JOIN-MEET and MEET-JOIN PRESERVING MAPS Yong Chan Kim Department of Mathematics Gangneung-Wonju University Gangneung, Gangwondo 210-702, KOREA

Home , Join and meet

International Journal of Pure and Applied Mathematics Volume 93 No. 1 2014, 121-134 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v93i1.10 ijpam.eu

JOIN-MEET AND MEET-JOIN PRESERVING MAPS Yong Chan Kim Department of Mathematics Gangneung-Wonju University Gangneung, Gangwondo 210-702, KOREA

Abstract: We investigate the properties of join-meet and meet-join preserving maps in complete residuated lattice. In particular, we give their examples.

AMS Subject Classiﬁcation: 03E72, 03G10, 06A15, 06F07 Key Words: join-meet and meet-join preserving maps, fuzzy preorder, complete residuated lattices

1. Introduction

Pawlak [6,7] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. Hájek [2] introduced a complete residuated lattice which is an algebraic structure for many valued logic. By using the concepts of lower and upper approximation operators, information systems and decision rules are investigated in complete residuated lattices [1,2,8,9]. Bˇelohlávek [1] developed the notion of fuzzy contexts using Galois connections with R ∈ LX×Y on a complete residuated lattice. Zhang [10,11] introduced the fuzzy complete lattice which is defined by join and meet on fuzzy posets. It is an important mathematical tool for algebraic structure of fuzzy contexts [1,4-6]. Kim [3] show that join (resp. meet, meet join, join meet) preserving maps and upper (resp. lower, meet join, join meet) approximation maps are equivalent in complete residuated lattices. In this paper, we investigate the properties of join-meet and meet-join preserving maps in complete residuated lattice. In particular, we give their examples. c 2014 Academic Publications, Ltd. Received: January 13, 2014 url: www.acadpubl.eu 122 Y.C. Kim

2. Preliminaries

Definition 1. [1,2] A structure (L, ∨, ∧, ⊙, →, ⊥, ⊤) is called a complete residuated lattice iff it satisfies the following properties: (L1) (L, ∨, ∧, ⊥, ⊤) is a complete lattice where ⊥ is the bottom element and ⊤ is the top element; (L2) (L, ⊙, ⊤) is a monoid; (L3) adjointness properties,i.e.

x ≤ y → z iﬀ x ⊙ y ≤ z.

A map ∗ : L → L deﬁned by a∗ = a → ⊥ is called strong negations if a∗∗ = a.

⊤, if y = x, ∗ ⊥, if y = x, ⊤x(y) = ⊤ (y) = ⊥, otherwise. x ⊤, otherwise. In this paper, we assume that (L, ∨, ∧, ⊙, →,∗ , ⊥, ⊤) be a complete residuated lattice with a strong negation ∗.

Definition 2. [10,11] Let X be a set. A function eX : X × X → L 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, (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 (simply, fuzzy poset).

Example 3. (1) We define a function eL : L×L → L as eL(x, y) = x → y. Then (L, eL) is a fuzzy poset. X X (2) We define a function eLX : L ×L → L as eLX (A, B) = x∈X (A(x) → X B(x)). Then (L , eLX ) is a fuzzy poset from Lemma 10(9). V X Definition 4. [10,11] Let (X, eX ) be a fuzzy poset and A ∈ L . (1) A point x0 is called a join of A, denoted by x0 = ⊔A, if it satisfies (J1) A(x) ≤ eX (x, x0), (J2) x∈X (A(x) → eX (x, y)) ≤ eX (x0, y). A pointV x1 is called a meet of A, denoted by x1 = ⊓A, if it satisfies (M1) A(x) ≤ eX (x1, x), (M2) x∈X (A(x) → eX (y, x)) ≤ eX (y, x1). V X Remark 5. Let (X, eX ) be a fuzzy poset and A ∈ L . JOIN-MEET AND MEET-JOIN PRESERVING MAPS 123

