Bonds Between $L$-Fuzzy Contexts Over Different Structures of Truth-Degrees

Abstract

We consider the problem of bonds between $L$-fuzzy contexts over different complete residuated lattices. For this purpose we define $(l,k)$-connection and dual $(l,k)$-connection – pairs of mappings between the residuated lattices based on Krupka’s results on factorizations of complete residuated lattices. We show that the bonds defined using the dual $(l,k)$-connection have very natural properties.

1 Introduction

We study the problem of bonding formal fuzzy contexts over different structures of truth-degrees. This problem was addressed in [12]¹ where the authors used residuation-preserving isotone Galois connections between complete residuated lattices to define bonds. We find the definition of residuation-preserving isotone Galois connection unnecessarily strict for its purpose and we take a new look at it.

Similarly as in [12] we look for an isotone Galois connection between two complete residuated lattices. We apply Krupka’s results on factorization of residuated lattices [13] to find looser and more flexible requirements for the correspondence. As a result we obtain two interrelated correspondences between complete residuated concept lattices — $(l,k)$-connection and dual $(l,k)$-connection. Both of them can be considered to be a variant of the residuation-preserving isotone Galois connection from [12]. Using the dual $(l,k)$-connection we define bonds between formal fuzzy contexts over different complete residuated lattices.

The paper is organized as follows. In Sect. 2, we recall fundamental notions used in the paper. Sections 3 and 4 introduce the $(l,k)$-connection and dual the $(l,k)$-connection, respectively, and describe their properties. In Sect. 5 we utilize the new connections in formal concept analysis to define bonds between formal fuzzy contexts over different residuated lattices. Finally, Sect. 6 summarizes our conclusions and ideas for future research in this area.

2 Preliminaries

2.1 Residuated Lattices, Fuzzy Sets, and Fuzzy Relations

We use complete residuated lattices as basic structures of truth-degrees. A complete residuated lattice is a structure $\mathbf{{L}}=\langle { L,\wedge ,\vee ,\otimes ,\rightarrow ,0,1}\rangle$ such that

• (i) $\langle L,\wedge ,\vee ,0,1\rangle$ is a complete lattice, i.e. a partially ordered set in which arbitrary infima and suprema exist;

• (ii) $\langle L,\otimes ,1\rangle$ is a commutative monoid, i.e. $\otimes$ is a binary operation which is commutative, associative, and $a \otimes 1 = a$ for each $a \in L$;

• (iii) $\otimes$ and $\rightarrow$ satisfy adjointness, i.e. $a\otimes b\leqslant c$ iff $a\leqslant b\rightarrow c$.

$0$ and $1$ denote the least and greatest elements. The partial order of L is denoted by $\leqslant$. Throughout this work, L denotes an arbitrary complete residuated lattice.

Elements $a$ of $L$ are called truth degrees. Operations $\otimes$ (multiplication) and $\rightarrow$ (residuum) play the role of (truth functions of) “fuzzy conjunction” and “fuzzy implication”. Furthermore, we define the complement of $a \in L$ as

$$\begin{aligned} \lnot a = a \rightarrow 0, \end{aligned}$$

(1)

and binary operation of biresiduum $\leftrightarrow$ as

$$\begin{aligned} a \leftrightarrow b = (a \rightarrow b) \wedge (b \rightarrow a)\quad \text {for each } a,b\in L \end{aligned}$$

(2)

An L-set (or L-fuzzy set) $A$ in a universe set $X$ is a mapping assigning to each $x \in X$ some truth degree $A(x)\in L$. The set of all L-sets in a universe $X$ is denoted $L^X$.

The operations with L-sets are defined componentwise. For instance, the intersection of L-sets $A,B \in L^X$ is an L-set $A \cap B$ in $X$ such that $(A \cap B)(x) = A(x) \wedge B(x)$ for each $x \in X$, etc. An L-set $A \in L^X$ is also denoted $\{{}^{A(x)\!}/x\mid x\in X \}$. If for all $y \in X$ distinct from $x_1, x_2, \dots , x_n$ we have $A(y) = 0$, we also write

$$\begin{aligned} \{ {}^{A(x_1)\!}/x_1, {}^{A(x_2)\!}/x_2, \dots , {}^{A(x_n)\!}/x_n \}. \end{aligned}$$

An L-set $A\in L^X$ is called crisp if $A(x)\in \{0,1\}$ for each $x\in X$. Crisp L-sets can be identified with ordinary sets. For a crisp $A$, we also write $x\in A$ for $A(x)=1$ and $x\not \in A$ for $A(x)=0$. An L-set $A\in L^X$ is called empty (denoted by $\emptyset$) if $A(x)=0$ for each $x\in X$.

Binary L-relations (binary L-fuzzy relations) between $X$ and $Y$ can be thought of as L-sets in the universe $X \times Y$. That is, a binary L-relation $I\in L^{X\times Y}$ between a set $X$ and a set $Y$ is a mapping assigning to each $x\in X$ and each $y\in Y$ a truth degree $I(x,y)\in L$ (a degree to which $x$ and $y$ are related by $I$).

Various composition operators for binary L-relations were extensively studied by [7]; we will use the following three composition operators, defined for relations $A\in L^{X\times F}$ and $B\in L^{F\times Y}$:

$$\begin{aligned} (A\circ B)(x,y) = \bigvee _{f \in F} A(x,f)\otimes B(f,y),\end{aligned}$$

(3)

$$\begin{aligned} (A\triangleleft B)(x,y) = \bigwedge _{f \in F} A(x,f)\rightarrow B(f,y),\end{aligned}$$

(4)

$$\begin{aligned} (A\triangleright B)(x,y) = \bigwedge _{f \in F} B(f,y) \rightarrow A(x,f). \end{aligned}$$

(5)

All of them have natural verbal descriptions. For instance, $(A\circ B)(x,y)$ is the truth degree of the proposition “there is a factor $f$ such that $f$ applies to object $x$ and attribute $y$ is a manifestation of $f$”; $(A\triangleleft B)(x,y)$ is the truth degree of “for every factor $f$ , if $f$ applies to object $x$ then attribute $y$ is a manifestation of $f$”. Note also that for $L=\{0,1\}$, $A\circ B$ coincides with the well-known composition of binary relations.

