Join Preserving Maps and Various Concepts

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 5, 010, no. 5, Join Preserving Maps and Various Concepts Yong Chan Kim Department of Mathematics, Gangneung-Wonju University Gangneung, Gangwondo 10-70, Korea Jin Won Park Department of Mathematics Educations, Cheju National University Jeju, Jejudo, , Korea Abstract In this paper, we investigate the properties of isotone (antitone) Galois connection. Moreover, we show that join preserving maps induce formal, attribute oriented and object oriented concepts on a complete residuated lattice. Mathematics Subject Classification: 03E7, 03G5 Keywords: Complete residuated lattices, join preserving maps, isotone (antitone) Galois connetion, formal (resp. attribute oriented, object oriented) concepts 1 Introduction Formal concept analysis is an important mathematical tool for data analysis and knowledge processing [1-4,7,9]. A formal concept consists of (X, Y, R) where X is a set of objects, Y is a set of attributes and R is a relation between X and Y. Bělohlávek [1-4] developed the notion of formal concepts with R L X Y on a complete residuated lattice L. In this paper, we investigate the properties of isotone (antitone) Galois connections. Using their properties, we define formal, attribute oriented and object oriented concepts on a complete residuated lattice. Moreover, we show that join preserving maps induce formal, attribute oriented and object oriented concepts on a complete residuated lattice.

2 44 Yong Chan Kim and Jin Won Park Preliminaries Definition.1 [4,8,9] A triple (L,, ) is called a complete residuated lattice iff it satisfies the following conditions: (L1) L =(L,, 1, 0) is a complete lattice where 1 is the universal upper bound and 0 denotes the universal lower bound; (L) (L,, 1) is a commutative monoid; (L3) is distributive over arbitrary joins, i.e. ( a i ) b = i b). i Γ i Γ(a Define an operation as a b = {c L a c b}, for each a, b L. Example. [4,8,9] (1) Each frame (L,, ) is a complete residuated lattice. () The unit interval with a left-continuous t-norm t, ([0, 1],,t), is a complete residuated lattice. (3) Define a binary operation on [0, 1] by x y = max{0,x+ y 1}. Then ([0, 1],, ) is a complete residuated lattice. Let (L,, ) be a complete residuated lattice. A order reversing map : L L defined by a = a 0 is called a strong negation if a = a for each a L. In this paper, we assume (L,,, ) is a complete residuated lattice with a strong negation. Lemma.3 [4,8,9] For each x, y, z, x i,y i L, we have the following properties. (1) If y z, x y x z, x y x z and z x y x. () x y x y. (3) x ( i Γ y i )= i Γ(x y i ). (4) ( i Γ x i ) y = i Γ(x i y). (5) i Γ y i =( i Γ y i ) and i Γ y i =( i Γ y i ). (6) (x y) z = x (y z) =y (x z). (7) x y =(x y ) and x y = y x. 3 Join preserving maps and various concepts Definition 3.1 Let X and Y be two sets. Let ω,φ,ξ : L X L Y and ω,φ,ξ : L Y L X be operators. (1) The pair (φ,φ ) is called an isotone Galois connection between X and Y if for μ L X and ρ L Y, φ (μ) ρ iff μ φ (ρ). Moreover, the