(1) If x0 is a join of A, then it is unique because eX (x0, y) = eX (y0, y) for all y ∈ X, put y = x0 or y = y0, then eX (x0, y0) = eX (y0, x0) = ⊤ implies x0 = y0. Similarly, if a meet of A exist, then it is unique. (2) x0 is a join of A iff x∈X (A(x) → eX (x, y)) = eX (x0, y). (3) x1 is a meet of A iffV x∈X (A(x) → eX (y, x)) = eX (y, x1). V L Remark 6. Let (L, eL) be a fuzzy poset and A ∈ L . (1) Since x0 is a join of A iff x∈L(A(x) → eL(x, y)) = x∈L(A(x) → (x ⇒ y)) = x∈L(x ⊙ A(x)) → y V= eL(x0, y) = x0 → y, thenV x0 = ⊔A = x∈L(x ⊙ A(xW)). W (2) Since x0 is a join of A iff x∈L(A(x) → eL(x, y) = x∈L(A(x) → (y → x)) = x∈L(y → (A(x) → x))V = y → x∈L(A(x) → x)V = y → ⊓A, then ⊓A = Vx∈L(A(x) → x). V V X LX Remark 7. Let (L , eLX ) be a fuzzy poset and Φ ∈ L . (1) Since A∈LX (Φ(A) → eLX (A, B)) = eLX ( A∈LX (Φ(A) ⊙ A),B) = eLX (⊔Φ,B), thenV ⊔Φ = A∈LX (Φ(A) ⊙ A). W (2) Since A∈LX (Φ(WA) → eLX (B,A) = A∈LX eLX (B, (Φ(A) → A)) = eLX (B, A∈LXV(Φ(A) → A)), then ⊓Φ = A∈LVX (Φ(A) → A). V X V Y Definition 8. [10,11] Let (L , eLX ) and (L , eLY ) be fuzzy posets. (1) K : LX → LY is a join-meet preserving map if K(⊔Φ) = ⊓K→(Φ) for LX → all Φ ∈ L , where K (Φ)(B) = K(A)=B Φ(A). (2) M : LX → LY is a meet-joinW preserving map if M(⊓Φ) = ⊔M→(Φ) for LX → all Φ ∈ L , where M (Φ)(B) = M(A)=B Φ(A). W X Y Theorem 9. [3] Let X and Y be two sets. Let (L , eLX ) and (L , eLY ) be fuzzy posets. Then the following statements are equivalent: (1) K : LX → LY is a join-meet preserving map iff K(α ⊙ A) = α → K(A) X and K( i∈I Ai) = i∈I K(Ai) for all A, Ai ∈ L , and α ∈ L. (2) WM : LX →VLY is a meet-join preserving map iff M(α → A) = α → X J (A) and M( i∈I Ai) = i∈I M(Ai) for all A, Ai ∈ L , and α ∈ L. Lemma 10.V [1,2] LetW (L, ∨, ∧, ⊙, →,∗ , ⊥, ⊤) be a complete residuated lat- ∗ tice with a strong negation . For each x, y, z, xi, yi ∈ L, the following properties hold. (1) ⊙ is isotone in both arguments. (2) → is antitone in the first and isotone in the second argument. (3) x → y = ⊤ iff x ≤ y. (4) x → ⊤ = ⊤ and ⊤ → x = x. (5) x ⊙ y ≤ x ∧ y.

(6) x ⊙ ( i∈Γ yi) = i∈Γ(x ⊙ yi) and ( i∈Γ xi) ⊙ y = i∈Γ(xi ⊙ y). W W W W 124 Y.C. Kim

(7) x → ( i∈Γ yi) = i∈Γ(x → yi) and ( i∈Γ xi) → y = i∈Γ(xi → y). (8) i∈Γ xVi → i∈Γ yVi) ≥ i∈Γ(xi → yi).W V (9) (Wx → y) ⊙ xW≤ y and (Vy → z) ⊙ (x → y) ≤ (x → z). (10) x → y ≤ (y → z) → (x → z). ∗ ∗ ∗ ∗ (11) i∈Γ xi = ( i∈Γ xi) and i∈Γ xi = ( i∈Γ xi) . ∗ ∗ (12) (Vx ⊙ y) → zW= x → (y → zW) = y → (xV→ z) and (x ⊙ y) = x → y . (13) x∗ → y∗ = y → x and (x → y)∗ = x ⊙ y∗.

3. Join-Meet and Meet-Join Preserving Maps

X −1 X X Theorem 11. Let (L , eLX ) be a fuzzy poset. Let K, K : L → L be −1 join-meet preserving maps such that K (⊤x)(y) = K(⊤y)(x) for each x, y ∈ X. −1 X X −1 ∗ Let M, M : L → L be meet preserving maps such that M (⊤x)(y) = ∗ ∗ ∗ M(⊤y)(x) and K (⊤x)(y) = M(⊤x)(y) for each x, y ∈ X. For x, y ∈ X, α ∈ L and A, B ∈ LX , we have the following properties. (1) K(A)(y) = x(A(x) → K(⊤x)(y) and V −1 −1 K (A)(y) = (A(x) → K (⊤x)(y) = (A(x) → K(⊤y)(x)) ^x ^x

∗ ∗ −1 ∗ (2) M(A)(y) = x(A (x) ⊙ M(⊤x)(y)) and M (A)(y) = x(A (x) ⊙ −1 ∗ M (⊤x)(y)). W W (3) M(⊤) = M−1(⊤) = ⊥ and M(⊥) = K−1(⊥) = ⊤. (4) M(A) = (K(A∗))∗ and M(A) = (M(A∗))∗. (5) K(α → A) ≥ α ⊙ K(A) and M(α ⊙ A) ≤ α → M(A). ∗ ∗ ∗ ∗ (6) M(⊤x → α)(y) = M(⊤x)(y) ⊙ α = K (⊤x ⊙ α )(y). (7) α∈L((A(y) → α) → α) = A(y). ∗ ∗ (8) VK(A ) = α∈L(M(A ⊙ α ) → α). ∗ ∗ (9) K(α ⊙ A)V = α → K(A) = M(A ) → α . (10) M(α ⊙ A) ≤ K(A∗) → α∗. −1 (11) eLX (B, K(A)) = eLX (A, K (B)) and

−1 eLX (M(A),B) = eLX (M (B),A).

−1 ∗ ∗ (12) eLX (B, K(A)) = eLX (M (B ),A ) and

∗ −1 ∗ eLX (M(A),B) = eLX (A , K (B )).

