odot \alpha (A)\right),$$

(90)

$$G(\alpha )(A)=\underset{B\in {\tau }_{e_X}}{\bigwedge }(S(B,A)\to \alpha (B))=\underset{B\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(A(h(z))\to B(z))\to \alpha (B)\right).$$

(91)

4. Fuzzy Dual Residuated Connections on Alexandrov $\mathit{L}$-Topologies

Theorem 9.

Let $(X,e_X)$ and $(Y,e_Y)$ be fuzzy posets. Let ${\tau }_{e_X}$ and ${\tau }_{e_Y}$ be Alexandrov L-topologies. Then the following hold:

(1) $(e_X,f,g,e_Y)$ is a dual residuated connection. That is, $e_Y(y,f(x))=e_X(g(y),x)$ for all $x,y\in X$ if and only if there exist maps $R:{\tau }_{e_X}\times {\tau }_{e_Y}\to L$ and $S:{\tau }_{e_Y}\times {\tau }_{e_X}\to L$ by

$$\begin{array}{cc}\hfill R(A,B)& =\underset{x\in X}{\bigwedge }(B(f(x))\to A(x)),S(B,A)=\underset{y\in Y}{\bigwedge }(B(y)\to A(g(y)))\hfill \end{array}$$

(92)

with isotone maps $f:X\to Y$, $g:Y\to X$ such that $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_X}})$ is a dual residuated frame.

(2) In $(1)$,

$$R(A,B)=e_{{\tau }_{e_X}}(f^{\leftarrow }(B),A)=e_{{\tau }_{e_Y}}(B,F(A))=e_{{\tau }_{e_X}}(G(B),A)$$

(93)

where $F(A)(y)={\bigwedge }_{z\in X}(e_Y(y,f(z))\to A(z))$ and $G(B)={\bigvee }_{y\in Y}(e_Y(y,f(z))\odot B(y))$.

$$S(B,A)=e_{{\tau }_{e_Y}}(B,g^{\leftarrow }(A))=e_{{\tau }_{e_Y}}(B,F_1(A))=e_{{\tau }_{e_X}}(G_1(B),A)$$

(94)

where $F_1(A)(w)={\bigwedge }_{z\in X}(e_Y(g(w),z)\to A(z))$ and $G_1(B)(z)={\bigvee }_{w\in Y}(e_Y(g(w),z)\odot B(w)).$

Proof. 

(1) $(\Rightarrow )$ Let $A\in {\tau }_{e_X}$. Since $A(g(f(x)))\odot e_X(g(f(x)),x)\le A(x)$ and $B(y)\odot e_Y(y,f(g(y)))\le B(f(g(y)))$ and $e_X(g(f(x)),x))=e_Y(y,f(g(y)))=\top$ by Theorem 2, we have

$$\begin{array}{cc}S(B,A)\hfill & =\underset{y\in X}{\bigwedge }(B(y)\to A(g(y)))\le \underset{x\in X}{\bigwedge }(B(f(x))\to A(g(f(x)))\odot e_X(g(f(x)),x))\hfill \\ & \le \underset{x\in X}{\bigwedge }(B(f(x))\to A(x))=R(A,B)\hfill \end{array}$$

(95)

and

$$\begin{array}{cc}R(A,B)\hfill & =\underset{x\in X}{\bigwedge }(B(f(x))\to A(x))\le \underset{y\in X}{\bigwedge }(B(f(g(y)))\to A(g(y)))\hfill \\ & \le \underset{y\in X}{\bigwedge }(B(y)\odot e_Y(y,f(g(y)))\to A(g(y)))\hfill \\ & =S(B,A).\hfill \end{array}$$

(96)

