### Fuzzy Connections and Relations in Complete Residuated Lattices

• #### ABSTRACT

In this paper, we investigate the properties of fuzzy Galois (dual Galois, residuated, and dual residuated) connections in a complete residuated lattice L. We give their examples. In particular, we study fuzzy Galois (dual Galois, residuated, dual residuated) connections induced by L-fuzzy relations.

• #### KEYWORD

Fuzzy Galois connections , Fuzzy posets , Fuzzy isotone maps , Fuzzy antitone maps

• ### 1. Introduction

A Galois connection is an important mathematical tool for algebraic structure, data analysis, and knowledge processing [1-10]. Hájek [11] introduced a complete residuated lattice L that is an algebraic structure for many-valued logic. A context consists of (U, V, R), where U is a set of objects, V is a set of attributes, and R is a relation between U and V. Bĕlohlávek [1-3] developed a notion of fuzzy contexts using Galois connections with RLX×Y on L.

In this paper, we investigate properties of fuzzy Galois (dual Galois, residuated, and dual residuated) connections in L and give their examples. In particular, we study fuzzy Galois (dual Galois, residuated, and dual residuated) connections induced by L-fuzzy relations.

Definition 1.1. [11, 12] An algebra is called a complete residuated lattice if it satisfies the following conditions:

(C1) is a complete lattice with the greatest element 1 and the least element 0; (C2) (L, , 1) is a commutative monoid; (C3) x y ≤ z iff x ≤ y → z for x, y, z ∈ L.

Remark 1.2. [11, 12] (1) A completely distributive lattice is a complete residuated lattice defined by

In particular, the unit interval is a complete residuated lattice defined by

(2) The unit interval with a left-continuous t-norm , , is a complete residuated lattice defined by

In this paper, we assume that is a complete residuated lattice with the law of double negation, i.e., a = a** where a = a → 0.

Lemma 1.3. [12] For each x, y, z, xi, yiL, we have the following properties.

Definition 1.4. [4, 7] Let X denote a set. A function eX : X × XL is called:

(E1) reflexive if eX(x, x) = 1 for all x ∈ X, (E2) transitive if eX(x, y) eX(y, z) ≤ eX(x, z), for all x, y, z ∈ X, and (E3) if eX(x, y) = eX(y, x) = 1, then x = y.

If e satisfies (E1) and (E2), (X, eX) is a fuzzy preorder set. If e satisfies (E1), (E2), and (E3), (X, eX) is a fuzzy partially order set (for simplicity, fuzzy poset).

Example 1.5. (1) We define a function

eLx : LX × LXL

as

eLx(A, B) = (A(x) → B(x)).

Then, (LX, eLx) is a fuzzy poset from Lemma 1.3 (10, 11).

(2) If (X, eX) is a fuzzy poset and we define a function (x, y) = eX(y, x), then (X, ) is a fuzzy poset.

### 2. Fuzzy Connections and Relations in Complete Residuated Lattices

Definition 2.1. Let (X, eX) and (Y, eY) denote fuzzy posets and f : XY and g : YX denote maps.

(1) (eX, f, g, eY) is called a Galois connection if for all xX, yY,

eY(y, f(x)) = eX(x, g(y)).

(2) (eX, f, g, eY) is called a dual Galois connection if for all xX, yY,

eY(f(x), y) = eX(g(y), x).

(3) (eX, f, g, eY) is called a residuated connection if for all xX, yY,

eY(f(x), y) = eX(x, g(y)).

(4) (eX, f, g, eY) is called a dual residuated connection if for all xX, yY,

eY(y, f(x)) = eX(g(y), x).

(5) f is an isotone map if eY(f(x1), f(x2)) ≥ eX(x1, x2) for all x1, x2X.

(6) f is an antitone map if eY(f(x1), f(x2)) ≥ eX(x2, x1) for all x1, x2X.

(7) f is an embedding map if eY(f(x1), f(x2)) = eX(x1, x2) for all x1, x2X.

If X = Y and eX = eY, we simply denote (eX, f, g) for (eX, f, g, eY). (X, (eX, f, g)) is called a Galois (resp. residuated, dual Galois, and dual residuated) pair.

Remark 2.2. Let (X, eX) and (Y, eY) denote a fuzzy poset and f : XY and g : YX denote maps.

(1) (eX, f, g, eY) is a Galois (resp. dual Galois) connection iff (eY, g, f, eX) is a Galois (resp. dual Galois) connection. (2) (eX, f, g, eY) is a Galois (resp. residuated) connection iff (, f, g, ) is a dual (resp. dual residuated) Galois connection. (3) (eX, f, g, eY ) is a residuated (resp. dual residuated) connection iff (, g, f, ) is a residuated (resp. dual residuated) connection. (4) (eX, f, g, eY) is a Galois (resp. dual Galois) connection iff (eX, f, g, ) is a residuated (resp. dual residuated) connection. (5) (eX, f, g, eY) is a residuated connection iff (eY, g, f, eX) is a dual residuated connection.

Theorem 2.3. Let (X, eX) and (Y, eY) denote a fuzzy poset and f : XY and g : YX denote maps.

(1) (eX, f, g, eY) is a Galois connection if f, g are antitone maps and eY(y, f(g(y))) = eX(x, g(f(x))) = 1. (2) (eX, f, g, eY) is a dual Galois connection if f, g are antitone maps and eY(f(g(y)), y) = eX(g(f(x)), x) = 1. (3) (eX, f, g, eY) is a residuated connection if f, g are isotone maps and eY(f(g(y)), y) = eX(x, g(f(x))) = 1. (4) (eX, f, g, eY) is a dual residuated connection if f, g are isotone maps and eY(y, f(g(y))) = eX(g(f(x)), x) = 1.