(13) eLX (A, B) ≤ eLX (K(B), K(A)). (14) eLX (A, B) ≤ eLX (M(B), M(A)). JOIN-MEET AND MEET-JOIN PRESERVING MAPS 125

Proof. (1) For A = x∈X (A(x) ⊙ ⊤x), by Theorem 9(1), we have W K(A)(y) = K( x∈X (A(x) ⊙ ⊤x))(y) = x∈X (A(x) → K(⊤x)(y)) −1 −1 −1 K (A)(y) = K W( x∈X (A(x) ⊙ ⊤x))(y) =V x∈X (A(x) → K (⊤x)(y)) = x∈XW(A(x) → K(⊤y)(x)). V W ∗ ∗ (2) For A = x∈X (A (x) → ⊤x), by Theorem 9(2), we have

V ∗ ∗ ∗ ∗ M(A)(y) = M( x∈X (A (x) → ⊤x))(y) = x∈X (A (x) ⊙ M(⊤x)(y)) −1 ∗ −1 ∗ M (A)(y) = x∈VX (A (x) ⊙ M (⊤x)(y)) W W ∗ ∗ (3) M(⊤)(y) = x(⊤ (x) ⊙ M(⊤x)) = ⊥ and other cases are similarly proved. W (4) By Lemma 10 (11), we have ∗ ∗ ∗ ∗ (K(A )(y)) = x∈X (A (x) → K(⊤x)(y) ∗ ∗ ∗ ∗ = x∈X (A (x) ⊙V K (⊤x)(y)) = x∈X (A (x) ⊙ M(⊤x)(y)) ∗ ∗ = MW ( x∈X (A (x) → ⊤x)(y) = MW (A)(y). V ∗ ∗ ∗ ∗ ∗ ∗ (M(A )(y)) = x∈X (A(x) ⊙ M(⊤x)(y)) = x∈X (A(x) → M (⊤x)(y)) = x∈X (A(x) → KW(⊤x)(y)) = K(A)(y). V V (5) Since α ⊙ (α → A(x)) ⊙ (A(x) → K(⊤x)(y)) ≤ A(x) ⊙ (A(x) → K(⊤x)(y)) ≤ K(⊤x)(y) iﬀ α ⊙ (A(x) → K(⊤x)(y)) ≤ (α → A(x)) → K(⊤x)(y), then α ⊙ K(A) ≤ K(α → A). ∗ ∗ ∗ ∗ ∗ Since α ⊙ (α → A ) ⊙ M(⊤x)(y) ≤ A ⊙ M(⊤x)(y) iﬀ (α → A ) ⊙ ∗ ∗ ∗ M(⊤x)(y) ≤ α → A ⊙ M(⊤x)(y), then M(α ⊙ A) ≤ α → M(A). ∗ ∗ (6) By (4), M(⊤x → α)(z) = K (⊤x ⊙ α )(z) and

∗ ∗ M(⊤x → α)(z) = y∈X (M(⊤y)(z) ⊙ (⊤x → α) (y)) ∗ ∗ ∗ ∗ = y∈X (M(⊤y)(z)W⊙ (⊤x ⊙ α )(y)) = M(⊤x)(z) ⊙ α ∗ ∗ ∗ ∗ ∗ ∗ = WK (⊤x)(z) ⊙ α = (α → K(⊤x)(z)) = K (⊤x ⊙ α )(z).

(7) Since A(y) ⊙ (A(y) → α) ≤ α iﬀ A(y) ≤ (A(y) → α) → α, we have

((A(y) → α) → α) ≥ A(y). α^∈L

Put α = A(y). Then α∈L((A(y) → α) → α) ≤ (A(y) → A(y)) → A(y)) = A(y). Hence α∈L((A(Vy) → α) → α) = A(y). V 126 Y.C. Kim

(8)

∗ α∈L(M(A ⊙ α )(x) → α) ∗ V= α∈L( x∈X (M(⊤x)(y) ⊙ (A → α)(x)) → α) ∗ = Vα∈L Wx∈X (M(⊤x)(y) → ((A(x) → α) → α)) ∗ = Vx∈X (VK (⊤x)(y) → α∈L((A(x) → α) → α)) ∗ ∗ = Vx∈X (K (⊤x)(y) → AV(x)) = x∈X (A (x) → K(⊤x)(y)) ∗ = KV(A )(x). V (9)

α → K(A)(z) = K(α ⊙ A)(z) = x∈X ((α ⊙ A)(x) → K(⊤x)(z)) ∗ ∗ = Vx∈X (K (⊤x)(z) → (α ⊙ A) (x)) ∗ ∗ ∗ = Vx∈X (K (⊤x)(z) ⊙ A (x)) → α ∗ ∗ = Wx∈X (M(⊤x)(z) ⊙ A(x)) → α ∗ ∗ = MW (A )(z) → α ∗ (10) Since (A(x) → α ) ⊙ M(⊤x)(y) ⊙ (M(⊤x)(y) → A(x)) ≤ (A(x) → ∗ ∗ ∗ ∗ α ) ⊙ A(x) ≤ α iﬀ (A(x) → α ) ⊙ M(⊤x)(y) ≤ (M(⊤x)(y) → A(x)) → α , we have ∗ ∗ M(A ⊙ α) = x∈X ((A ⊙ α) (x) ⊙ M(⊤x)(y)) ∗ ∗ ∗ ∗ = x∈X ((A(xW) → α ) ⊙ M(⊤x)(y)) ≤ x∈X ((M(⊤x)(y) → A(x)) → α ) ∗ ∗ ∗ ∗ = Wx∈X ((K (⊤x)(y) → A(x)) → α ) ≤W( x∈X (K (⊤x)(y) → A(x)) → α ∗ ∗ ∗ = (W x∈X (A (x) → K(⊤x)(y))) → α = K(VA ) → α . (11)V By Lemma 10 (12), we have

eLX (B, K(A)) = y∈X (B(y) → K(A)(y)) = Vy∈X (B(y) → x∈X (A(x) → K(⊤x)(y))) = Vy∈X x∈X (B(Vy) → (A(x) → K(⊤x)(y))) = Vx∈X (VA(x) → y∈X (B(y) → K(⊤x)(y))) −1 = VeLX (A, K (B))V.