Thus $S=R^{-1}$. For all $A,A_1\in {\tau }_{e_X}$ and $B,B_1\in {\tau }_{e_Y}$, we have

$$\begin{array}{c}{e}_{{\tau }_{e_X}}^{-1}(A,A_1)\odot R(A_1,B_1)\odot e_{{\tau }_{e_Y}}^{-1}(B_1,B)\hfill \\ \le e_{{\tau }_{e_X}}(A_1,A)\odot \underset{x\in X}{\bigwedge }(B_1(f(x))\to A_1(x))\odot \underset{x\in X}{\bigwedge }(B(f(x))\to B_1(f(x)))\hfill \\ \le \underset{x\in X}{\bigwedge }(B(f(x))\to A(x))=R(A,B).\hfill \end{array}$$

(97)

$(\Leftarrow )$ For all ${(e_X)}_x^{-1\ast }\in {\tau }_{e_X}$ and ${(e_Y)}_y\in {\tau }_{e_Y}$, we have

$$\begin{array}{cc}R({(e_X)}_x^{-1\ast },{(e_Y)}_y)\hfill & =\underset{z\in X}{\bigwedge }({(e_Y)}_y(f(z))\to {(e_X)}_x^{-1\ast }(z))\le {(e_Y)}_y(f(x))\to {(e_X)}_x^{-1\ast }(x)\hfill \\ & =e_Y{(y,f(x))}^{\ast }.\hfill \end{array}$$

(98)

Since

$$e_Y(y,f(z))\odot e_X(z,x)\le e_Y(y,f(z))\odot e_Y(f(z),f(x))\le e_Y(y,f(x)),$$

(99)

we have $e_X(x,z)\to e_Y^{\ast }(y,f(z))\ge e_Y^{\ast }(y,f(x))$. Hence $R({(e_X)}_x^{-1\ast },{(e_Y)}_y)=e_Y^{\ast }(y,f(x))$. Additionally,

$$\begin{array}{cc}S({(e_Y)}_y,{(e_X)}_x^{-1\ast })\hfill & =\underset{z\in X}{\bigwedge }({(e_Y)}_y(z)\to {(e_X)}_x^{-1\ast }(g(z))\hfill \\ & \le {(e_Y)}_y(y)\to {(e_Y)}_x^{-1\ast }(g(y))=e_X{(g(y),x)}^{\ast }.\hfill \end{array}$$

(100)

Since

$$e_X(g(z),x)\odot e_Y(y,z)\le e_X(g(z),x)\odot e_X(g(y),g(z))\le e_X(g(y),x),$$

(101)

we have $e_Y(y,z)\to e_X^{\ast }(g(z),x)\ge e_X^{\ast }(g(y),x)$. Hence $S({(e_Y)}_y,{(e_X)}_x^{-1\ast })=e_X^{\ast }(g(y),x)$. Since

$$e_Y^{\ast }(y,f(x))=R({(e_X)}_x^{-1\ast },{(e_Y)}_y)=S({(e_Y)}_y,{(e_X)}_x^{-1\ast })=e_X^{\ast }(g(y),x),$$

(102)

we have that $(e_X,f,g,e_Y)$ is a dual residuated connection. □

Example 4.

Let $(e_{L^X},F,G,e_{L^Y})$ be a dual residuated connection for $R\in L^{X\times Y}$ defined by

$$F(A)(y)=\underset{x\in X}{\bigwedge }(R(x,y)\to A(x)),G(B)(x)=\underset{y\in Y}{\bigvee }(R(x,y)\odot B(y)),$$

(103)

and${\tau }_{e_{L^X}}=\{\alpha \in L^{L^X}|\alpha (A)\odot e_{L^X}(A,B)\le \alpha (B)\}$and${\tau }_{e_{L^Y}}=\{\beta \in L^{L^Y}|\beta (A)\odot e_{L^Y}(A,B)\le \beta (B)\}$. Two maps $T_1,S_1:{\tau }_{e_{L^X}}\times {\tau }_{e_{L^Y}}\to L$ are defined by

$$T_1(\alpha ,\beta )=\underset{A\in L^X}{\bigwedge }(\beta (F(A))\to \alpha (A)),S_1(\beta ,\alpha )=\underset{B\in L^X}{\bigwedge }(\beta (B)\to \alpha (G(B))).$$

(104)

Then $(e_{{\tau }_{e_{L^X}}},T_1,S_1,e_{{\tau }_{e_{L^Y}}})$ is a dual residuated frame.

