[IEEE 2010 Third International Workshop on Advanced Computational Intelligence (IWACI 2010) - Suzhou, Jiangsu (2010.08.25-2010.08.27)] Third International Workshop on Advanced Computational Intelligence - L-Chu correspondences and Galois functor

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  • L-Chu Correspondences and Galois Functor

    Ondrej Krdlo and Stanislav Krajci

    Abstract This paper provides a categorical view on L-fuzzyconcept lattices inspired by paper [8]. Hideo Mori proved thata category of Chu correspondences between classical formalcontexts called ChuCors is equal to a category of join preservingmappings between complete lattices called Slat . Mori provedthe equality by introducing a Galois functor from ChuCors toSlat , that is an equivalence functor. We show that the Galoisfunctor for L-fuzzy extension of the category ChuCors is notthe equivalence functor to Slat .

    I. INTRODUCTION

    FORMAL CONCEPT ANALYSIS (FCA) introduced byGanter and Wille [5] as applied Lattice theory [4],dealing with conceptual and hierarchical structures providedby objectattribute tables. L-fuzzy extension of basic notionsof FCA introduced by Belohlavek [1], [2] by substitutingclassical relation between objects-attribute sets by L-fuzzybinary relation, i.e. by mapping from set of all object-attribute pairs into a structure of truth values, completeresiduated lattice.

    II. PRELIMINARIESDefinition 1: An algebra L,,,,, 0, 1 is said to be

    a complete residuated lattice if1) L,,, 0, 1 is a complete bounded lattice with the

    least element 0 and the greatest element 1,2) L,, 1 is a commutative monoid,3) and are adjoint, i.e. a b c if and only if

    a b c, for all a, b, c L, where is the orderingin the lattice generated from and .

    We will use the notation Y X to refer to the set of mappingsfrom X to Y .

    Let L be a complete residuated lattice, an L-fuzzy contextis a triple B,A, r consisting of a set of objects B, a set ofattributes A and an L-fuzzy binary relation r, i.e. a mappingr : BA L, which can be alternatively understood as anL-fuzzy subset of B A

    Consider an L-fuzzy context B,A, r. A pair of mappings : LB LA and : LA LB can be defined for everyf LB and g LA as follows:

    f(a) =oB

    (f(o) r(o, a)

    )(1)

    g(o) =aA

    (g(a) r(o, a)

    )(2)

    Lemma 1: Let L be a complete residuated lattice, let r LBA be an L-fuzzy relation between B and A. Then the

    Ondrej Krdlo and Stanislav Krajci are with the Institute of ComputerScience, The University of P. J. Safarik of Kosice, Slovakia (email:o.kridlo@gmail.com, stanislav.krajci@upjs.sk).

    This work was supported by a VEGA grant 1/0131/09.

    pair of operators and form a Galois connection betweenLB ; and LA;, that is, : LB LA and : LA LB are antitonic and, furthermore, for all f LB and g LA we have f f and g g.

    Definition 2: Consider an L-fuzzy context C = B,A, r.An L-fuzzy set of objects f LB (resp. an L-fuzzy set ofattributes g LA) is said to be closed in C iff f = f(resp. g = g).

    Lemma 2: Under the conditions of Lemma 1, the fol-lowing equalities hold for arbitrary f LB and g LA, f = f and g = g, that is, both f and g areclosed in C.

    Definition 3: An L-fuzzy concept is a pair f, g suchthat f = g, g = f . The first component f is said to bethe extent of the concept, whereas the second component gis the intent of the concept.

    The set of all L-fuzzy concepts associated to a fuzzycontext (B,A, r) will be denoted as CLL(B,A, r).

    An ordering between L-fuzzy concepts is defined as fol-lows: f1, g1 f2, g2 if and only if f1 f2 if and onlyif g1 g2.

    Theorem 1: The poset (CLL(B,A, r),) is a completelattice where

    jJfj , gj =

    jJ

    fj , (

    jJfj)

    jJfj , gj =

    (

    jJgj),

    jJ

    gj

    .

    III. L-CHU CORRESPONDENCES BETWEEN L-CONTEXTS

    A notion of Chu correspondence introduced by Mori [8]as a pair of multifunctions between object sets of two formalcontexts and attribute sets in opposite way. The L-fuzzyextension of all basic notions of Chu correspondences isdescribed in [12].