2.2 Formal Fuzzy Concept Analysis

An L-context is a triplet $\langle {X,Y,I}\rangle$ where $X$ and $Y$ are (ordinary nonempty) sets and $I \in L^{X\times Y}$ is an L-relation between $X$ and $Y$. Elements of $X$ are called objects, elements of $Y$ are called attributes, $I$ is called an incidence relation. $I(x,y)=a$ is read: “The object $x$ has the attribute $y$ to degree $a$.”

Consider the following pair $\langle {{}^{\uparrow },{}^{\downarrow }}\rangle$ of operators ${}^\uparrow : L^X\rightarrow L^Y$ and ${}^\downarrow : L^Y\rightarrow L^X$ induced by an L-context $\langle {X,Y,I}\rangle$ as

$$\begin{aligned} A^{\uparrow }(y) = \bigwedge _{x \in X} A(x)\rightarrow I(x,y)\quad \text {and}\quad B^{\downarrow }(x)= \bigwedge _{y \in Y} B(y)\rightarrow I(x,y) \end{aligned}$$

(6)

for all $A\in L^X$ and $B\in L^Y$.

Furthermore, denote the set of fixed points of $\langle {{}^{\uparrow },{}^{\downarrow }}\rangle$ by ${\mathcal {B}}^{\uparrow \downarrow }(X,Y,I)$, i.e.

$$\begin{aligned} {\mathcal {B}}^{\uparrow \downarrow }(X,Y,I)&= \{ \langle {A,B}\rangle \in L^X \times L^Y \mid A^\uparrow =B,\, B^\downarrow =A \}. \end{aligned}$$

(7)

The set of fixed points endowed with $\leqslant$, defined by

$$\langle {A_1,B_1}\rangle \leqslant \langle {A_2,B_2}\rangle \quad \text {if } A_1 \subseteq A_2 \text { (equivalently } B_2 \subseteq B_1) $$

is a complete lattice [2, 15], called a standard $\mathbf{{L}}$ -concept lattice associated with $I$, and its elements are called formal concepts. In a formal concept $\langle {A,B}\rangle$, the $A$ is called an extent, and $B$ is called an intent. The set of all extents and the set of all intents are denoted by ${\text {Ext}}^{\uparrow \downarrow }$ and ${\text {Int}}^{\uparrow \downarrow }$, respectively. That is,

$$\begin{aligned} \begin{aligned} {\text {Ext}}^{\uparrow \downarrow }(X,Y,I)&= \{ A\in L^X \mid \langle {A,B}\rangle \in {\mathcal {B}}^{\uparrow \downarrow }(X,Y,I) \text { for some } B \},\\ {\text {Int}}^{\uparrow \downarrow }(X,Y,I)&= \{ B\in L^Y \mid \langle {A,B}\rangle \in {\mathcal {B}}^{\uparrow \downarrow }(X,Y,I) \text { for some } A \}. \end{aligned} \end{aligned}$$

(8)

An L-relation $\beta \in L^{X_1\times Y_2}$ is called an L -bond ² from L-context $\langle {X_1,Y_1,I_1}\rangle$ to L-context $\langle {X_2,Y_2,I_2}\rangle$ if

$$\begin{aligned} \begin{aligned} {\text {Ext}}^{\uparrow \downarrow }(X_1,Y_2,\beta )&\subseteq {\text {Ext}}^{\uparrow \downarrow }(X_1,Y_1,I_1),\\ {\text {Int}}^{\uparrow \downarrow }(X_1,Y_2,\beta )&\subseteq {\text {Int}}^{\uparrow \downarrow }(X_2,Y_2,I_2). \end{aligned} \end{aligned}$$

(9)

3 $(l,k)$-Connections Between Complete Residuated Lattices

Similarly as in [12] we look for a pair of mappings $\lambda : \mathbf{{L}}_1 \rightarrow \mathbf{{L}}_2$ and $\kappa : \mathbf{{L}}_2 \rightarrow \mathbf{{L}}_1$ which form an isotone Galois connection. Set of its fixpoints with order defined as

$$\begin{aligned} \begin{aligned} \langle {a_1,a_2}\rangle \leqslant \langle {b_1,b_2}\rangle \quad \text {if{}f}\quad a_1\leqslant b_1 \quad (\text {or equivalently }a_2 \leqslant b_2) \end{aligned} \end{aligned}$$

(10)

is a complete lattice. We denote it as $\mathbf{{L}}_{\langle {\lambda ,\kappa }\rangle }$. We need to assure that an adjoint pair exists in $\mathbf{{L}}_{\langle {\lambda ,\kappa }\rangle }$ and this pair is related to adjoint pairs of both, $\mathbf{{L}}_1$ and $\mathbf{{L}}_2$. To this purpose we apply Krupka’s results on factorization of residuated lattices [13]. In fact, the problem can be reformulated as finding an isomorphism between some factorizations of $\mathbf{{L}}_1$ and $\mathbf{{L}}_2$ as depicted in Fig. 1.

Let us recollect Krupka’s approach to factorization of complete residuated lattices. Krupka defines the factorization by cuts of biresiduum as follows. Consider a complete residuated lattice $\mathbf{{L}}$, a truth degree $e\in L$, and mappings

$$\begin{aligned} a^e&= \bigvee \{ b \in L \mid a \leftrightarrow b \geqslant e \} = e \rightarrow a, \end{aligned}$$

(11)

$$\begin{aligned} a_e&= \bigwedge \{ b \in L \mid a \leftrightarrow b \geqslant e\} = e \otimes a. \end{aligned}$$

(12)

For each $a \in L$ define intervals

$$\begin{aligned}{}[a]_e&= [a_e,(a_e)^e] = [e \otimes a, e\rightarrow (e \otimes a)],\\ [a]^e&= [(a^e)_e,a^e] = [e\otimes (e \rightarrow a),e \rightarrow a]. \end{aligned}$$

Denote $L/e = \{ [a]^e \mid a \in L \} (= \{ [a]_e \mid a \in L \})$. Then we have the following result.