Definition 8.

Let $(e_X,f,g,e_X)$ and $(e_Z,\tilde{f},\tilde{g},e_Z)$ be dual residuated connections. An injective function $k:(e_X,f,g,e_X)\to (e_Z,\tilde{f},\tilde{g},e_Z)$ is a DR-DR embedding if

$$e_X(x,y)=e_Z(k(x),k(y)),e_X(y,f(x))=e_Z(k(y),\tilde{f}(k(x))),e_X(g(y),x)=e_Z(\tilde{g}(k(y)),k(x)).$$

(105)

If k is a bijective DR-DR embedding map, then k is called a DR-DR isomorphism.

Theorem 10.

Let $(e_X,f,g,e_X)$ be a dual residuated connection, ${\tau }_{e_X}$ be an Alexandrov L-topology and ${\tau }_{e_{{\tau }_{e_X}}}=\{\alpha \in L^{{\tau }_{e_X}}|\alpha (A)\odot e_{{\tau }_{e_X}}(A,B)\le \alpha (B)\}$. Define a map $h:X\to {\tau }_{e_{{\tau }_{e_X}}}$ by $h(x)(A)=\widehat{x}(A)=A(x).$ Then $h:(e_X,f,g,e_X)\to (e_{{\tau }_{e_{{\tau }_{e_X}}}},F,G,e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a DR-DR embedding map with $e_X(x,y)=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},\widehat{y})$, $F(h(x))(B)=F(\widehat{x})(B)=\widehat{f(x)}(B)$ and $G(h(y))(A)=G(\widehat{y})(A)=\widehat{g(y)}(A)$ for all $A\in {\tau }_{e_X}$ where

$$\begin{array}{cc}R(A,B)\hfill & =\underset{x\in X}{\bigwedge }(B(f(x))\to A(x)),S(B,A)=\underset{y\in X}{\bigwedge }(B(y)\to A(g(y))),\hfill \\ F(\alpha )(B)\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(R(A,B)\to \alpha (A)),G(\alpha )(A)=\underset{B\in {\tau }_{e_X}}{\bigvee }(S(B,A)\odot \alpha (B)).\hfill \end{array}$$

(106)

Moreover, $e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{y},F(\widehat{x}))=e_{{\tau }_{e_{{\tau }_{e_X}}}}(G(\widehat{y}),\widehat{x})$.

Proof. 

By Theorem 9, $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_X}})$ is a dual residuated frame where

$$\begin{array}{cc}\hfill R(A,B)& =\underset{x\in X}{\bigwedge }(B(f(x))\to A(x)),S(B,A)=\underset{y\in X}{\bigwedge }(B(y)\to A(g(y))).\hfill \end{array}$$

(107)

By Theorem 4(1), $(e_{{\tau }_{e_{{\tau }_{e_X}}}},F,G,e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a dual residuated connection where

$$\begin{array}{cc}F(\alpha )(B)\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(R(A,B)\to \alpha (A))=\underset{A\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(B(f(z))\to A(z))\to \alpha (A)\right),\hfill \\ G(\alpha )(A)\hfill & =\underset{B\in {\tau }_{e_X}}{\bigvee }(S(B,A)\odot \alpha (B))=\underset{B\in {\tau }_{e_X}}{\bigvee }\left(\underset{z\in X}{\bigwedge }(B(z)\to A(g(z)))\odot \alpha (B)\right).\hfill \end{array}$$

(108)