3 Join preserving maps and various concepts 45 pair (ξ,ξ ) is called an isotone Galois connection between X and Y if for μ L X and ρ L Y, ξ (ρ) μ iff ρ ξ (μ). () The pair (ω,ω ) is called antitone Galois connection between X and Y if for μ L X and ρ L Y, ρ ω (μ) iffμ ω (ρ). Theorem 3. Let φ : L X L Y and φ : L Y L X be operators. Let (φ,φ ) be an isotone Galois connection between X and Y. For each μ, μ i L X and ρ, ρ j L Y, the following properties hold: (1) μ φ (φ (μ)) and φ (φ (ρ)) ρ. () If μ 1 μ, then φ (μ 1 ) φ (μ ). Moreover, if ρ 1 ρ, then φ (ρ 1 ) φ (ρ ). (3) φ (φ (φ (ρ))) = φ (ρ) and φ (φ (φ (μ))) = φ (μ). (4) φ ( i I μ i )= i I φ (μ i ) and φ ( j J ρ j )= j J φ (ρ j ). (5) If φ (φ (μ i )) = μ i, then φ (φ ( i I μ i )) = i I μ i, (6) If (φ (φ (ρ j )) = ρ j, then φ (φ ( j J ρ j )) = j J ρ j. Proof. (1) Since φ (μ) φ (μ), we have μ φ (φ (μ)). Since φ (ρ) φ (ρ), we have φ (φ (ρ)) ρ. () Since μ 1 μ φ (φ (μ )), φ (μ 1 ) φ (μ ). Since φ (φ (ρ 1 )) ρ 1 ρ, φ (ρ 1 ) φ (ρ ). (3) It easily proved from (1) and (). (4) By (), φ ( i I μ i ) i I φ (μ i ). Since φ (μ i ) i I φ (μ i ) implies μ i φ ( i I φ (μ i )), we have i I μ i φ ( i I φ (μ i )). Hence φ ( i I μ i ) i I φ (μ i ). (5) By (1), φ (φ ( μ i )) μ i and by (), μ i = φ (φ (μ i )) φ (φ ( μ i )). i I i I i I Hence, φ (φ ( i I μ i )) = i I μ i. (6) It is similarly proved as (5). Definition 3.3 Let ω,φ,ξ : L X L Y and ω,φ,ξ : L Y L X be functions. A pair (μ, ρ) L X L Y is called: (1) formal concept if ρ = ω (μ) and μ = ω (ρ) where (ω,ω )isan antitone Galios connection, () an attribute oriented concept if ρ = φ (μ) and μ = φ (ρ) where (φ,φ ) is an isotone Galios connection, (3) an object oriented concept if ρ = ξ (μ) and μ = ξ (ρ) where (ξ,ξ ) is an isotone Galios connection. Definition 3.4 An operator φ : L X L Y is called a join-generating operator, denoted by φ J(X, Y ), if φ( i Γ λ i )= i Γ φ(λ i ), for {λ i } i Γ L X. An operator ψ : L X L Y is called a meet-generating operator, denoted by φ M(X, Y ), if ψ( i Γ λ i )= i Γ ψ(λ i ), for {λ i } i Γ L X.

4 46 Yong Chan Kim and Jin Won Park Theorem 3.5 For φ J(X, Y ), Define functions ωφ,ξ φ : LX L Y and φ,ωφ,ξ φ : LY L X as follows:, for all λ L X,ρ L Y, φ (ρ) = {λ L X φ (λ) ρ}, ω φ (λ) =(φ (λ)), ω φ (ρ) =φ (ρ ), ξ φ (ρ) = {λ L X φ (λ ) ρ }, ξ φ (λ) = {ρ L Y ξ φ (ρ) λ}, Then the following properties hold: (1) The pair (φ,φ ) is an isotone Galois connection and (φ (φ (λ)),φ (λ)) for each λ L X are attribute oriented concepts with φ M(Y,X). () If α φ (λ) φ (α λ) for λ L X, then α φ (ρ) φ (α ρ) for ρ L Y. (3) The pair (ω φ,ω φ ) is an antitone Galois connection and (ω φ (ω φ (λ)),ω φ (λ)) for λ L X are formal concepts. (4) If ω φ (α μ) α ω φ (μ), then ω φ (α ρ) α ω φ (ρ). (5) ξ φ J(Y,X) such that ξ φ (ρ) =(φ (ρ )) and φ (λ) ρ λ φ (ρ) ξ φ (ρ ) λ If α φ (λ) φ (α λ), then α ξφ (ρ) ξ φ (α ρ). (6) ξφ M(X, Y ) such that ξ φ (λ) =(φ (λ )) and φ(λ) ρ λ φ (ρ) ξ φ (ρ ) λ ρ ξ φ (λ ) If α ξφ (ρ) ξ φ (α ρ), then α ξ φ (λ) ξ φ (α λ). (7) The pair (ξφ,ξ φ ) is an isotone Galois connection and (ξ φ (ρ),ξ φ (ξ φ (ρ))) for each ρ L Y, are object oriented concepts. Proof. (1) Since φ is a join preserving map and φ (ρ) = {λ L X φ (λ) ρ}, we have φ (λ) ρ λ φ (ρ). Hence (φ,φ ) is an isotone Galois connection and, by Theorem 3.(3), (φ (φ (λ)),φ (λ)) are attribute oriented concepts. Moreover, φ ( i Γ ρ i )= i Γ φ (ρ i ) from i Γ φ (ρ i ) λ φ (ρ i ) λ, i Γ φ (λ) ρ i, i Γ φ (λ) i Γ ρ i, φ ( i Γ ρ i ) λ. () Let α φ (λ) φ (α λ) for λ L X. For μ α φ (ρ), μ α φ (ρ) iffφ (μ α) ρ. Then α φ (μ) ρ iff φ (μ) α ρ iff μ φ (α ρ). Hence α φ (ρ) φ (α ρ).