Theorem 1

([13]). $\mathbf{{L}}/e = \langle {L/e, \wedge , \vee , \otimes , \rightarrow , 0,1}\rangle$, where $\wedge$ and $\vee$ are given by the order

$$\begin{aligned} B_1 \leqslant B_2 \quad \text {if{}f}\quad \bigvee B_1 \leqslant \bigvee B_2 \end{aligned}$$

and

$$\begin{aligned} B_1 \otimes B_2&= [\bigvee B_1 \otimes \bigvee B_2]_e,\\ B_1 \rightarrow B_2&= [\bigvee B_1 \rightarrow \bigvee B_2]_e,\\ 0&= [0, e\rightarrow 0],\\ 1&= [e, 1] \end{aligned}$$

for each $B_1,B_2 \in L/e$, is a complete residuated lattice.

Following lemma shows alternative ways to define $\otimes$ and $\rightarrow$ in $\mathbf{{L}}/e$

Lemma 1

([13]). For any $B_1,B_2 \in L/e$ we have

$$\begin{aligned} \bigvee B_1 \otimes \bigwedge B_2&= \bigwedge B_1 \otimes B_2,\\ \bigvee B_1 \rightarrow \bigvee B_2&= \bigvee (B_1 \rightarrow B_2),\\ \bigwedge B_1 \rightarrow \bigwedge B_2&= \bigvee (B_1 \rightarrow B_2).\\ \end{aligned}$$

Note that the operators (11) and (12) form an isotone Galois connection on the complete residuated lattice $\mathbf{{L}}$. We extend this approach to have an isotone Galois connection between two (different) complete residuated lattices.

Fig. 1.

Six-element residuated lattice, with $\otimes$ and $\rightarrow$ as showed in the bottom part (011010:00A0B0BCAB in [6]), factorized by $c$-cuts of biresiduum (left), five-element Łukasiewicz chain (111:000AB in [6]) factorized by $0.5$-cuts of biresiduum (right), and their common lattice of factors (middle).

Definition 1

Let $\mathbf{{L}}_1\!=\!\langle {L_1,\! \wedge _1,\! \vee _1,\! \otimes _1,\! \rightarrow _1,0_1,1_1}\rangle , \mathbf{{L}}_2 \!=\! \langle {L_2,\! \wedge _2,\! \vee _2,\! \otimes _2,\rightarrow _2,0_2,1_2}\rangle$ be complete residuated lattices, let $l\in L_1, k \in L_2$ and let $\lambda : L_1\rightarrow L_2, \kappa : L_2\rightarrow L_1$ be mappings, such that

• 1. $\langle {\lambda ,\kappa }\rangle$ is an isotone Galois connection between $\mathbf{{L}}_1$ and $\mathbf{{L}}_2$,

• 2. $\kappa \lambda (a_1) = l\rightarrow _1 (l \otimes _1 a_1)$ for each $a_1 \in L_1$,

• 3. $\lambda \kappa (a_2) = k\otimes _2 (k \rightarrow _2 a_2)$ for each $a_2 \in L_2$.

We call $\langle {\lambda ,\kappa }\rangle$ an $(l,k)$ -connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$.

Fig. 2.

$(c,0.5)$-connection between the residuated lattices from Fig. 1.

Figure 2 shows an example of $(l,k)$-connection corresponding to the factorizations in Fig. 1.

Remark 1

• (a) A pair of identities $\langle {\mathrm {id},\mathrm {id}}\rangle$ on a complete residuated lattice $\mathbf{{L}}$ is $(1,1)$-connection from $\mathbf{{L}}$ to $\mathbf{{L}}$.

• (b) It is worth noting that an $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$ is not uniquely given by the pair of truth degrees $l\in L_1, k\in L_2$ as more than one isomorphism between $\mathbf{{L}}_1/l$ and $\mathbf{{L}}_2/k$ can exist. For example, consider four-element complete residuated lattice $\mathbf{{L}}$ in Fig. 3 (left) with $\otimes = \wedge$ and $\rightarrow$ as in Fig. 3 (right) and mapping $f:L \rightarrow L$ given by $f(0)=0, f(a)=b, f(b)=a,$ and $f(1)=1$. Then $\langle {\mathrm {id}_L,\mathrm {id}_L}\rangle$ and $\langle {f,f}\rangle$ are both $(1,1)$-connections from $\mathbf{{L}}$ to $\mathbf{{L}}$.

Utilizing Theorem 1 we can find particular adjoint pairs in the lattice of fixed points of $\langle {\lambda ,\kappa }\rangle$.

Theorem 2

Denote by $L_{\langle {\lambda ,\kappa }\rangle }$ the set of all fixed points of $(l,k)$-connection $\langle {\lambda ,\kappa }\rangle$ between $\mathbf{{L}}_1$ and $\mathbf{{L}}_2$.

• 1. The algebra $\langle {L_{\langle {\lambda ,\kappa }\rangle }, \wedge ,\vee ,\otimes ,\rightarrow , 0,1}\rangle$ where $\wedge$, $\vee$, $0$, and $1$ are given by the order (10) and

  $$\begin{aligned} \langle {a_1,a_2}\rangle \rightarrow \langle {b_1,b_2}\rangle&= \langle {a_1 \rightarrow _1 b_1, \lambda (a_1 \rightarrow _1 b_1)}\rangle ,\\ \langle {a_1,a_2}\rangle \otimes \langle {b_1,b_2}\rangle&= \langle {l \rightarrow ( l \otimes _1 a_1 \otimes _1 b_1), \lambda (a_1 \otimes _1 b_1) }\rangle \end{aligned}$$

  is a complete residuated lattice.

• 2. The algebra $\langle {L_{\langle {\lambda ,\kappa }\rangle }, \wedge ,\vee , \boxtimes , \searrow , 0,1}\rangle$ where $\wedge$, $\vee$, $0$, and $1$ are given by the order (10) and

  $$\begin{aligned} \langle {a_1,a_2}\rangle \searrow \langle {b_1,b_2}\rangle&= \langle { \kappa (k \otimes _2 (a_2 \rightarrow _2 b_2)) , k \otimes _2 (a_2 \rightarrow _2 b_2) }\rangle \\&= \langle { \kappa ((k \rightarrow _2 b_2) \rightarrow _2 (k \rightarrow _2 b_2)) , (k \rightarrow _2 a_2) \rightarrow _2 (k \rightarrow _2 b_2) }\rangle , \\ \langle {a_1,a_2} \rangle \boxtimes \langle {b_1,b_2}\rangle&= \langle { \kappa (a_2 \otimes _2 (k \rightarrow _2 b_2)) , a_2 \otimes _2 (k \rightarrow _2 b_2) }\rangle \\&= \langle { \kappa ((k \rightarrow _2 a_2) \otimes _2 b_2) , (k \rightarrow _2 a_2) \otimes _2 b_2 }\rangle \end{aligned}$$

  is a complete residuated lattice.