By Theorem 6, a map $h:X\to {\tau }_{e_{{\tau }_{e_X}}}$ by $h(x)(A)=\widehat{x}(A)=A(x)$ is embedding. That is, $e_X(x,y)=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},\widehat{y})$. For all $B\in {\tau }_{e_X}$, we have

$$\begin{array}{cc}F(\widehat{x})(B)\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(R(A,B)\to \widehat{x}(A))=\underset{A\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(B(f(z))\to A(z))\to A(x)\right)\hfill \\ & \ge \underset{A\in {\tau }_{e_X}}{\bigwedge }\left((B(f(x))\to A(x))\to A(x)\right)\ge B(f(x))=\widehat{f(x)}(B).\hfill \end{array}$$

(109)

Since f is isotone and $B\in {\tau }_{e_X}$, we have

$$B(f(x))\odot e_X(x,y)\le B(f(x))\odot e_X(f(x),f(y))\le B(f(y)).$$

(110)

Hence $f^{\leftarrow }(B)\in {\tau }_{e_X}$.

Let $A=f^{\leftarrow }(B)$. For all $A,B\in {\tau }_{e_X}$,

$$\begin{array}{cc}F(\widehat{x})(B)\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(B(f(z))\to A(z))\to A(x)\right)\le \underset{z\in X}{\bigwedge }(B(f(z))\to B(f(z))\to B(f(x)))\hfill \\ & =\top \to B(f(x))=B(f(x))=\widehat{f(x)}(B)\hfill \end{array}$$

(111)

and

$$\begin{array}{cc}G(\widehat{y})(A)\hfill & =\underset{B\in {\tau }_{e_X}}{\bigvee }(S(B,A)\odot \widehat{y}(B))=\underset{B\in {\tau }_{e_X}}{\bigvee }\left(\underset{z\in X}{\bigwedge }(B(z)\to A(g(z)))\odot B(y)\right)\hfill \\ & \le \underset{B\in {\tau }_{e_X}}{\bigwedge }\left(B(y)\to A(g(y)))\odot B(y)\right)\le A(g(y))=\widehat{g(y)}(A).\hfill \end{array}$$

(112)

Let $B(y)=g^{\leftarrow }(A)(y)=A(g(y))$ for all $y\in X$. Since

$$g^{\leftarrow }(A)(y)\odot e_Y(y,w)\le A(g(y))\odot e_X(g(y),g(w))\le A(g(w)),$$

(113)

we have $g^{\leftarrow }(A)\in {\tau }_{e_X}$. Moreover,

$$\begin{array}{cc}G(\widehat{y})(B)=\underset{A\in {\tau }_{e_X}}{\bigvee }\left(\underset{z\in X}{\bigwedge }(B(z)\to A(g(z)))\odot B(y)\right)\hfill & \ge \underset{z\in X}{\bigwedge }(A(g(z))\to A(g(z)))\odot A(g(y)))\hfill \\ & =\top \odot A(g(y))=A(g(y))=\widehat{g(y)}(A).\hfill \end{array}$$

(114)

Moreover,

$$\begin{array}{cc}\hfill e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{y},F(\widehat{x}))& =e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{y},\widehat{f(x)})=e_X(y,f(x))=e_X(g(y),x)=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{g(y)},\widehat{x})=e_{{\tau }_{e_{{\tau }_{e_X}}}}(G(\widehat{y}),\widehat{x}).\hfill \end{array}$$

(115)

□

Definition 9.

Let $(e_X,R,S,e_X)$ and $(e_Z,\tilde{R},\tilde{S},e_Z)$ be dual residuated frames. An injective map $k:(e_X,R,S,e_X)\to (e_Z,\tilde{R},\tilde{S},e_Z)$ is a DR-DR frame embedding if

$$e_X(x,y)=e_Z(k(x),k(y)),R(x,y)=\tilde{R}(k(x),k(y)),S(x,y)=\tilde{S}(k(x),k(y)).$$

(116)

If k is a bijective DR-DR frame embedding map, then k is called a DR-DR frame isomorphism.

Theorem 11.