    Definition 4: Consider two L-fuzzy contexts Ci =Bi, Ai, ri, (i = 1, 2), then the pair = (l, r) is calleda correspondence from C1 to C2 if l and r are L-multifunctions, respectively, from B1 to B2 and from A2to A1 (that is, l : B1 LB2 and r : A2 LA1 ).

    459

    Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

    978-1-4244-6337-4/10/$26.00 @2010 IEEE

  • The L-correspondence is said to be a weak L-Chucorrespondence if the equality

    1 (r(a2))(o1) =2 (l(o1))(a2) (3)

    holds for all o1 B1 and a2 A2. A weak Chu correspon-dence is an L-Chu correspondence if l(o1) is closed inC2 and r(a2) is closed in C1 for all o1 B1 and a2 A2.We will denote the set of all Chu correspondences from C1to C2 by L-ChuCors(C1, C2).

    IV. CONNECTIONS OF RIGHT AND LEFT SIDES OF L-CHUCORRESPONDENCES

    Now some technical properties of left and right side ofany L-Chu correspondence, which will be very helpful fordefining a category of L-Chu correspondences between L-fuzzy formal contexts.

    Definition 5: Consider a mapping $ : X LY . Letsdefine new mappings $ : LX LY and $ : LY LXfor all f LX and g LY put

    1) $(f)(y) =

    xX(f(x)$(x)(y))2) $(g)(x) =

    yY $(x)(y) g(y)

    Lemma 3: Let Ci = Bi, Ai, ri for i = 1, 2 be L-fuzzycontexts. Let = (l, r) L-ChuCors (C1, C2). Then forall f LB1 and g LA2 holds

    2 (l(f)) = r(1 (f)) (4)

    1 (r(g)) = l (2 (g)) (5)Proof: Let a2 A2

    2 (l(f))(a2)

    =

    o2B2

    ( o1B1

    (f(o1) l(o1)(o2)

    ) r2(o2, a2)

    )((iI

    ai) a =iI

    (ai a))

    =

    o2B2

    o1B1

    ((f(o1) l(o1)(o2)

    ) r2(o2, a2)

    )((a b) c = (a (b c)))

    =

    o2B2

    o1B1

    (f(o1)

    (l(o1)(o2) r2(o2, a2)

    ))=

    o1B1

    (f(o1)

    o2B2

    (l(o1)(o2) r2(o2, a2)

    ))=

    o1B1

    (f(o1)2 (l(o1))(a2)

    )(by definition of L-Chu correspondence)

    =

    o1B1

    (f(o1)1 (r(a2))(o1)

    )=

    o1B1

    (f(o1)

    a1A1

    (r(a2)(a1) r1(o1, a1)

    ))=

    o1B1

    a1A1

    (f(o1)

    (r(a2)(a1) r1(o1, a1)

    ))

    =

    o1B1

    a1A1

    ((f(o1) r(a2)(a1)

    ) r1(o1, a1)

    )=

    o1B1

    a1A1

    ((r(a2)(a1) f(o1)

    ) r1(o1, a1)

    )=

    o1B1

    a1A1

    (r(a2)(a1)

    (f(o1) r1(o1, a1)

    ))=

    a1A1

    (r(a2)(a1)

    o1B1

    (f(o1) r1(o1, a1)

    ))=

    a1A1

    (r(a2)(a1)1 (f)(a1)

    )= r(1 (f)))(a2)

    Second equation can be proved symmetrically.

    Corollary 1: Let Ci = Bi, Ai, ri for i = 1, 2 be L-contexts. If = (l, r) L-ChuCors (C1, C2), then forall o1 B1 and a2 A2 holds

    l(o1) =2 (r(1 (o1)))

    and

    r(a2) =1 (l (2 (a2))).Proof: By definition of (.) operator holds l(o1) =

    l(o1). By lemma 3 2 (l(o1)) = r(1 (o1)). Bydefinition of Chu correspondences l(o1) is closed in C2.l(o1) =2 (2 (l(o1))) =2 (r(1 (o1))).