Fig. 3.

Complete residuated lattice from Remark 1(b); (0:a0b in [6])

Proof

Directly from Definition 1, Theorem 1 and Lemma 1.   $\square$

Remark 2

For sake of completeness, we show how the meet, join, $0$, and $1$ given by the order (10) are defined in $L_{\langle {\lambda ,\kappa }\rangle }$:

$$\begin{aligned} \langle {a_1,a_2}\rangle \wedge \langle {b_1,b_2}\rangle&= \langle {a_1 \wedge _1 b_1, k \otimes _2 ((k \rightarrow _2 a_2) \wedge _2 (k \rightarrow _2 b_2))}\rangle ,\end{aligned}$$

(13)

$$\begin{aligned} \langle {a_1,a_2}\rangle \vee \langle {b_1,b_2}\rangle&= \langle {l \rightarrow _1 ((l \otimes _1 a_1) \vee _1 (l \otimes _1 b_1)), a_2 \vee _2 b_2}\rangle ,\end{aligned}$$

(14)

$$\begin{aligned} 0&=\langle {l \rightarrow 0_1, 0_2}\rangle ,\end{aligned}$$

(15)

$$\begin{aligned} 1&=\langle {1_1, k}\rangle . \end{aligned}$$

(16)

It is easy to see, that the two ajdoint pairs, $\langle {\otimes ,\rightarrow }\rangle$ and $\langle {\boxtimes ,\searrow }\rangle$, from Theorem 2 can be different. As an example consider $\mathbf{{L}}_1$ being three-element Łukasiewicz chain, $\mathbf{{L}}_2$ being three-element Gödel chain and $\lambda$ and $\kappa$ being identities on $L_1=L_2$. The related factorizations, $\mathbf{{L}}_1/1$ and $\mathbf{{L}}_2/1$ are the three-element Łukasiewicz chain and the three-element Gödel chain, respectively, again. Clearly, their adjoint pairs are different.

We call the $(l,k)$-connections whose factorizations produce the same adjoint pair residuation-preserving. The following corollary shows that for residuation-preserving $(l,k)$-connection $\langle {\lambda ,\kappa }\rangle$ we can specify the adjoint pair on the lattice of its fixed points without the mappings $\langle {\lambda ,\kappa }\rangle$.

Corollary 1

Let $\langle {\lambda ,\kappa }\rangle$ be a residuation-preserving $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$. The algebra $\mathbf{{L}}_{\langle {\lambda ,\kappa }\rangle } = \langle {L_{\langle {\lambda ,\kappa }\rangle }, \wedge ,\vee , \otimes , \rightarrow , 0,1}\rangle$ where $\wedge , \vee , 0, 1$ are given by the order (10) and

$$\begin{aligned} \langle {a_1,a_2} \rangle \rightarrow \langle {b_1,b_2}\rangle&= \langle {a_1 \rightarrow _1 b_1, k \otimes _2 (a_2 \rightarrow _2 b_2)}\rangle \end{aligned}$$

(17)

$$\begin{aligned}&= \langle { a_1 \rightarrow _1 b_1, k \otimes _2 ((k \rightarrow _2 a_2) \rightarrow _2 (k \rightarrow _2 b_2)) }\rangle , \end{aligned}$$

(18)

$$\begin{aligned} \langle {a_1,a_2}\rangle \otimes \langle {b_1,b_2}\rangle&= \langle {l \rightarrow _1( l \otimes _1 a_1 \otimes _1 b_1) a_2 \otimes _2 (k \rightarrow _2 b_2) }\rangle \end{aligned}$$

(19)

$$\begin{aligned}&= \langle {l \rightarrow _1 ( l \otimes _1 a_1 \otimes _1 b_1), (k \rightarrow _2 a_2) \otimes _2 b_2 }\rangle \end{aligned}$$

(20)

is a complete residuated lattice.

Proof

Directly from Theorem 2 and the property of residuation-preservation, that is $\otimes = \boxtimes$ and $\rightarrow = \searrow$.   $\square$

The following theorem provides more practical characterization of residuation-preserving $(l,k)$-connections.

Theorem 3

Let $\langle {\lambda ,\kappa }\rangle$ be an $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$. The following statements are equivalent

• (a) $\langle {\lambda ,\kappa }\rangle$ is residuation-preserving.

• (b) $\kappa (k \otimes _2 (\lambda (a) \rightarrow _2 \lambda (b))) = \kappa \lambda (a)\rightarrow _1\kappa \lambda (b)$ holds true for any $a,b \in L_1$.

• (c) $k \otimes _2 (\lambda \kappa (a) \rightarrow _2 \lambda \kappa (b)) = \lambda (\kappa (a)\rightarrow _1\kappa (b))$ holds true for any $a,b \in L_2$.

Proof

(sketch) Follows from the fact, that pairs in $L_{\langle {\lambda ,\kappa }\rangle }$ are exactly pairs $\langle {\kappa \lambda (a_1),\lambda (a_1)}\rangle$ for $a_1 \in L_1$ and exactly pairs $\langle {\kappa (a_2), \lambda \kappa (a_2)}\rangle$ for $a_2 \in L_2$.   $\square$

Note that left-hand sides of the equations in (b) and (c) of Theorem 3 contain an inconvenient multiplication by $k$. This leads to a quite cumbersome definition when we try to use them to define bonds between formal fuzzy context over different residuated lattices. In the next section we provide an alternative to $(l,k)$-connection which avoids this inconvenience.