eLX (M(A),B) = y∈X (M(A)(y) → B(y)) ∗ ∗ = Vy∈X ( x∈X (A (x) ⊙ M(⊤x)(y)) → B(y)) ∗ ∗ = Vy∈X Wx∈X (M(⊤x)(y) → (A (x) → B(y))) ∗ ∗ = Vy∈X Vx∈X (M(⊤x)(y) → (B (y) → A(x))) ∗ ∗ = Vx∈X (V y∈X (B (y) ⊙ M(⊤x)(y)) → A(x)) −1 = Vx∈X (MW (B)(y)) → A(x)) −1 = VeLX (M (B),A). JOIN-MEET AND MEET-JOIN PRESERVING MAPS 127

(12)

eLX (B, K(A)) = y∈X (B(y) → K(A)(y)) = Vy∈X (B(y) → x∈X (A(x) → K(⊤x)(y))) ∗ ∗ = Vy∈X x∈X (B(Vy) → (K (⊤x)(y) → A (x))) ∗ ∗ = Vx∈X (V y∈X (B(y) ⊙ K (⊤x)(y)) → A (x)) −1 ∗ = Vx∈X (Wy∈X (B(y) ⊙ M (⊤y)(x)) → A (x)) −1 ∗ ∗ = VeLX (MW (B ),A ).

eLX (M(A),B) = y∈X (M(A)(y) → B(y)) ∗ ∗ = Vy∈X ( x∈X (A (x) ⊙ M(⊤x)(y)) → B(y)) ∗ ∗ = Vy∈X Wx∈X (A (x) → (M(⊤x)(y) → B(y))) ∗ ∗ ∗ ∗ = Vy∈X Vx∈X (A (x) → (B (y) → M (⊤x)(y))) ∗ ∗ = Vx∈X (VA (x) → y∈X (B (y) → K(⊤x)(y))) ∗ −1 = eVLX (A , K (B))V. (12)

K(A)(y) ⊙ eLX (B,A) = x(A(x) → K(⊤x)(y)) ⊙ z∈X (B(z) → A(z)) ≤ x∈X ((A(x) → K(⊤x)(Vy)) ⊙ (B(x) → A(x)) V ≤ Vx∈X ((B(x) → K(⊤x)(y)) = K(B)(y). V Thus, eLX (B,A) ≤ K(A)(y) → K(B)(y). Hence eLX (B,A) ≤ eLX (K(A), K(B)). (13)

∗ ∗ ∗ M (⊤x)(y) ⊙ A (x)) ⊙ eLX (B,A) ∗ ∗ ∗ ≤ M (⊤x)(y) ⊙ A (x)) ⊙ (B(x) → A(x)) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ = M (⊤x)(y) ⊙ A (x)) ⊙ (A (x) → B ) ≤ M (⊤x)(y) ⊙ B (x). ∗ ∗ ∗ ∗ ∗ ∗ Thus,eLX (B,A) ≤ M (⊤x)(y) ⊙ A (x)) → M (⊤x)(y) ⊙ B (x). Hence

eLX (A, B) ≤ eLX (M(B), M(A)).

X −1 X Theorem 12. Let (L , eLX ) be a fuzzy poset. Let K, K : L → X −1 L be join-meet preserving maps such that K (⊤x)(y) = K(⊤y)(x) for all x, y ∈ X. Let M, M−1 : LX → LX be meet-join preserving maps such that −1 ∗ ∗ X M (⊤x)(y) = M(⊤y)(x) for all x, y ∈ X. For x, y, z ∈ X and A ∈ L , we have the following properties. ∗ ∗ ∗ −1 (1) If ⊤x ≤ M(⊤x), then A ≤ M(A) and A ≤ M (A). 128 Y.C. Kim