5 Join preserving maps and various concepts 47 (3) It follows from ωφ (ω φ (ω φ (λ))) = ω φ (ω φ (φ (λ) )) = ωφ (φ (φ (λ))) = (φ (φ (φ (λ)))) =(φ (λ)) = ωφ (λ) and ρ ω φ (λ) ρ (φ (λ)) φ (λ) ρ λ φ (ρ )=ω φ (ρ). (4) Let ωφ (α μ) =(φ (α μ)) α ωφ (μ) =α (φ (μ)) = (α φ (μ)) from Lemma.3(7). So, α φ (λ) φ (α λ). By (), α φ (ρ) φ (α ρ). ωφ (α ρ) =φ ((α ρ) )=φ (α ρ ) α φ (ρ )=α ωφ (ρ). (5) By Lemma.3(5), we have ξ φ (ρ) = {λ L X φ (λ ) ρ } = ( {λ L X λ φ (ρ )} ) =(φ (ρ )). It follows ξ φ J(Y,X) and φ (λ) ρ λ φ (ρ) ξ φ (ρ ) λ. By () and Lemma.3(7), we have: ξ φ (α ρ) = (φ ((α ρ) )) =(φ (α ρ )) (α φ (ρ )) = α (φ (ρ )) = α ξ φ (ρ). ξ φ (6) ξφ (λ) = {ρ L Y ξφ (ρ) λ} = {ρ L Y φ (λ ) ρ } =(φ(λ )). M(X, Y ) and other cases are similarly proved as (1) and (5). (7) ξφ (ξφ (ξφ (ρ))) = ξφ (ξφ (φ (ρ )) )=ξφ ((φ (φ (ρ ))) ) =(φ (φ (φ (ρ )))) =(φ (ρ )) = ξφ (ρ). Other cases are easily proved. Theorem 3.6 Let X and Y be sets and R L X Y. Define a function φ R : LX L Y as φ R (λ)(y) = x X(λ(x) R(x, y)). Then we have the following properties: (1) the pair (φ R,φ R ) is an isotone Galois connection which (φ R (φ R (λ)),φ R (λ)), for each λ L X, are attribute oriented concepts such that φ R (ρ)(x) = (R(x, y) ρ(y)). y Y () α φ R (λ) =φ R (α λ), α φ R (ρ) =φ R (α ρ) =φ α R (ρ). (3) The pair (ξφ R,ξφ R ) is an isotone Galois connection which (ξφ R (ρ),ξφ R (ξφ R (ρ))), for each ρ L X, are object oriented concepts with ξφ R (ρ)(x) = (R(x, y) ρ(y)), ξφ R (μ)(y) = (R(x, y) μ(x)). y X x X

6 48 Yong Chan Kim and Jin Won Park (4) ωφ R (μ) =(φ R (μ)) = ξφ R (μ ), ωφ R (ρ) =φ R (ρ )=(ξφ R (ρ)), ωφ R (α μ) =α ωφ R (μ) and ωφ R (α ρ) =α ωφ R (ρ) where ωφ R (μ)(y) = (μ(x) R (x, y)), ωφ R (ρ)(x) = (ρ(y) R (x, y)). y Y (5) The pair (ω φ R,ω φ R ) is an antitone Galois connection where (ω φ R (ω φ R (λ)),ω φ R (λ)) for each λ L X are attribute concepts. y Y Proof. (1) φ R is join preserving because φ R ( i λ i )(y) = x X( i λ i (x)) R(x, y) = i( x X λ i (x) R(x, y)) = i φ R (λ i)(y). By Theorem 3.5, we obtain φ R as follows φ R (ρ)(x) = {λ(x) L X φ R (λ) ρ} = {λ(x) L X λ(x) (R(x, y) ρ(y))} = (R(x, y) ρ(y)) Thus, (φ R,φ R ) is an isotone Galois connection. () From Lemma.3(3,6), φ R (α ρ)(x) = y Y (R(x, y) (α ρ(y))) = y Y (α (R(x, y) ρ(y))) = α y Y (R(x, y) ρ(y)) = α φ R (ρ)(x) = y Y (α R(x, y) ρ(y)) = φ α R(ρ)(x). (3) From Theorem 3.5(5) and Lemma.3(6), φ R (λ )(y) ρ (y) λ (x) R(x, y) ρ (y) x X R(x, y) λ (x) ρ (y) λ(x) y Y R(x, y) ρ(y). It follows ξ φ R (ρ)(x) = {λ(x) φ R (λ ) ρ } = y Y R(x, y) ρ(y) Since ξ φ R (λ) = ( φ R (λ ) ) from Theorem 3.5(6), by Lemma.3(6,7), we have from: ξ φ R (λ) = ( φ R (λ ) ) = ( x X(R(x, y) λ (x)) ) = x X(R(x, y) λ (x)) = x X(R(x, y) λ(x)). Other cases follows from Theorem 3.5 (5-7).