4 Dual $(l,k)$-Connections Between Complete Residuated Lattices

We defined $(l,k)$-connections as an isotone Galois connection to assure that the set of its fixed points is a complete lattice and that it preserves order of both $\mathbf{{L}}_1$ and $\mathbf{{L}}_2$. But another property of isotone Galois connection, namely its non-duality, is undesired for our purpose, that is bonding fuzzy contexts over different residuated lattices. To fix this, we make a small trick with the $(l,k)$-connections. Instead of connecting upper bounds of intervals from $\mathbf{{L}}_1/l$ with lower bounds of intervals in $\mathbf{{L}}_2/k$, we simply connect upper bounds with upper bounds. To do that we need to drop the requirement of being an isotone Galois connection.

Definition 2

Let $\mathbf{{L}}_1\!=\!\langle {L_1,\! \wedge _1,\!\vee _1,\!\otimes _1,\!\rightarrow _1\!,0_1,\!1_1}\rangle ,\, \mathbf{{L}}_2\! =\!\!\langle {L_2,\! \wedge _2,\!\vee _2,\!\otimes _2,\!\rightarrow _2,\!0_2,\!1_2}\rangle$ be complete residuated lattices, let $l\in L_1, k \in L_2$ and let ${\lambda '}: L_1\rightarrow L_2, \kappa ': L_2\rightarrow L_1$ be mappings, such that

• $\lambda '$ and $\kappa '$ are order-preserving,

• $\lambda '\kappa '\lambda '(a_1) = \lambda '(a_1)$ and $\kappa '\lambda '\kappa '(a_2) = \kappa '(a_2)$ for each $a_1\in L_1$ and $a_2 \in L_2$,

• $\kappa '\lambda '(a_1) = l\rightarrow _1 (l \otimes _1 a_1)$ for each $a_1 \in L_1$,

• ${\lambda '}\kappa '(a_2) = k \rightarrow _2 (k \otimes _2 a_2)$ for each $a_2 \in L_2$.

We call the pair $\langle {\lambda ',\kappa '}\rangle$ ³ a dual $(l,k)$ -connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$.

Fig. 4.

Dual $(c,0.5)$-connection between the residuated lattices from Fig. 1

The notion of dual and non-dual $(l,k)$-connections are related in following way.

Theorem 4

• (a) For each $(l,k)$-connection $\langle {\lambda ,\kappa }\rangle$ from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$ there is a dual $(l,k)$-connection $\langle {\lambda ',\kappa '}\rangle$ from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$, such that for each $a_1\in L_1,a_2\in L_2$,

  $$\begin{aligned} \begin{aligned}&\langle {a_1,a_2}\rangle \in L_{\langle {\lambda ,\kappa }\rangle } \text { implies } \langle {a_1,k\rightarrow _2 a_2}\rangle \in L_{\langle {\lambda ',\kappa '}\rangle },\\&\langle {a_1,a_2}\rangle \in L_{\langle {\lambda ',\kappa '}\rangle } \text { implies } \langle {a_1,k\otimes _2 a_2}\rangle \in L_{\langle {\lambda ,\kappa }\rangle }. \end{aligned} \end{aligned}$$

  (21)

• (b) For each dual $(l,k)$-connection $\langle {\lambda ',\kappa '}\rangle$ from $\mathbf{{L}}_1$ and $\mathbf{{L}}_2$ there is an $(l,k)$-connection $\langle {\lambda ,\kappa }\rangle$ from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$ such that (21) is satisfied.

Proof

• (a) Let $\langle {\lambda ,\kappa }\rangle$ be an $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$. We show that $\langle {\lambda ',\kappa '}\rangle$ defined as

  $$\begin{aligned} \lambda ' = k \rightarrow _2 \lambda (a_1) \quad \text {and}\quad \kappa ' = \kappa (k \otimes _2 a_2) \end{aligned}$$

  (22)

  for each $a_1\in L_1,a_2 \in L_2$ is a dual $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$ which satisfies (21). Since $\lambda$ and $\kappa$ are order-preserving and $\rightarrow$ and $\otimes$ are both monotone in the second argument, the mapping $\lambda '$ and $\kappa '$ are order-preserving as well. We have for each $a_1 \in L_1, a_2 \in L_2$

  $$ \kappa '\lambda '(a_1) = \kappa (k \otimes _2 (k \rightarrow _2 (\lambda (a_1)))) = \kappa \lambda \kappa \lambda (a_1) = \kappa \lambda (a_1) = l \rightarrow _1 (l \otimes _1 a_1) $$

  and

  $$ \lambda '\kappa '(a_2)\! =\! k \rightarrow _2 \lambda \kappa (k \otimes _2 a_2) = k \rightarrow _2 (k \otimes _2 (k \rightarrow _2 (k \otimes _2 a_2))) = k \rightarrow _2 (k \otimes _2 a_2). $$

  Finally, we have for each $a_1 \in L_1, a_2 \in L_2$

  $$ \lambda '\kappa '\lambda '(a_1)= k \rightarrow _2 (k \otimes _2 (k \rightarrow _2 \lambda (a_1))) = k \rightarrow _2 \lambda (a_1) = \lambda '(a_1) $$

  and

  $$ \kappa '\lambda '\kappa '(a_2) = \kappa '\lambda '(\kappa (k \otimes _2 a_2)= \kappa \lambda \kappa (k \otimes _2 a_2) = \kappa (k \otimes _2 a_2) = \kappa '(a_2). $$

  Thus, $\langle {\lambda ',\kappa '}\rangle$ is a dual $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$. Now we show that $\langle {\lambda ',\kappa '}\rangle$ satisfies (21). Let $\langle {a_1,a_2}\rangle \in L_{\langle {\lambda ,\kappa }\rangle }$; from that we have

  $$\begin{aligned} \langle {a_1,k \rightarrow a_2}\rangle&= \langle {\kappa (a_2),k \rightarrow _2 \lambda (a_1)}\rangle \\&= \langle {\kappa \lambda (a_1), k \rightarrow _2 \lambda (a_1)}\rangle \\&= \langle {\kappa '\lambda '(a_1), \lambda '(a_1)}\rangle \end{aligned}$$

  showing $\langle {a_1,k \rightarrow _2 a_2}\rangle \in L_{\langle {\lambda ',\kappa '}\rangle }$. The other part can be showed similarly.