∗ ∗ −1 ∗ (2) If K(⊤x) ≤ ⊤x, then K(A) ≤ A and K (A) ≤ A . ∗ ∗ ∗ ∗ (3) y∈X K (⊤x)(y)⊙K (⊤y)(z) ≤ K (⊤x)(z) for all x, z ∈ X iff K(K (⊤x)) −1 ∗ ∗ −1 ≥ K(⊤Wx) iff K (K(⊤x)) ≥ K (⊤x) iff K(K (A)) ≥ K(A) iff K (K(A)) ≥ K∗(A). ∗ ∗ ∗ −1∗ (4) x∈X K (⊤x)(y)⊙K (⊤x)(z) ≤ K (⊤y)(z) for all x, z ∈ X iff K(K (⊤y)) −1 −1∗ −1∗ −1 ≥ K(⊤yW) iff K (K(⊤y)) ≥ K (⊤y) iff K(K (A)) ≤ K(A) iff K (K(A)) ≤ K−1∗(A). −1 ∗ (5) x∈X K(⊤y)(x)⊙K(⊤z)(x) ≤ K(⊤y)(z) for all x, z ∈ X iff K (K (⊤y)) ∗ −1 ∗ ∗ ≥ K(⊤yW) iff K(K(⊤y)) ≥ K (⊤y) iff K (K (A)) ≥ K(A) iff K(K(A)) ≥ K (A). ∗ ∗ ∗ ∗ ∗ (6) y∈X M(⊤x)(y)⊙M(⊤y)(z) ≤ M(⊤x)(z) for all y, z ∈ X iff M(M (⊤x)) ∗ −1 −1∗ ∗ −1 ∗ ∗ ≤ M(⊤Wx) iff M (M (⊤x)) ≤ M (⊤x) iff M(M (A)) ≤ M(A) iff M−1(M−1∗(A)) ≤ M−1(A). ∗ ∗ ∗ (7) x∈X M(⊤x)(y) ⊙ M(⊤x)(z) ≤ M(⊤y)(z) for all x, y ∈ X iff −1∗ ∗ ∗ −1∗ ∗ −1 ∗ −1∗ M(M W(⊤x)) ≤ M(⊤x) iff M(M (⊤z)) ≤ M (⊤z) iff M(M (A)) ≤ M(A) iff M(M−1∗(A)) ≤ M−1(A). ∗ ∗ ∗ (8) x∈X M(⊤y)(x) ⊙ M(⊤z)(x) ≤ M(⊤y)(z) for all y, z ∈ X iff −1 ∗ ∗ −1 ∗ −1 ∗ ∗ ∗ −1 ∗ M (MW(⊤z)) ≤ M (⊤z) iff M (M (⊤y)) ≤ M(⊤y) iff M (M (A)) ≤ M−1(A) iff M−1(M∗(A)) ≤ M(A). ∗ ∗ ∗ (9) If K (⊤x)(y) = M(⊤x)(y) for all x, y ∈ X, then y∈X K (⊤x)(y) ⊙ ∗ ∗ ∗ K (⊤y)(z) ≤ K (⊤x)(z) for all x, z ∈ X iff M(K(⊤xW)) ≤ K (⊤x) iff −1 −1 −1∗ ∗ −1 −1 −1∗ M (K (⊤z)) ≤ K (⊤z) iff M(K(A)) ≤ K (A) iff M (K (A)) ≤ K (A). ∗ ∗ ∗ (10) If K (⊤x)(y) = M(⊤x)(y) for all x, y ∈ X, then x∈X K (⊤x)(y) ⊙ ∗ ∗ K (⊤x)(z) ≤ K (⊤y)(z) for all x, z W ∈ X iff −1 ∗ −1 −1∗ −1 ∗ M(K (⊤y)) ≤ K (⊤y) iff M(K (⊤z)) ≤ K (⊤z) iff M(K (A)) ≤ K (A) iff M(K−1(A)) ≤ K−1∗(A). ∗ ∗ ∗ (11) If K (⊤x)(y) = M(⊤x)(y) for all x, y ∈ X, then x∈X K (⊤y)(x) ⊙ ∗ ∗ −1 ∗ K (⊤z)(x) ≤ K (⊤y)(z) for all x, z ∈ X iff M (K(⊤Wy)) ≤ K (⊤y) iff −1 −1∗ −1 ∗ −1 −1∗ M (K(⊤z)) ≤ K (⊤z) iff M (K(A)) ≤ K (A) iff M (K(A)) ≤ K (A). ∗ ∗ Proof. (1) For A = x∈X (A (x) → ⊤x), we have V ∗ ∗ ∗ ∗ M(A)(y) = M( x∈X (A (x) → ⊤x))(y) = x∈X (A (x) ⊙ M(⊤x)(y)) ∗ ∗ ∗ ≥ x∈VX (A (x) ⊙ ⊤x(y)) = A (y)W. W Similarly, we have M−1(A) ≥ A∗ for each A ∈ LX . (2) For A = x∈X (A(x) ⊙ ⊤x), we have W K(A)(y) = K( x∈X (A(x) ⊙ ⊤x))(y) = x∈X (A(x) → K(⊤x)(y)) ∗ ∗ ≤ xW∈X (A(x) → ⊤x(y)) = A (Vy). V Similarly, we have K−1(A) ≥ A∗ for each A ∈ LX . JOIN-MEET AND MEET-JOIN PRESERVING MAPS 129

∗ ∗ ∗ (3) Since y∈X K (⊤x)(y) ⊙ K (⊤y)(z) ≤ K (⊤x)(z) for all x, z ∈ X, then ∗ ∗ ∗ y∈X K (⊤x)(Wy) → K(⊤y)(z) = K( y∈X K (⊤x)(y) ⊙ ⊤y) = K(K (⊤x))(z) ≥ KV(⊤x)(z) for all x, z ∈ X. W ∗ ∗ ∗ Since y∈X K (⊤x)(y) ⊙ K (⊤y)(z) ≤ K (⊤x)(z) for all x, z ∈ X, then ∗ ∗ ∗ K (⊤x)(y)W≤ K (⊤y)(z) → K (⊤x)(z). ∗ ∗ ∗ K (⊤x)(y) ≤ y∈X (K (⊤y)(z) → K (⊤x)(z)) = y∈X (K(⊤x)(z) → −1 −1 K(⊤y)(z)) = y∈VX (K(⊤x)(z) → K (⊤z)(y)) = K V(K(⊤x))(y). V ∗ ∗ ∗ K(K (A))(z) = K( y∈X (K (A)(y) ⊙ ⊤y)(z) = y∈X (K (A)(y) → K(⊤x)(z) ∗ = yW∈X (( x∈X (A(x) → K(⊤x)(yV))) → K(⊤y)(z) ∗ = Vy∈X ( Vx∈X (A(x) ⊙ K (⊤x)(y)) → K(⊤y)(z) ∗ ∗ ∗ = Vy∈X Wx∈X (K (⊤x)(y) ⊙ K (⊤y)(z) → A (x)) ∗ ∗ ≥ Vy∈X (VK (⊤x)(z) → A (x)) = KV(A)(z).