    Second equation can be proved similarly.Lemma 4: Let Ci = Bi, Ai, ri be an L-context and

    L-ChuCors (C1, C2). Then

    2 (l(11 (iI

    fi)))(a2) =2 (iI

    l(fi))(a2) (6)

    1 (r(22 (iI

    gi)))(o1) =1 (iI

    r(gi))(o1) (7)

    Proof:

    2 (l(11 (iI

    fi)))(a2)

    =2 (l(1 (iI1 (fi))))(a2)

    by lemma 3

    = r(11 (iI1 (fi)))(a2)

    iI1 (fi) is closed in C1

    = r(iI1 (fi))(a2)

    =

    a1A1

    (r(a2)(a1)iI1 (fi)(a1))

    =iI

    a1A1

    (r(a2)(a1)1 (fi)(a1))

    460

  • =iI

    r(1 (fi))(a2)

    =iI2 (l(fi))(a2)

    =2 (iI

    l(fi))(a2)

    V. CATEGORY L-ChuCors

    Now a category of L-Chu correspondences between L-fuzzy formal contexts will be showed. objects L-fuzzy formal contexts arrows L-Chu correspondences identity arrow : C C of L-context C = B,A, r

    l(o) = (o), for all o B r(a) = (a), for all a A

    composition 2 1 : C1 C3 of arrows1 : C1 C2, 2 : C2 C3, where Ci = Bi, Ai, ribe L-fuzzy formal contexts for i {1, 2}

    (21)l : B1 LB3 and (21)r : A3 LA1 (2 1)l(o1) =33 (2l(1l(o1)))

    (2 1)r(a3) =11 (1r(2r(a3)))

    where

    2l(1l(o1))(o3) =

    o2B2

    1l(o1)(o2) 2l(o2)(o3)

    and

    1r(2r(a3))(a1) =

    a2A2

    2r(a3)(a2)1r(a2)(a1)

    associativity of composition3 (2 1) = (3 2) 1 for all 1, 2, 3 L-ChuCors such that i : Ci Ci+1 for i {1, 2, 3},where C1, C2, C3, C4 are L-fuzzy contexts.

    Proposition 1: Let C = B,A, r be the L-fuzzy formalcontext. The identity arrow : C C defined above is anL-Chu correspondence from C to C.

    Proof:

    (l(o))(a)

    =bB

    ( (o)(b) r(b, a)) = (o)(a) = (o)(a)

    =bB

    (o(b) r(b, a)) = r(o, a)

    =dA

    (a(d) r(o, d)) = (a)(o)

    = (a)(o) =dA

    ( (a)(d) r(o, d))

    = (r(a))(o)

    Hence is a weak chu correspondence, but l(o) and r(a)are closed in C for all o B and all a A, so L-ChuCors (C,C).

    Proposition 2: Let Ci = B,A, r, i {1, 2}be the L-fuzzy formal contexts. For identity arrowsof L-ChuCors i : Ci Ci, i {1, 2} and L-ChuCors (C1, C2) holds: (2 ) = and ( 1) = .

    Proof: (2 )l(o2) =22 (2l(l(o1)))(o2)

    2 (2l(l(o1)))(a2)

    =

    o2B2

    (2l(l(o1))(o2) r2(o2, a2))

    =

    o2B2

    (

    b2B2

    (2l(b2)(o2) l(o1)(b2)) r2(o2, a2))

    =

    o2B2

    b2B2

    ((2l(b2)(o2) l(o1)(b2)) r2(o2, a2))

    =

    o2B2

    b2B2

    (l(o1)(b2) (2l(b2)(o2) r2(o2, a2)))

    =

    b2B2

    (l(o1)(b2)

    o2B2

    (2l(b2)(o2) r2(o2, a2)))

    =

    b2B2

    (l(o1)(b2)2 (2l(b2))(a2))

    =

    b2B2

    (l(o1)(b2)222 ({b2})(a2))

    =

    b2B2

    (l(o1)(b2)2 ({b2})(a2))

    =

    b2B2

    (l(o1)(b2) r2(b2, a2))

    =2 (l(o1))(a2)

    So (2 )l(o2) =22 (l(o1)) = l(o1). Second equationcan be proved similarly.