• (b) Similarly as in (a) we can show that $\langle {\lambda ,\kappa }\rangle$ defined as

  $$\begin{aligned} \lambda = k \otimes _2 \lambda '(a_1) \quad \text {and}\quad \kappa = \kappa '(k \rightarrow _2 a_2) \end{aligned}$$

  (23)

  for each $a_1\in L_1,a_2 \in L_2$ is a $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$ which satisfies (21).

   $\square$

What we get from this trick are more convenient operations $\wedge$ and $\rightarrow$ in the complete residuated lattice $\mathbf{{L}}_{\langle {\lambda ,\kappa }\rangle }$ of fixed points of $\langle {\lambda ,\kappa }\rangle$. That is important for definition of bonds because concept-forming operators $\langle {\uparrow ,\downarrow }\rangle$ are defined using the operations $\wedge$ and $\rightarrow$.

Theorem 5

The $(l,k)$-connections from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$ are in one-to-one correspondence with dual $(l,k)$-connections from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$.

Proof

From proof of Theorem 4 we have (22) and (23) providing ways to get a dual $(l,k)$-connection from an $(l,k)$-connection and vice versa. We only need to show, that they are mutually inverse. Let $\langle {\lambda ,\kappa }\rangle$ be an $(l,k)$-connection from $\mathbf{{L}}_1$ and $\mathbf{{L}}_2$ and let $\langle {\lambda ',\kappa '}\rangle$ be a dual $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$ defined by (22). Applying (23) we get

$$ \lambda ''(a_1) = k \otimes _2 \lambda '(a_1) = k \otimes _2 (k \rightarrow _2 \lambda (a_1)) = \lambda \kappa \lambda (a_1) = \lambda (a_1) $$

for each $a_1 \in L_1$ and

$$ \kappa ''(a_2) = \kappa '(k \rightarrow _2 a_2) = \kappa (k \otimes _2 (k \rightarrow _2 a_2)) = \kappa \lambda \kappa (a_2) = \kappa (a_2) $$

for each $a_2 \in L_2$. Similarly, the other composition can be showed to be an identity.   $\square$

From the above one-to-one correspondence we obtain the following theorem.

Theorem 6

Let be $\langle {\lambda ',\kappa '}\rangle$ dual $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$.

• 1. The algebra $\langle {L_{\langle {\lambda ',\kappa '}\rangle }, \wedge ,\vee ,\otimes _1,\rightarrow _1, 0,1}\rangle$ where $\wedge$ and $\vee$ are given by the order (10) and

  $$\begin{aligned} \langle {a_1,a_2}\rangle \rightarrow \langle {b_1,b_2}\rangle&= \langle {a_1 \rightarrow _1 b_1, \lambda '(a_1 \rightarrow _1 b_1)}\rangle \\ \langle {a_1,a_2} \rangle \otimes \langle {b_1,b_2}\rangle&= \langle {l \rightarrow _1( l \otimes _1 a_1 \otimes _1 b_1), \lambda '(a_1 \otimes _1 b_1))) }\rangle \end{aligned}$$

  is a complete residuated lattice.

• 2. The algebra $\langle {L_{\langle {\lambda ',\kappa '}\rangle }, \wedge ,\vee , \boxtimes ,\searrow , 0,1}\rangle$ where $\wedge$ and $\vee$ are given by the order (10) and

  $$\begin{aligned} \langle {a_1,a_2}\rangle \searrow \langle {b_1,b_2}\rangle&= \langle { \kappa '(a_2 \rightarrow _2 b_2) , a_2 \rightarrow _2 b_2 } \rangle \\ \langle {a_1,a_2}\rangle \boxtimes \langle {b_1,b_2}\rangle&= \langle { \kappa '( a_2 \otimes _2 b_2) , k \rightarrow _2 (k \otimes _2 a_2 \otimes _2 b_2) }\rangle \end{aligned}$$

  is a complete residuated lattice.

Proof

Directly from Theorems 2 and 4 and its proof, and Theorem 5.   $\square$

For sake of completeness, we also show how $\wedge ,\vee ,0$ and $1$ are defined in the complete residuated lattice from the previous theorem:

$$\begin{aligned} \langle {a_1,a_2}\rangle \wedge \langle {b_1,b_2}\rangle&= \langle {a_1 \wedge _1 b_1, a_2 \wedge _2 b_2}\rangle ,\end{aligned}$$

(24)

$$\begin{aligned} \langle {a_1,a_2}\rangle \vee \langle {b_1,b_2}\rangle&= \langle {l \rightarrow _1 (l \otimes _1 (a_1 \vee _1 b_1)), k \rightarrow _2 (k \otimes _2 (a_2 \vee _2 b_2))}\rangle , \end{aligned}$$

(25)

and $0=\langle {l\rightarrow 0_1,k\rightarrow 0_2}\rangle$, $1=\langle {1_1,1_2}\rangle$.

Again, we want the two adjoint pairs from Theorem 6 to be equal. We define the notion of residuation-preserving dual $(l,k)$-connection analogously, as in the non-dual case.

Theorem 7

Let $\langle {\lambda ',\kappa '}\rangle$ be a dual $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$. The following statements are equivalent

• (a) $\langle {\lambda ',\kappa '}\rangle$ is residuation-preserving.

• (b) $\kappa '(\lambda '(a_1) \rightarrow _2 \lambda '(b_1)) = \kappa '\lambda '(a_1) \rightarrow _1 \kappa '\lambda '(b_1)$ holds true for any $a_1, b_1 \in L_1$.

• (c) $\lambda '(\kappa '(a_2) \rightarrow _1 \kappa '(b_2)) = \lambda '\kappa '(a_2) \rightarrow _2 \lambda '\kappa '(b_2)$ holds true for any for any $a_2,b_2 \in L_2$.

Proof

Similar as proof of Theorem 3.   $\square$

Theorem 8

A dual $(l,k)$-connection $\langle {\lambda ',\kappa '}\rangle$ from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$ is residuation-preserving if and only if its associated $(l,k)$-connection is residuation-preserving.