Let $(e_X,R,S,e_X)$ be a dual residual frame, ${\tau }_{e_X}$ be an Alexandrov L-topology and ${\tau }_{e_{{\tau }_{e_X}}}=\{\alpha \in L^{{\tau }_{e_X}}|\alpha (A)\odot e_{{\tau }_{e_X}}(A,B)\le \alpha (B)\}$. Define a map $k:X\to {\tau }_{e_{{\tau }_{e_X}}}$ by $k(x)(A)=\widehat{x}(A)=A(x).$ Then the map $k:(e_X,R,S,e_X)\to (e_{{\tau }_{e_{{\tau }_{e_X}}}},\widehat{R},\widehat{S},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a DR-DR frame embedding map with $e_X(x,y)=e_{{\tau }_{e_{{\tau }_{e_X}}}}(k(x),k(y))$, $R(x,y)=\widehat{R}(\widehat{x},\widehat{y})$ and $S(x,y)=\widehat{S}(\widehat{x},\widehat{y})$ where

$$\begin{array}{cc}F(A)(y)\hfill & =\underset{x\in X}{\bigwedge }(R(x,y)\to A(x)),G(B)(x)=\underset{x\in X}{\bigvee }(S(y,x)\odot B(y)),\hfill \\ \widehat{R}(\alpha ,\beta )\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(\beta (F(A))\to \alpha (A)),\widehat{S}(\beta ,\alpha )=\underset{B\in {\tau }_{e_X}}{\bigwedge }(\beta (B)\to \alpha (G(B))).\hfill \end{array}$$

(117)

Proof. 

By Theorem 4(1), $({\tau }_{e_X},F,G,{\tau }_{e_X})$ is a dual residuated connection where

$$\begin{array}{cc}\hfill F(A)(y)& =\underset{x\in X}{\bigwedge }(R(x,y)\to A(x)),G(B)(x)=\underset{y\in Y}{\bigvee }(R(x,y)\odot B(y)).\hfill \end{array}$$

(118)

By Theorem 9, $(e_{{\tau }_{e_{{\tau }_{e_X}}}},\widehat{R},\widehat{S},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a dual residuated frame where

$$\begin{array}{cc}\hfill \widehat{R}(\alpha ,\beta )& =\underset{A\in {\tau }_{e_X}}{\bigwedge }(\beta (F(A))\to \alpha (A)),\widehat{S}(\beta ,\alpha )=\underset{B\in {\tau }_{e_X}}{\bigwedge }(\beta (B)\to \alpha (G(B))).\hfill \end{array}$$

(119)

By Theorem 6, $e_X(x,y)=e_{{\tau }_{e_{{\tau }_{e_X}}}}(\widehat{x},\widehat{y})$. Moreover,

$$\begin{array}{cc}\widehat{R}(\widehat{x},\widehat{y})\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(\widehat{y}(F(A))\to \widehat{x}(A))=\underset{A\in {\tau }_{e_X}}{\bigwedge }(\underset{z\in X}{\bigwedge }(R(z,y)\to A(z))\to A(x))\hfill \\ & \ge \underset{A\in {\tau }_{e_X}}{\bigwedge }((R(x,y)\to A(x))\to A(x))\ge R(x,y).\hfill \end{array}$$

(120)

Let $R_y^{-1}(z)=R(z,y)$. Since $e_X^{-1}\circ R\le e_X^{-1}\circ R\circ e_X^{-1}\le R$, we have

$$R_y^{-1}(x)\odot e_X(x,z)=e_X^{-1}(z,x)\odot R(x,y)\le R_y^{-1}(z).$$

(121)

Thus $R_y^{-1}\in {\tau }_{e_X}$, and so

$$\begin{array}{cc}\widehat{R}(\widehat{x},\widehat{y})=\underset{A\in {\tau }_{e_X}}{\bigwedge }(\underset{z\in X}{\bigwedge }(R(z,y)\to A(z))\to A(x))\hfill & \\ \hfill \le & \underset{z\in X}{\bigwedge }((R(z,y)\to R_y^{-1}(z))\to R_y^{-1}(x))=R(x,y),\hfill \end{array}$$