    Proposition 3: Let Ci = Bi, Ai, ri be an L-fuzzy for-mal context for all i {1, 2, 3}. Let j : Cj Cj+1be an L-Chu correspondence for all j {1, 2}, thencomposition 2 1 : C1 C3 defined above is an L-Chucorrespondence.

    Proof: Let o1 B1 be an arbitrary object of C1 anda3 A3 be an arbitrary attribute of C3

    1 ((2 1)r(a3))(o1)=1 (11 (1r(2r(a3))))(o1)

    =

    a1A1

    (11 (1r(2r(a3)))(a1) r1(o1, a1))

    =111 (1r(2r(a3)))(o1)=1 (1r(2r(a3)))(o1)

    =

    a1A1

    (1r(2r(a3))(a1) r1(o1, a1))

    =

    a1A1

    (

    a2A2

    (2r(a3)(a2) 1r(a2)(a1)) r1(o1, a1))

    461

  • =

    a1A1

    a2A2

    ((2r(a3)(a2) 1r(a2)(a1)) r1(o1, a1))

    =

    a1A1

    a2A2

    (2r(a3)(a2) (1r(a2)(a1) r1(o1, a1)))

    =

    a2A2

    (2r(a3)(a2)

    a1A1

    (1r(a2)(a1) r1(o1, a1)))

    =

    a2A2

    (2r(a3)(a2)1 (1r(a2))(o1))

    =

    a2A2

    (2r(a3)(a2)2 (1l(o1))(a2))

    =

    a2A2

    (2r(a3)(a2)

    o2B2

    (1l(o1)(o2) r2(o2, a2)))

    =

    a2A2

    o2B2

    (2r(a3)(a2) (1l(o1)(o2) r2(o2, a2)))

    =

    a2A2

    o2B2

    ((2r(a3)(a2) 1l(o1)(o2)) r2(o2, a2))

    =

    a2A2

    o2B2

    ((1l(o1)(o2) 2r(a3)(a2)) r2(o2, a2))

    =

    a2A2

    o2B2

    (1l(o1)(o2) (2r(a3)(a2) r2(o2, a2)))

    =

    o2B2

    (1l(o1)(o2)

    a2A2

    (2r(a3)(a2) r2(o2, a2)))

    =

    o2B2

    (1l(o1)(o2)2 (2r(a3))(o2))

    =

    o2B2

    (1l(o1)(o2)3 (2l(o2))(a3))

    =

    o2B2

    (1l(o1)(o2)

    o3B3

    (2l(o2)(o3) r3(o3, a3)))

    =

    o2B2

    o3B3

    (1l(o1)(o2) (2l(o2)(o3) r3(o3, a3)))

    =

    o2B2

    o3B3

    ((1l(o1)(o2) 2l(o2)(o3)) r3(o3, a3))

    =

    o3B3

    (

    o2B2

    (1l(o1)(o2) 2l(o2)(o3)) r3(o3, a3))

    =

    o3B3

    (2l(1l(o1))(o3) r3(o3, a3))

    =3 (2l(1l(o1)))(a3)=333 (2l(1l(o1)))(a3)

    =

    o3B3

    (33 (2l(1l(o1)))(o3) r3(o3, a3))

    =3 ((2 1)l(o1))(a3)

    Hence the composition is weak L-Chu correspondence. Butleft or right side is closed set of objects in C3 or closed set ofattributes in C1 respectively. So the composition is an L-Chucorrespondence.

    Proposition 4: Let Ci = Bi, Ai, ri for i {1, 2, 3, 4}be the L-fuzzy contexts. i : Ci Ci+1 for i {1, 2, 3}be the Chu correspondences. Then the associativity of com-

    position holds 3 (2 1) = (3 2) 1.Proof:

    ((3 2) 1)l(o1)=44 ((3 2)l(1l(o1)))=44 (44 (3l(2l(1l(o1)))))by the property of closure operator=44 (3l(2l(1l(o1))))by lemma 3=44 (3l(33 (2l(1l(o1)))))=44 (3l((2 1)l(o1))) == (3 (2 1))l(o1)

    VI. GALOIS FUNCTOR

    Lets define a functor that assigns to every L-formal contextC a complete lattice Gal(C), the concept lattice of C andto every L-Chu correspondence assigns a join preservingmapping between concept lattices of contexts.