Proof. (1) Let (f, g) denote a Galois connection. Since

eY(y, f(x)) = eX(x, g(y)),

we have

1 = eY(f(x), f(x)) = eX(x, g(f(x)))

and

eY(y, f(g(y))) = eX(g(y), g(y)) = 1.

Furthermore,

Conversely,

Similarly, eY(y, f(x)) ≤ eX(x, g(y)).

(2) Since eY(f(x), y) = eX(g(y), x), we have

1 = eY(f(x), f(x)) = eX(g(f(x)), x)

and

eY(f(g(y)), y) = eX(g(y), g(y)) = 1.

Furthermore,

Conversely,

Similarly, eY(f(x), y) ≤ eX(g(y), x).

(3) Since eY(f(x), y) = eX(x, g(y)), we have

1 = eY(f(x), f(x)) = eX(x, g(f(x)))

and

eY(f(g(y)), y) = eX(g(y), g(y)) = 1.

Furthermore,

Conversely,

Moreover,

(4) It is similarly proved as (3).

Example 2.4. Let X = {a, b, c} denote a set and f : XX denote a function as f(a) = b, f(b) = a, f(c) = c. Define a binary operation (called Łukasiewicz conjunction) on L = [0, 1] using

x y = max{0, x + y − 1},

xy = min{1 − x + y, 1}.

(1) Let (X = {a, b, c}, e1) denote a fuzzy poset as follows:

Since e1(x, y) = e1(f(x), f(y)),

e1(x, f(f(x))) = e1(f(f(x)), x) = 1,

then, (e1, f, f) are both residuated and dual residuated connections. Since 0.7 = e1(c, a) e1(f(a, f(c)) = e1(b, c) = 0.5, f is not an antitone map. Hence, (e1, f, f) are neither Galois nor dual Galois connections.

(2) Let (X = {a, b, c}, e2) denote a fuzzy poset as follows:

Since e2(x, y) = e2(f(y), f(x)),

e2(x, f(f(x))) = e2(f(f(x)), x) = 1,

then, (e2, f, f) are both Galois and dual Galois connections. Since 0.7 = e2(c, a) e2(f(c), f(a)) = e2(c, b) = 0.5, f is not an isotone map. Hence, (e2, f, f) are neither residuated nor dual residuated connections.

Definition 2.5. Let RLX×Y denote a fuzzy relation. For each ALX and BLY, we define operations R−1(y, x) = R(x, y) and : LYLX as follows:

Theorem 2.6. Let RLX×Y denote a fuzzy relation. For each ALX and BLY,

Proof. (1) From Lemma 1.3 (13,14), we have

(2)

(3) From Lemma 1.3 (7), we have

From Lemma 1.3 (8), we have

(4) From Lemma 1.3 (8), we have

(6)

(7)

(9) from:

(10) from:

Similarly, from:

Other cases are similarly proved.

Theorem 2.7. Let RLX×Y denote a fuzzy relation, (LX, eLx) and (LY, eLʏ) denote fuzzy posets. We have the following properties.

(1) and are residuated connections.

(2) and are dual residuated connections.

(3) and are Galois connections.

(4) and are dual Galois connections.

Proof. (1) For each CLX, BLY,

(2) For each CLX, BLY,

(3) For each CLX, BLY,

(4) For each CLX, BLY,

For each CLX, BLY,

Other cases are similarly proved.

### 3. Conclusion

In this paper, we investigated the properties of fuzzy Galois (dual Galois, residuated, and dual residuated) connections in a complete residuated lattice L. In particular, we studied fuzzy Galois (dual Galois, residuated, and dual residuated) connections induced by L-fuzzy relations.

In the future, we will investigate the properties using fuzzy connections on algebraic structures and study the fuzzy concept lattices.

### Conflict of Interest

• 1. B？lohlavek R. 1999 “Fuzzy Galois connections,” [Mathematical Logic Quarterly] Vol.45 P.497-504
• 2. B？lohlavek R. 2001 “Lattices of fixed points of fuzzy Galois connections,” [Mathematical Logic Quarterly] Vol.47 P.111-116
• 3. B？lohlavek R. 2004 “Concept lattices and order in fuzzy logic,” [Annals of Pure and Applied Logic] Vol.128 P.277-298
• 4. Garcia J. G., Mardones-Prez I., de Prada Vicente M. A., Zhang D. 2010 “Fuzzy Galois connections categorically,” [Mathematical Logic Quarterly] Vol.56 P.131-147
• 5. Georgescu G., Popescu A. 2004 “Non-dual fuzzy connections,” [Archive for Mathematical Logic] Vol.43 P.1009-1039
• 6. Georgescu G., Popescu A. 2003 “Non-commutative fuzzy Galois connections,” [Soft Computing] Vol.7 P.458-467
• 7. Lai H., Zhang D. 2009 “Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory,” [International Journal of Approximate Reasoning] Vol.50 P.695-707
• 8. Melton A., Schmidt D. A., Strecker G. E., Pitt D., Abramsky S., Poign A., Rydeheard D. 1986 “Galois connections and computer science applications,” [Category Theory and Computer Programming, Lecture Notes in Computer Science] Vol.240 P.299-312
• 9. Wille R., Rival I. 1982 “Restructuring lattice theory: an approach based on hierarchies of concepts,”;NATO Advanced Study Institutes Series [Proceedings of the NATO Advanced Study Institute] Vol.83 P.445-470
• 10. Yao W., Lu L. X. 2009 “Fuzzy Galois connections on fuzzy posets,” [Mathematical Logic Quarterly] Vol.55 P.105-112
• 11. Hajek P. 1998 Metamathematics of Fuzzy Logic
• 12. Turunen E. 1999 Mathematics Behind Fuzzy Logic
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