Proof

We have

$$\begin{aligned} \kappa '(\lambda '(a_1) \rightarrow _2 \lambda '(b_1))&= \kappa (k\otimes _2 ((k \rightarrow _2 \lambda (a_1)) \rightarrow _2 (k \rightarrow _2 \lambda (b_1))))\\&= \kappa (k\otimes _2 ((k \otimes _2 (k \rightarrow _2 \lambda (a_1))) \rightarrow _2 \lambda (b_1)))\\&= \kappa (k\otimes _2 ((\lambda \kappa \lambda (a_1)) \rightarrow _2 \lambda (b_1)))\\&= \kappa (k\otimes _2 (\lambda (a_1) \rightarrow _2 \lambda (b_1))) \end{aligned}$$

and

$$\begin{aligned} \kappa '\lambda '(a_1) \rightarrow _2 \kappa '\lambda '(b_1)&= \kappa \lambda (a_1) \rightarrow _2 \kappa \lambda (b_1) \end{aligned}$$

showing that the condition Theorem 7(b) is equivalent to Theorem 3(b). The statement of Theorem 8 then follows from Theorems 3 and 7.   $\square$

Remark 3

In the previous approach [12], the residuation-preserving Galois connections are defined as isotone Galois connections, as in the case of $(l,k)$-connections. In the same time, they have to satisfy conditions similar to Theorem 7(b) and (c), as in the case of dual $(l,k)$-connections. This is where we see the unnecessary strictness of the previous approach. Loosely speaking, the residuation-preserving isotone Galois connections were wanted to be both, $(l,k)$-connections and dual $(l,k)$-connections.

5 $\langle {\lambda ,\kappa }\rangle$-Bonds

In this section, we define bond between formal fuzzy contexts over different complete residuated lattices $\mathbf{{L}}_1$ and $\mathbf{{L}}_2$ and describe their properties. More specifically, we propose new bonds, called $\langle {\lambda ,\kappa }\rangle$-bonds, which are based directly on dual $(l,k)$-connections⁴ from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$. In this section we omit proofs due to page limit.

Below, we define the $\langle {\lambda ,\kappa }\rangle$-bonds as a special $\mathbf{{L}}_{\langle {\lambda ,\kappa }}\rangle$-relation $\beta$ between $X_1$ and $Y_2$ and we define concept-forming operators $\vartriangle :L_1^{X_1} \rightarrow L_2^{Y_2}$ and $\triangledown :L_2^{Y_1} \rightarrow L_2^{X_2}$ induced by $\langle {\lambda ,\kappa }\rangle$-bond by⁵

$$\begin{aligned} \begin{aligned} A^\vartriangle (y_2)&= \mathop {{\bigwedge }_{2}}\limits _{{x_1 \in X_1}} \lambda (A(x_1)) \rightarrow _2 {\mathrm {proj}}_2(\beta (x_1,y_2)),\\ B^\triangledown (x_1)&= \mathop {{\bigwedge }_{1}}\limits _{{y_2 \in Y_2}} \kappa (B(y_2)) \rightarrow _1 {\mathrm {proj}}_1(\beta (x_1,y_2)). \end{aligned} \end{aligned}$$

(26)

Thus we can express the concept-forming operators $\langle {\vartriangle ,\triangledown }\rangle$ using the classic ones, i.e. $\langle {\uparrow ,\downarrow }\rangle$, as

$$\begin{aligned} A^\vartriangle = (\lambda (A))^{\uparrow _{{\mathrm {proj}}_2(\beta )}} \quad \text {and} \quad B^\triangledown = (\kappa (B))^{\downarrow _{{\mathrm {proj}}_1(\beta )}} \end{aligned}$$

for each $A \in L_1^{X_1}$ and $B \in L_2^{Y_2}$.

Remark 4

The definition of concept-forming operators (26) actually follows as a corollary of particular setting in the framework of supremum-preserving aggregation structures. The framework was introduced in [3] and studied further in [4] (see also [1, 5, 10, 14] for related works). We will bring detailed explanation in the full version of this paper.

Definition 3

Let $\mathbf{{L}}_1,\mathbf{{L}}_2$ be complete residuated lattices, $\langle {\lambda ,\kappa }\rangle$ be dual $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$, and let $\langle {X_1,Y_1,I_1}\rangle$ and $\langle {X_2,Y_2,I_2}\rangle$ be $\mathbf{{L}}_1$-context and $\mathbf{{L}}_2$-context, respectively. We call $\beta \in L_{\langle {\lambda ,\kappa }\rangle }^{X_1 \times Y_2}$ a $\langle {\lambda ,\kappa }\rangle$ -bond from $\langle {X_1,Y_1,I_1}\rangle$ to $\langle {X_2,Y_2,I_2}\rangle$ if the following inclusions hold:

$$\begin{aligned} {\text {Ext}}^{{\vartriangle }{\triangledown }}(X_1,Y_2,\beta )&\subseteq {\text {Ext}}^{\uparrow \downarrow }(X_1,Y_1,\kappa \lambda (I_1)),\end{aligned}$$

(27)

$$\begin{aligned} {\text {Int}}^{{\vartriangle }{\triangledown }}(X_1,Y_2,\beta )&\subseteq {\text {Int}}^{\uparrow \downarrow }(X_2,Y_2,\lambda \kappa (I_2)). \end{aligned}$$

(28)

Obviously, when $\mathbf{{L}}_1=\mathbf{{L}}_2 = \mathbf{{L}}$ the pair of identities $\langle {\mathrm {id},\mathrm {id}}\rangle$ on $L$ is a $(1,1)$-connection between them and the $\langle {\mathrm {id},\mathrm {id}}\rangle$-bonds correspond with L-bonds. The following theorem explains the relationship of $\langle {\lambda ,\kappa }\rangle$-bonds with the $\mathbf{{L}}$-bonds more generally.

Theorem 9

Let $\beta \in L_{\langle {\lambda ,\kappa }\rangle }^{X_1 \times Y_2}$. The following statements are equivalent.