−1 −1 K (K(A))(z) = K ( y∈X (K(A)(y) ⊙ ⊤y)(z) = y∈X (K(A)(y) −1 →W K (⊤y)(z)) V −1 = y∈X (( x∈X (A(x) → K(⊤x)(y))) → K (⊤y)(z) −1∗ ∗ = Vy∈X (KV (⊤y)(z) → ( x∈X (A(x) ⊙ K (⊤x)(y)))) V W Since

∗ ∗ ∗ ∗ K (⊤x)(z) ⊙ K (⊤z)(y) ⊙ (K (⊤x)(y) → A (x)) ∗ ∗ ∗ ∗ ≤ K (⊤x)(y) ⊙ (K (⊤x)(y) → A (x)) ≤ A (x) ∗ ∗ ∗ ∗ ∗ iﬀ K (⊤z)(y) ⊙ (K (⊤x)(y) → A (x)) ≤ K (⊤x)(z) → A (x)

∗ ∗ −1 −1∗ ∗ K (K(A))(z) = y∈X (K (⊤y)(z) → ( x∈X (A(x) ⊙ K (⊤x)(y)))) −1∗ ∗ ∗ = Vy∈X (K (⊤y)(z) ⊙ x∈WX (K (⊤x)(y) → A (x))) −1∗ ∗ ∗ ≤ Wy∈X x∈X (K (⊤y)(Vz) ⊙ (K (⊤x)(y) → A (x))) ∗ ∗ ≤ Wx∈X (WK (⊤x)(z) → A (x)) = K(A)(z) W Other cases are similarly proved. ∗ ∗ ∗ (4) Since x∈X K (⊤x)(y) ⊙ K (⊤x)(z) ≤ K (⊤x)(z) for all x, z ∈ X, then ∗ −1∗ x∈X K (⊤x)(Wy) → K(⊤x)(z) = K( x∈X K (⊤y)(x) ⊙ ⊤x)(z) = −1∗ KV(K (⊤y))(z) ≥ K(⊤y)(z) for allWy, z ∈ X. ∗ ∗ ∗ Since y∈X K (⊤x)(y) ⊙ K (⊤x)(z) ≤ K (⊤y)(z) for all x, z ∈ X, then ∗ ∗ ∗ K (⊤x)(y)W≤ K (⊤x)(z) → K (⊤y)(z). ∗ ∗ ∗ K (⊤x)(y) ≤ z∈X (K (⊤x)(z) → K (⊤y)(z)) = y∈X (K(⊤y)(z) V V 130 Y.C. Kim

−1 −1 → K(⊤x)(z)) = y∈X (K(⊤y)(z) → K (⊤z)(x)) = K (K(⊤y))(x). V −1∗ −1∗ −1∗ K(K (A))(z) = K( y∈X (K (A)(y) ⊙ ⊤y)(z) = y∈X (K (A)(y) →W K(⊤x)(z) V −1 ∗ = y∈X (( x∈X (A(x) → K (⊤x)(y))) → K(⊤y)(z) −1∗ = Vy∈X ( Vx∈X (A(x) ⊙ K (⊤x)(y)) → K(⊤y)(z) ∗ ∗ ∗ = Vx∈X Wy∈X (K (⊤y)(x) ⊙ K (⊤y)(z) → A (x)) ∗ ∗ ≥ Vx∈X (VK (⊤x)(z) → A (x)) = KV(A)(z).

−1 −1 K (K(A))(z) = K ( y∈X (K(A)(y) ⊙ ⊤y)(z) = y∈X (K(A)(y) −1 →W K (⊤y)(z)) V −1 = y∈X (( x∈X (A(x) → K(⊤x)(y))) → K (⊤y)(z) −1∗ ∗ = Vy∈X (KV (⊤y)(z) → ( x∈X (A(x) ⊙ K (⊤x)(y)))) V W Since ∗ ∗ ∗ ∗ K (⊤z)(x) ⊙ K (⊤z)(y) ⊙ (K (⊤x)(y) → A (x)) ∗ ∗ ∗ ∗ ≤ K (⊤x)(y) ⊙ (K (⊤x)(y) → A (x)) ≤ A (x) ∗ ∗ ∗ ∗ ∗ iﬀ K (⊤z)(y) ⊙ (K (⊤x)(y) → A (x)) ≤ K (⊤z)(x) → A (x)

∗ ∗ −1 −1∗ ∗ K (K(A))(z) = y∈X (K (⊤y)(z) → ( x∈X (A(x) ⊙ K (⊤x)(y)))) −1∗ ∗ ∗ = Vy∈X (K (⊤y)(z) ⊙ x∈WX (K (⊤x)(y) → A (x))) −1∗ ∗ ∗ ≤ Wy∈X x∈X (K (⊤y)(Vz) ⊙ (K (⊤x)(y) → A (x))) ∗ ∗ −1 ≤ Wx∈X (WK (⊤z)(x) → A (x)) = K (A)(z) W Other cases are similarly proved. (5) It is similarly proved as (4). ∗ ∗ ∗ ∗ (6) For M(⊤x) = y∈X (M (⊤x)(y) → ⊤y), we have V ∗ ∗ ∗ y∈X M(⊤x)(y) ⊙ M(⊤y)(z) ≤ M(⊤x)(z) ∗ ∗ ∗ iﬀW M( y∈X (M(⊤x)(y) → ⊤y)(z) ≤ M(⊤x)(z) ∗ ∗ ∗ iﬀ M(VM (⊤x))(z) ≤ M(⊤x)(z)