(122)

$$\begin{array}{cc}\widehat{S}(\widehat{y},\widehat{x})\hfill & =\underset{B\in {\tau }_{e_X}}{\bigwedge }(\widehat{y}(B)\to \widehat{x}(G(B)))=\underset{B\in {\tau }_{e_X}}{\bigwedge }(B(y)\to \underset{z\in X}{\bigvee }(S(z,x)\odot B(z))\hfill \\ & \ge \underset{B\in {\tau }_{e_X}}{\bigwedge }(B(y)\to (S(y,x)\odot B(y))\ge S(y,x).\hfill \end{array}$$

(123)

For all $R_y^{-1}\in {\tau }_{e_X}$,

$$\begin{array}{cc}\widehat{S}(\widehat{y},\widehat{x})=\underset{B\in {\tau }_{e_X}}{\bigwedge }(\widehat{y}(B)\to \widehat{x}(G(B)))\hfill & \le (R_y^{-1}(y)\to \underset{z\in X}{\bigvee }(R(x,z)\odot R_y^{-1}(z))\hfill \\ & \le \top \to R(x,y)=R(x,y)=S(y,x).\hfill \end{array}$$

(124)

Hence $k:(e_X,R,S,e_X)\to (e_{{\tau }_{e_{{\tau }_{e_X}}}},\widehat{R},\widehat{S},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a $DR$-$DR$ frame embedding map. □

Example 5.

Let $X=\{a,b,c\}$ be a set. Let $f:X\to X$ a map and $([0,1],\odot )$ defined as in Example 3.

(1) Let $(X=\{a,b,c\},e_X)$ be a fuzzy poset defined as in Example 3(1). Since $(e_X,f,f,e_X)$ is a dual residuated connection, that is, $e_X(f(x),y)=e_X(x,f(y))$ for all $x,y\in X$, there exist maps $R:{\tau }_{e_X}\times {\tau }_{e_X}\to L$ and $S:{\tau }_{e_X}\times {\tau }_{e_X}\to L$ by

$$R(A,B)=\underset{x\in X}{\bigwedge }(B(f(x))\to A(x)),S(B,A)=\underset{y\in Y}{\bigwedge }(B(y)\to A(g(y)))$$

(125)

with an isotone map $f:X\to Y$ such that $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_X}})$ is a dual residuated frame. For all ${(e_X)}_a,{(e_X)}_b\in {\tau }_{e_X}$,

$$\begin{array}{cc}R({(e_X)}_a,{(e_X)}_b)\hfill & =\underset{x\in X}{\bigwedge }(e_X(b,f(x))\to e_X(a,x))=1,R({(e_X)}_b,{(e_X)}_a)=1,\hfill \\ R({(e_X)}_a,{(e_X)}_a)\hfill & =R({(e_X)}_b,{(e_X)}_b)=0.6,R({(e_X)}_c,{(e_X)}_c)=1,R({(e_X)}_a,{(e_X)}_c)=0.5,\hfill \\ R({(e_X)}_c,{(e_X)}_a)\hfill & =0.7,R({(e_X)}_b,{(e_X)}_c)=0.5,R({(e_X)}_c,{(e_X)}_b)=0.7,\hfill \\ S({(e_X)}_x,{(e_X)}_y)\hfill & =R({(e_X)}_y,{(e_X)}_x)\mathrm{for}\mathrm{all}x,y\in X.\hfill \end{array}$$

(126)

Moreover,

$$R({(e_X)}_a^{-1\ast },{(e_X)}_b)=\underset{x\in X}{\bigwedge }{({(e_X)}_b(f(x))\to e_X)}_a^{-1\ast }(x))=e_X^{\ast }(b,f(a)).$$

(127)

By Theorem 4(1), $(e_{{\tau }_{e_{{\tau }_{e_X}}}},F,G,e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a dual residuated connection where

$$\begin{array}{cc}F(\alpha )(B)\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(R(A,B)\to \alpha (A))=\underset{A\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(B(f(z))\to A(z))\to \alpha (A)\right),\hfill \\ G(\alpha )(A)\hfill & =\underset{B\in {\tau }_{e_X}}{\bigvee }(S(B,A)\odot \alpha (B))=\underset{B\in {\tau }_{e_X}}{\bigvee }\left(\underset{z\in X}{\bigwedge }(B(z)\to A(g(z)))\odot \alpha (B)\right).\hfill \end{array}$$

(128)

By a similar method used in Example 3, one can see that $F=G$.

