Fuzzy Connections and Relations in Complete Residuated Lattices

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  • 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

    image

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

    image

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

    image

    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.

    image

    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,

    image

    Conversely,

    image

    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,

    image

    Conversely,

    image

    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,

    image

    Conversely,

    image

    Moreover,

    image

    (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:

    image

    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:

    image

    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:

    image

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

    image

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

    image

    (2)

    image

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

    image

    From Lemma 1.3 (8), we have

    image

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

    image

    (6)

    image

    (7)

    image

    (9) from:

    image
    image
    image

    (10) from:

    image
    image
    image

    Similarly, from:

    image

    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,

    image

    (2) For each CLX, BLY,

    image

    (3) For each CLX, BLY,

    image

    (4) For each CLX, BLY,

    image

    For each CLX, BLY,

    image

    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

    No potential conflict of interest relevant to this article was reported.

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