∗ ∗ ∗ y∈X M(⊤x)(y) ⊙ M(⊤y)(z) ≤ M(⊤x)(z) −1 ∗ −1 ∗ ∗ iffW y∈X M (⊤y)(x) ⊙ M (⊤z)(y) ≤ M(⊤x)(z) −1 −1 ∗ ∗ −1 ∗ iff MW ( y∈X (M (⊤z)(y) → ⊤y)(x) ≤ M (⊤z)(x) −1 −1∗ ∗ −1 ∗ iff M (MV (⊤z))(x) ≤ M (⊤z)(x) JOIN-MEET AND MEET-JOIN PRESERVING MAPS 131

∗ ∗ M(M (A))(z) = M( y∈X (M(A)(y) → ⊤y)) ∗ = y∈VX (M(A)(y) ⊙ M(⊤y)(z)) ∗ ∗ ∗ = Wy∈X ( x∈X (A (x) ⊙ M(⊤x)(y)) ⊙ M(⊤y)(z)) ∗ ∗ ≤ Wx∈X (WA (x) ⊙ M(⊤x)(z)) = M(A)(z)) W ∗ ∗ ∗ ∗ (7) For M(⊤x) = y∈X (M (⊤x)(y) → ⊤y), we have V ∗ ∗ ∗ y∈X M(⊤x)(y) ⊙ M(⊤x)(z) ≤ M(⊤y)(z) −1 ∗ ∗ ∗ iﬀW M( x∈X (M (⊤y)(z) → ⊤x)(z) ≤ M(⊤y)(z) −1∗ ∗ ∗ iﬀ M(VM (⊤y))(z) ≤ M(⊤y)(z)

∗ ∗ ∗ x∈X M(⊤x)(y) ⊙ M(⊤x)(z) ≤ M(⊤y)(z) ∗ −1 ∗ ∗ iffW x∈X M(⊤x)(y) ⊙ M (⊤z)(x) ≤ M(⊤y)(z) −1 ∗ ∗ −1 ∗ iff MW ( x∈X (M (⊤z)(x) → ⊤x)(y) ≤ M (⊤z)(y) −1∗ ∗ −1 ∗ iff M(VM (⊤z))(y) ≤ M (⊤z)(y)

−1∗ −1 ∗ M(M (A))(z) = M( y∈X (M (A)(y) → ⊤y)) −1 ∗ = y∈VX (M (A)(y) ⊙ M(⊤y)(z)) ∗ −1 ∗ ∗ = Wy∈X ( x∈X (A (x) ⊙ M (⊤x)(y)) ⊙ M(⊤y)(z)) ∗ ∗ ≤ Wx∈X (WA (x) ⊙ M(⊤x)(z)) = M(A)(z)) W (8) It is similarly proved. ∗ ∗ ∗ (9) Since y∈X K (⊤x)(y)⊙M(⊤y)(z) = M( y∈X (K (⊤x)(y) → ⊤y))(z) = M(K(⊤x))(zW), we have V

∗ ∗ ∗ y∈X K (⊤x)(y) ⊙ K (⊤y)(z) ≤ K (⊤x)(z) ∗ ∗ ∗ iﬀW y∈X K (⊤x)(y) ⊙ M(⊤y)(z) ≤ K (⊤x)(z) ∗ iﬀ MW (K(⊤x))(z) ≤ K (⊤x)(z).

−1∗ −1 ∗ −1 −1∗ ∗ Since y∈X K (⊤z)(y)⊙M (⊤y)(x) = M ( y∈X (K (⊤z)(y) → ⊤y))(x) = −1 −1 M (WK (⊤z))(x), we have V

∗ ∗ ∗ y∈X K (⊤x)(y) ⊙ K (⊤y)(z) ≤ K (⊤x)(z) −1∗ −1∗ ∗ Wiff y∈X K (⊤z)(y) ⊙ K (⊤y)(x) ≤ K (⊤x)(z) ∗ ∗ −1∗ iff Wy∈X K (⊤x)(y) ⊙ M(⊤y)(z) ≤ K (⊤z)(x) −1 −1 −1∗ iff MW (K (⊤z))(x) ≤ K (⊤z)(x). 132 Y.C. Kim

∗ ∗ M(K(A))(z) = M( y∈X (K (A)(y) → ⊤y))(z) ∗ ∗ = y∈VX (K (A)(y) ⊙ M(⊤y)(z)) ∗ ∗ = Wy∈X ( x∈X (A(x) ⊙ K (⊤x)(y)) ⊙ K (⊤y)(z)) ∗ ∗ ≤ Wx∈X (WA(x) ⊙ K (⊤x)(z)) = K (A)(z)). W (10) ∗ ∗ ∗ y∈X K (⊤x)(y) ⊙ K (⊤x)(z) ≤ K (⊤y)(z) −1∗ ∗ ∗ Wiff y∈X K (⊤y)(x) ⊙ M(⊤x)(y) ≤ K (⊤y)(z) −1 ∗ iff MW (K (⊤y))(z) ≤ K (⊤y)(z). ∗ ∗ ∗ y∈X K (⊤x)(y) ⊙ K (⊤x)(z) ≤ K (⊤y)(z) −1∗ ∗ ∗ iffW y∈X K (⊤z)(x) ⊙ M(⊤x)(y) ≤ K (⊤y)(z) −1 −1∗ iff MW (K (⊤z))(y) ≤ K (⊤z)(y). Other cases and (11) are similarly proved.