7 Join preserving maps and various concepts 49 (4) ω φ R (μ)(y) = (φ R (μ)) (y) =( x X μ(x) R(x, y)) = x X(μ(x) R(x, y)) = x X(R(x, y) μ (x) =ξ φ R (μ )(y) = x X(μ(x) R (x, y)) (by Lemma.3(7)). ω φ R (ρ)(x) = φ R (ρ )(x) =( y Y (R(x, y) ρ (y)) =( y Y (ρ(y) R(x, y))) =(ξ R (ρ)) = x X(ρ(y) R (x, y)) (by Lemma.3(7)). Other cases follows from Theorem 3.5 (3). Example 3.7 Let X = {a, b} and Y = {u, v, w} be sets. Let (L = {0, 1, 1}, ) be a complete residuated lattice such that x y =(x + y 1) 1, x y =(1 x + y) 1, x =1 x and R L X Y as R = ( For λ L X,ρ L Y, we denote (λ(a),λ(b)) and (ρ(u),ρ(v),ρ(w)), we obtain; φ R (0, 0) = (0, 0, 0) φ R (0, 1 )=(1, 0, 0) φ R (0, 1) = (1, 0, 1 ) φ R ( 1, 0) = ( 1, 0, 1 ) φ R ( 1, 1 )=(1, 0, 1 ) φ R ( 1, 1) = (1, 0, 1 ) φ R (1, 0) = (1, 1, 1) φ R (1, 1 )=(1, 1, 1) φ R (1, 1) = (1, 1, 1) φ R (0, 0, 0) = (0, 0) φ R (0, 0, 1)=(0, 0) φ R (0, 0, 1) = (0, 0) φ R (0, 1, 0) = (0, 0) φ R (0, 1, 1)=(0, 0) φ R (0, 1, 1 )=(0, 0) φ R (0, 1, 0) = (0, 0) φ R (0, 1, 1)=(0, 0) φ R (0, 1, 1) = (0, 0) φ R ( 1, 0, 0) = (0, 1 ) φ R ( 1, 0, 1 )=(1, 1 ) φ R ( 1, 0, 1) = ( 1, 1 ) φ R ( 1, 1, 0) = (0, 1) φ R ( 1, 1, 1 )=(1, 1) φ R ( 1, 1, 1) = ( 1, 1) φ R ( 1, 1, 0) = (0, 1) φ R ( 1, 1, 1 )=(1, 1) φ R ( 1, 1, 1) = ( 1, 1) φ R (1, 0, 0) = (0, 1 ) φ R (1, 0, 1 )=(1, 1) φ R (1, 0, 1) = ( 1, 1 ) φ R (1, 1, 0) = (0, 1) φ R (1, 1, 1 )=(1, 1) φ R (1, 1, 1) = (1, 1) φ R (1, 1, 0) = (0, 1) φ R (1, 1, 1 )=(1, 1) φ R (1, 1, 1) = (1, 1) We obtain an attribute oriented concept family as follows: {((0, 0), (0, 0, 0)), ((0, 1 ), ( 1, 0, 0)), (( 1, 1 ), ( 1, 0, 1 )), (( 1, 1), (1, 0, 1 )), ((1, 1), (1, 1, 1))} )