(2) Let $(X=\{a,b,c\},e_X)$ be a fuzzy poset and $g,h:X\to X$ defined as in Example 3(3). Since $(e_X,h,g,e_X)$ is a dual residuated connection, that is, $e_X(h(x),y)=e_X(x,g(y))$ for all $x,y\in X$, there exist relations $R:{\tau }_{e_X}\times {\tau }_{e_X}\to L$ and $S:{\tau }_{e_X}\times {\tau }_{e_X}\to L$ by

$$\begin{array}{cc}\hfill R(A,B)& =\underset{x\in X}{\bigwedge }(B(h(x))\to A(x)),S(B,A)=\underset{y\in Y}{\bigwedge }(B(y)\to A(g(y)))\hfill \end{array}$$

(129)

such that $(e_{{\tau }_{e_X}},R,S,e_{{\tau }_{e_X}})$ is a dual residuated frame. By Theorem 4(1), $(e_{{\tau }_{e_{{\tau }_{e_X}}}},F,G,e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a dual residuated connection where

$$\begin{array}{cc}F(\alpha )(B)\hfill & =\underset{A\in {\tau }_{e_X}}{\bigwedge }(R(A,B)\to \alpha (A))=\underset{A\in {\tau }_{e_X}}{\bigwedge }\left(\underset{z\in X}{\bigwedge }(B(h(z))\to A(z))\to \alpha (A)\right),\hfill \\ G(\alpha )(A)\hfill & =\underset{B\in {\tau }_{e_X}}{\bigvee }(S(B,A)\odot \alpha (B))=\underset{B\in {\tau }_{e_X}}{\bigvee }\left(\underset{z\in X}{\bigwedge }(B(z)\to A(g(z)))\odot \alpha (B)\right).\hfill \end{array}$$

(130)

Example 6.

(1) Let $(X=\{a,b,c\},e_X)$ be a fuzzy poset where

$$e_X=\left(\begin{array}{ccc}1& 0.6& 0.5\\ 0.6& 1& 0.7\\ 0.5& 0.7& 1\end{array}\right).$$

(131)

Define a binary operation⊙ on $[0,1]$ by

$$x\odot y=max\{0,x+y-1\},x\to y=min\{1-x+y,1\}.$$

(132)

Then $(L=[0,1],\odot ,\to ,0,1)$ is a complete residuated lattice. Let

$$R=\left(\begin{array}{ccc}0.7& 0.4& 0.3\\ 0.6& 0.8& 0.5\\ 0.3& 0.5& 0.8\end{array}\right).$$

(133)

Since $(e_X,R,S,e_X)$ is a residuated frame, we have by Theorem 3(1) that $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_X}})$ is a residuated connection where

$$\begin{array}{cc}\hfill F(A)(y)& =\underset{x\in X}{\bigvee }(R(x,y)\odot A(x)),G(B)(x)=\underset{y\in X}{\bigwedge }(R(x,y)\to B(y)).\hfill \end{array}$$

(134)

By Theorem 11, $(e_{{\tau }_{e_{{\tau }_{e_X}}}},\widehat{R},\widehat{S},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a residuated frame where

$$\begin{array}{cc}\hfill \widehat{R}(\alpha ,\beta )& =\underset{A\in {\tau }_{e_X}}{\bigwedge }(\alpha (A)\to \beta (F(A))),\widehat{S}(\beta ,\alpha )=\underset{B\in {\tau }_{e_X}}{\bigwedge }(\alpha (G(B))\to \beta (B)).\hfill \end{array}$$