• (a) $\beta$ is a $\langle {\lambda ,\kappa }\rangle$ -bond from $\langle {X_1,Y_1,I_1}\rangle$ to $\langle {X_2,Y_2,I_2}\rangle$;

• (b) ${\mathrm {proj}}_1(\beta )$ is a $\mathbf{{L}}_1$-bond from $\langle {X_1,Y_1,\kappa \lambda (I_1)\rangle }$ to $\langle {X_2,Y_2,\kappa (I_2)\rangle }$;

• (c) ${\mathrm {proj}}_2(\beta )$ is a $\mathbf{{L}}_2$-bond from $\langle {X_1,Y_1,\lambda (I_1)\rangle }$ to $\langle {X_2,Y_2,\lambda \kappa (I_2)\rangle }$;

• (d) ${\mathrm {proj}}_1(\beta ) = \lambda \kappa (I_1) \triangleright _1 S_\mathrm {i}$ and ${\mathrm {proj}}_2(\beta ) = S_\mathrm {e}\triangleleft _2 \lambda \kappa (I_2)$ for some $S_\mathrm {e}\in L_1^{X_1\times X_2}$ and $S_\mathrm {i}\in L_2^{Y_1\times X_2}$.

From Theorem 9(a)$\Leftrightarrow$(d) we have the following corollary.

Corollary 2

Set of all $\langle {\lambda ,\kappa }\rangle$-bonds is an $\mathbf{{L}}_{\langle {\lambda ,\kappa }\rangle }$-closure system.

Fig. 5.

A $\mathbf{{L}}_1$-context $\mathbb {K}_1$ and $\mathbf{{L}}_2$-context $\mathbb {K}_2$ (top left and top right) with $\mathbf{{L}}_1,\mathbf{{L}}_2$ as in Fig. 1; $\mathbb {K}_1 \boxplus _{\langle {\lambda ,\kappa }\rangle } \mathbb {K}_2$ (middle) with $\langle {\lambda ,\kappa }\rangle$ as in Fig. 4.; Lattice of all $\langle {\lambda ,\kappa }\rangle$-bonds (bottom); the solid lined bonds are regular and the dotted lined bonds are irregular.

$\langle {\lambda ,\kappa }\rangle$ -direct products and regular $\langle {\lambda ,\kappa }\rangle$ -bonds

In this part, we assume that $\mathbf{{L}}_{\langle {\lambda ,\kappa }}\rangle$ satisfies the double negation law, that is

$$\begin{aligned} (a \rightarrow 0) \rightarrow 0 = a \quad \text {for each } a \in L_{\langle {\lambda ,\kappa }\rangle }. \end{aligned}$$

Note that it means

$$\begin{aligned} \langle {a_1,a_2}\rangle&= (\langle {a_1,a_2}\rangle \rightarrow \langle {l\rightarrow _1 0_1,k\rightarrow _2 0_2}\rangle )\rightarrow \langle {l\rightarrow _1 0_1,k\rightarrow _2 0_2}\rangle \\&= \langle {(a_1 \rightarrow _1 (l \rightarrow _1 0))\rightarrow _1 (l \rightarrow _1 0),(a_2 \rightarrow _2 (k\rightarrow _2 0_2)) \rightarrow _2 (k\rightarrow _2 0_2)\rangle } \end{aligned}$$

for each $\langle {a_1,a_2}\rangle \in L_{\langle {\lambda ,\kappa }\rangle }$.

Definition 4

Let $\mathbb {K}_1 = \langle {X_1,Y_1,I_1}\rangle$ be an $\mathbf{{L}}_1$-context, $\mathbb {K}_2= \langle {X_2,Y_2,I_2}\rangle$ be an $\mathbf{{L}}_2$-context, and $\langle {\lambda ,\kappa }\rangle$ be a dual $(l,k)$-connection from $\mathbf{{L}}_1$ to $\mathbf{{L}}_2$. We define $\langle {\lambda ,\kappa }\rangle$ -direct product $\mathbb {K}_1 \boxplus _{\langle {\lambda ,\kappa }\rangle } \mathbb {K}_2$ as $\mathbf{{L}}_{\langle {\lambda ,\kappa }\rangle }$-context $\langle {X_2 \times Y_1,X_1 \times Y_2, \varDelta }\rangle$ with

$$\begin{aligned} \varDelta (\langle {x_2,y_1}\rangle ,\langle {x_1,y_2}\rangle )&= \lnot \langle {\kappa \lambda I_1(x_1,y_1), \lambda I_1(x_1,y_1)}\rangle \rightarrow \langle {\kappa I_2(x_2,y_2),\lambda \kappa I_2(x_2,y_2)}\rangle \end{aligned}$$

for each $x_1\in X_1, y_1\in Y_1, x_2\in X_2, y_2\in Y_2$.

Extents of the $\langle {\lambda ,\kappa }\rangle$-direct product are $\langle {\lambda ,\kappa }\rangle$-bonds:

Theorem 10

Let $\mathbb {K}_1 \boxplus _{\langle {\lambda ,\kappa }\rangle } \mathbb {K}_2 = \langle {X_1 \times Y_2, X_2 \times Y_1, \varDelta }\rangle$ be a $\langle {\lambda ,\kappa }\rangle$-direct product. Extents in ${\text {Ext}}^{\uparrow \downarrow }(X_1 \times Y_2, X_2 \times Y_1, \varDelta )$ are $\langle {\lambda ,\kappa }\rangle$-bonds from $\mathbb {K}_1$ to $\mathbb {K}_2$.

Analogously, with the L-bonds there exist $\langle {\lambda ,\kappa }\rangle$-bonds which are not extents of the direct product $\mathbb {K}_1 \boxplus _{\langle {\lambda ,\kappa }\rangle } \mathbb {K}_2$ (see Fig. 5). A $\langle {\lambda ,\kappa }\rangle$-bond is called regular if it is extent of the direct product, otherwise it is called irregular.

6 Conclusions and Further Research

We revisited results on bonding formal fuzzy contexts in [12] and identified the main flaw: the residuation-preserving isotone Galois connections between complete residuated concept lattices had to fulfill two conflicting sets of requirements. In the present paper we studied two variants of residuation-preserving isotone Galois connections emerging by altering one of the two conflicting sets of requirements. One of the variants, namely dual $(l,k)$-connections, brought very convenient definition of bonds between formal fuzzy contexts with different structures of truth-degrees.

Our future research in this area includes:

• Extension of the present results to homogeneous bonds wrt. isotone concept-forming operators and heterogeneous bonds studied in [8, 9]. We are going to generalize our previous results on bonds. Our preliminary observations show that $(l,k)$-connections will be useful for homogeneous bonds wrt. isotone concept-forming operators and for heterogeneous bonds.

• Connections between complete residuated lattices based on antitone Galois connections.