8 50 Yong Chan Kim and Jin Won Park ξr (0, 0, 0) = (0, 0) ξ R (0, 0, 1 )=(1, 0) ξ R (0, 0, 1) = (1, 1) ξr (0, 1, 0) = (0, 0) ξ R (0, 1, 1 )=(1, 0) ξ R (0, 1, 1) = (1, 1) ξ R (0, 1, 0) = ( 1, 0) ξ R (0, 1, 1 )=(1, 0) ξ R (0, 1, 1) = (1, 1) ξr ( 1, 0, 0) = ( 1, 1 ) ξ R ( 1, 0, 1 )=(1, 1 ) ξ R ( 1, 0, 1) = (1, 1 ) ξ R ( 1, 1, 0) = ( 1, 1) ξ R ( 1, 1, 1 )=(1, 1) ξ R ( 1, 1, 1) = (1, 1) ξr ( 1, 1, 0) = ( 1, 1) ξ R ( 1, 1, 1 )=(1, 1) ξ R ( 1, 1, 1) = (1, 1) ξr (1, 0, 0) = (1, 1) ξr (1, 0, 1 )=(1, 1) ξ R (1, 0, 1) = (1, 1) ξr (1, 1, 0) = (1, 1) ξ R (1, 1, 1)=(1, 1) ξ R (1, 1, 1) = (1, 1) ξr (1, 1, 0) = (1, 1) ξ R (1, 1, 1)=(1, 1) ξ R (1, 1, 1) = (1, 1) ξr (0, 0) = (0, 1, 0) ξ R (0, 1)=(0, 1, 0) ξ R (0, 1) = (0, 1, 0) ξr ( 1, 0) = (0, 1, 1 ) ξ R ( 1, 1 )=(1, 1, 1 ) ξ R ( 1, 1) = ( 1, 1, 1 ) ξr (1, 0) = (0, 1, 1) ξ R (1, 1 )=(1, 1, 1) ξ R (1, 1) = (1, 1, 1) We obtain an object oriented concept family as follows: {((0, 0), (0, 1, 0)), (( 1, 0), (0, 1, 1 )), (( 1, 1 ), ( 1, 1, 1 )), ((1, 1), ( 1, 1, 1)), ((1, 1), (1, 1, 1))} R = ( ωφ R (0, 0) = (1, 1, 1) ωφ R (0, 1 )=(1, 1, 1) ω φ R (0, 1) = (0, 1, 1 ) ωφ R ( 1, 0) = ( 1, 1, 1) ω φ R ( 1, 1 )=(1, 1, 1) ω φ R ( 1, 1) = (0, 1, 1) ωφ R (1, 0) = (0, 1, 0) ω φ R (1, 1)=(0, 1, 0) ω φ R (1, 1) = (0, 1, 0) ωφ R (1, 1, 1) = (0, 0) ωφ R (1, 1, 1 )=(0, 0) ω φ R (1, 1, 0) = (0, 0) ω φ R (1, 1, 1) = (0, 0) ω φ R (1, 1, 1 )=(0, 0) ω φ R (1, 1, 0) = (0, 0) ωφ R (1, 0, 1) = (0, 0) ωφ R (1, 0, 1)=(0, 0) ω φ R (1, 0, 0) = (0, 0) ω φ R ( 1, 1, 1) = (0, 1) ω φ R ( 1, 1, 1 )=(1, 1) ω φ R ( 1, 1, 0) = ( 1, 1) ωφ R ( 1, 1, 1) = (0, 1) ω φ R ( 1, 1, 1 )=(1, 1) ω φ R ( 1, 1, 0) = ( 1, 1) ω φ R ( 1, 0, 1) = (0, 1) ω φ R ( 1, 0, 1 )=(1, 1) ω φ R ( 1, 0, 0) = ( 1, 1) ωφ R (0, 1, 1) = (0, 1 ) ω φ R (0, 1, 1 )=(1, 1) ω φ R (0, 1, 0) = ( 1, 1 ) ωφ R (0, 1, 1) = (0, 1) ω φ R (0, 1, 1 )=(1, 1) ω φ R (0, 1, 0) = (1, 1) ωφ R (0, 0, 1) = (0, 1) ω φ R (0, 0, 1 )=(1, 1) ω φ R (0, 0, 0) = (1, 1) We obtain a formal concept family as follows: {((0, 0), (1, 1, 1)), ((0, 1 ), ( 1, 1, 1)), (( 1, 1 ), ( 1, 1, 1 )), (( 1, 1), (0, 1, 1 )), ((1, 1), (0, 1, 0))} )

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