    Definition 6: Let Ci = Bi, Ai, ri be an L-formal con-text for i {1, 2}, L-ChuCors (C1, C2) and f, 1 (f),2 (g), g be L-concepts from Gal(C1) or Gal(C2) re-spectively. Define < : Gal(C1) Gal(C2) and > :Gal(C2) Gal(C1) by

    (2 (g), g) = 1 (r(g)), 11 (r(g)). (9)

    Lemma 5: Let Ci = Bi, Ai, ri be an L-fuzzy formalcontext for i {1, 2} and L-ChuCors (C1, C2).Assign b LB1A2 as a new L-relation defined by anL-bond ([12],[8]), namely, b(o1, a2) =2 (l(o1))(a2),for all o1 B1 and a2 A2. b, b are up- and down-arrowmappings defined on relation b. For all f LB1 ,g LA2 holds

    2 (l(f)) =b (f) and 1 (r(g)) =b (g).Proof:

    1 (r(g))

    =

    a1A1

    (r(g) r1(o1, a1))

    =

    a1A1

    (

    a2A2

    (r(a2)(a1) g(a2))) r1(o1, a1))

    =

    a1A1

    a2A2

    ((r(a2)(a1) g(a2)) r1(o1, a1))

    =

    a1A1

    a2A2

    ((g(a2) r(a2)(a1)) r1(o1, a1))

    =

    a1A1

    a2A2

    (g(a2) (r(a2)(a1) r1(o1, a1)))

    462

  • =

    a2A2

    (g(a2)

    a1A1

    (r(a2)(a1) r1(o1, a1)))

    =

    a2A2

    (g(a2)1 (r(a2))(o1))

    =

    a2A2

    (g(a2) (o1)(a2))

    =

    a2A2

    (g(a2) b(o1, a2))

    =b (g)(o1)

    The second equation can be proved similarly.

    Lemma 6: For all f LB1 and g LA2 holds

    f 1 (r(g)) g 2 (l(f)).Proof:

    1 (r(g))=b (g)

    =

    a2A2

    (g(a2) b(o1, a2))

    now precondition will be used

    a2A2

    (2 (l(f))(a2) b(o1, a2))

    from lemma 5

    =

    a2A2

    (b (f)(a2) b(o1, a2))

    =bb (f)(o1) f(o1)

    Similar.

    Proposition 5: Pair of mappings for any L-ChuCors (C1, C2), where Ci = Bi, Ai, ri be L-fuzzyformal context for i {1, 2}, forms a Galois connectionbetween concept lattices of C1 and C2, namely, for all f LB1 closed in C1 and g LA2 closed in C2 holds

    f, 1 (f) 1 (r(g)), 11 (r(g))

    22 (l(f)), 2 (l(f)) 2 (g), g

    .Proof: Equivalence above is equal to equivalence

    f 1 (r(g)) g 2 (l(f)),

    which holds from lemma 6.

    Lemma 7: Let Ci = Bi, Ai, ri be an L-context for i {1, 2}. For every L-ChuCors (C1, C2) holds

    1) < is supremum-preserving,2) > is infimum-preserving.

    Proof:1

    From lemma 4

    assigns a meet of L-concepts

    iI 1

    (r(gi)), 11 (

    iI gi) from CLL(C1).

    Lemma 8: Let C = B,A, r be an L-context. For L-ChuCors (C,C), : C C, an identity arrow of categoryL-ChuCors holds for all f LB and all g LA

    (l(f)) = (f) and (r(g)) = (g).Proof: Omited due to lack of space.

    Definition 7: Define a mapping Gal, which to every L-context C assigns a complete lattice of L-concepts of C, andto every L-Cu correspondence assigns asigns a supremum-preserving mapping

  • TABLE ITABLES WITH ORDER RELATION AND CONNECTIVES AND

    1 a b c d 01 a b c d 0

    1 a b c d 01 1 a b c d 0a a a c c d 0b b c b c 0 0c c c c c 0 0d d d 0 0 d 00 0 0 0 0 0 0

    1 a b c d 01 1 a b c d 0a 1 1 b b 0 0b 1 a 1 a d dc 1 1 1 1 d dd 1 1 b b 1 b0 1 1 1 1 1 1

    f, (f) CLL(C1)

    (2 1)

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