(135)

Since $(e_X,R,S,e_X)$ is a dual residuated frame, we have by Theorem 4(1) that $({\tau }_{e_X},F,G,{\tau }_{e_X})$ is a dual residuated connection where

$$\begin{array}{cc}\hfill F(A)(y)& =\underset{x\in X}{\bigwedge }(R(x,y)\to A(x)),G(B)(x)=\underset{y\in Y}{\bigvee }(R(x,y)\odot B(y)).\hfill \end{array}$$

(136)

By Theorem 11, $(e_{{\tau }_{e_{{\tau }_{e_X}}}},\widehat{R},\widehat{S},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a dual residuated connection where

$$\begin{array}{cc}\hfill \widehat{R}(\alpha ,\beta )& =\underset{A\in {\tau }_{e_X}}{\bigwedge }(\beta (F(A))\to \alpha (A)),\widehat{S}(\beta ,\alpha )=\underset{B\in {\tau }_{e_X}}{\bigwedge }(\beta (B)\to \alpha (G(B))).\hfill \end{array}$$

(137)

(2) Let

$$e_X=\left(\begin{array}{ccc}1& 0.7& 0.5\\ 0.4& 1& 0.3\\ 0.3& 0.5& 1\end{array}\right).$$

(138)

Then

$$e_X\circ R\circ e_X=\left(\begin{array}{ccc}0.7& 0.5& 0.3\\ 0.6& 0.8& 0.5\\ 0.3& 0.5& 0.8\end{array}\right),$$

(139)

and so $R<e_X\circ R\circ e_X$. Hence $(e_X,R,S,e_X)$ is not residuated frame. Since $G({(e_X)}_b^{-1\ast })(a)\odot e_X(a,b)=R^{\ast }(a,b)\odot e_X(a,b)=0.6\odot 0.7=0.3\nleqq 0.2=R^{\ast }(b,b)=G({(e_X)}_b^{-1\ast })(b),$ we have $G({(e_X)}_b^{-1\ast })\notin {\tau }_{e_X}$. However, since $R=e_X^{-1}\circ R\circ e_X^{-1}$, we have that $(e_{{\tau }_{e_X}},F,G,e_{{\tau }_{e_X}})$ is a dual residuated connection defined by

$$F(A)(y)=\underset{x\in X}{\bigwedge }(R(x,y)\to A(x)),G(B)(x)=\underset{y\in Y}{\bigvee }(R(x,y)\odot B(y)).$$

(140)

By Theorem 11, $(e_{{\tau }_{e_{{\tau }_{e_X}}}},\widehat{R},\widehat{S},e_{{\tau }_{e_{{\tau }_{e_X}}}})$ is a dual residuated frame where

$$\begin{array}{cc}\hfill \widehat{R}(\alpha ,\beta )& =\underset{A\in {\tau }_{e_X}}{\bigwedge }(\beta (F(A))\to \alpha (A)),\widehat{S}(\beta ,\alpha )=\underset{B\in {\tau }_{e_X}}{\bigwedge }(\beta (B)\to \alpha (G(B))).\hfill \end{array}$$

(141)

5. Conclusions

As an extension of residuated frames for classical relational semantics, we have introduced (dual) residuated frames for fuzzy logics. As a generalization of the classical Tarski’s fixed point theorem, we have shown that an Alexandrov L-topology is a fuzzy complete lattice with residuated connections. By using residuated connections, we have constructed fuzzy rough sets and have solved fuzzy relation equations on the Alexandrov L-topology. Moreover, as a generalization of the Dedekind–MacNeille completion, we have introduced R-R (resp. $DR$-$DR$) embedding maps and R-R (resp. $DR$-$DR$) frame embedding maps.

In the future, by using the concepts of (dual) residuated connections and frames, we plan to investigate fuzzy contexts, information systems and decision rules on Alexandrov L-topologies.