Example 13. Let (L = [0, 1], ⊙, →,∗ ) be a complete residuated lattice with the law of double negation deﬁned by

x ⊙ y = (x + y − 1) ∨ 0, x → y = (1 − x + y) ∧ 1, x∗ = 1 − x.

Let X = {x, y, z} and A, B ∈ LX as follows:

A(x) = 0.9,A(y) = 0.8,A(z) = 0.3,B(x) = 0.3,A(y) = 0.7,A(z) = 0.8

∗ ∗ Deﬁne K (1x)(y) = M(1x)(y) as follows

∗ ∗ ∗ K (1x)(x) = 1 K (1x)(y) = 0.8 K (1x)(z) = 0.6 ∗ ∗ ∗  K (1y)(x) = 0.7 K (1y)(y) = 1 K (1y)(z) = 0.3  K∗(1 )(x) = 0.5 K∗(1 )(y) = 0.6 K∗(1 )(z) = 1  z z z  ∗ ∗ ∗ ∗ (1) We have y∈X (K (1x)(y) ⊙ K (1y)(z) = K (1x)(z) and 1x ≤ K (1x) for all x, y ∈ X.W Since K(A)(y) = x∈X (A(x) → K(1x)(y)) and M(A)(y) = ∗ ∗ x∈X (M(1x)(y) ⊙ A (x)), we have V W K(A) = (0.1, 0.2, 0.5), K(B) = (0.6, 0.3, 0.2),

K(A∗) = (0.8, 0.7, 0.3), K(B∗) = (0.3, 0.5, 0.7), M(A) = (0.2, 0.3, 0.7), M(B) = (0.7, 0.5, 0.3), JOIN-MEET AND MEET-JOIN PRESERVING MAPS 133

M(A∗) = (0.9, 0.8, 0.5), M(B∗) = (0.4, 0.7, 0.8). Furthermore, by Theorem 12(3,6,9), K(K∗(A)) = K(A), M(M∗(A)) = M(A) ∗ ∗ and M(K(A)) = K (A), for all A ∈ X. For K (1x) = (1, 0.8, 0.6), by Theo- ∗ ∗ rem 12(2,6), K(K (1x) = K(1x) = (0, 0.2, 0.4). For M(1x) = (1, 0.8, 0.6), by ∗ ∗ ∗ Theorem 12(2,6), M(M (1x) = M(1x). −1∗ −1 ∗ ∗ (2) We obtain K (1x)(y) = M (1y)(x) = K (1y)(x) as follows

−1∗ −1∗ −1∗ K (1x)(x) = 1 K (1x)(y) = 0.7 K (1x)(z) = 0.5 −1∗ −1∗ −1∗  K (1y)(x) = 0.8 K (1y)(y) = 1 K (1y)(z) = 0.6  K−1∗(1 )(x) = 0.6 K−1∗(1 )(y) = 0.3 K−1∗(1 )(z) = 1  z z z  −1∗ −1∗ −1∗ −1∗ We have y∈X (K (1x)(y) ⊙ K (1y)(z) = K (1x)(z) and 1x ≤ K (1x) for all x, yW∈ X.

K−1(A) = (0.1, 0.2, 0.6), K−1(B) = (0.5, 0.3, 0.2).

K−1(A∗) = (0.7, 0.8, 0.3), K−1(B∗) = (0.3, 0.6, 0.8), M−1(A) = (0.3, 0.2, 0.7), M−1(B) = (0.7, 0.4, 0.2), M−1(A∗) = (0.9, 0.8, 0.4), M−1(B∗) = (0.5, 0.7, 0.8). Furthermore, by Theorem 12(3,6,9), K−1(K(A)) = K∗(A), M−1(M−1∗(A)) = −1 −1 −1 −1∗ ∗ M (A) and M (K (A)) = K (A), for all A ∈ X. For K (1x) = (1, 0.8, 0.6), −1 ∗ −1 ∗ by Theorem 12(2,6), K (K(1x) = K (1x). For M (1x) = (1, 0.7, 0.5), by −1 −1∗ ∗ −1 ∗ Theorem 12(2,6), M (M (1x) = M (1x). ∗ ∗ ∗ (3) Since 0.6 = x∈X (K (1x)(y) ⊙ K (1x)(z) 6≤ K (1y)(z) = 0.3, then

W −1∗ (0.3, 0, 0.7) = K(1y) 6≤ K(K (1y)) = (0.2, 0, 0.4)

−1∗ −1 (0.8, 1, 0.6) = K (1y) 6≤ K (K(1y)) = (0.7, 1, 0.3) ∗ −1 (0.7, 1, 0.3) = K (1y) 6≥ M(K (1y)) = (0.8, 1, 0.6) ∗ −1 ∗ (0.7, 1, 0.3) = M(1y) 6≥ M(M (1y)) = (0.8, 1, 0.6).

(4) Since 0.6 = x∈X (K(1y)(x) ⊙ H(1z)(x) 6≤ K(1y)(z) = 0.3, then W − ∗ (0.3, 0, 0.7) = K(1y) 6≤ K (K (1y)) = (0.2, 0, 0.4)

∗ (0.7, 1, 0.3) = K (1y) 6≤ K(K(1y )) = (0.7, 0.7, 0.3) ∗ −1 (0.7, 1, 0.3) = K (1y) 6≥ M (K(1y)) = (0.8, 1, 0.6) ∗ −1 ∗ ∗ (0.7, 1, 0.3) = M(1y) 6≥ M (M (1y)) = (0.8, 1, 0.6). 134 Y.C. Kim

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