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The generalized f-derivations of lattices.

[section]1. Introduction

The concept of derivation for BCI-algebra was introduced by Y. B. Jun and X. L. Xin [3]. Further, in 2009, C. Prabprayak and U. Leerawat [7] also studied the derivation of BCC-algebra. In 2005, J. Zhan and Y. L. Liut [7] introduced the concept of a /-derivation for BCI-algebra and obtained some related properties. In 2008, L. X. Xin, T. Y. Li and J. H. Lu [6] studied derivation of lattice and investigated some of its properties. In 2010, N. O. Alshehri introduced the concept of a generalized derivation and investigated some of its properties. In 2011, S. Harmaitree and U. Leerawat studied the /-derivation of lattice and investigated some of its properties. The purpose of this paper, we applied the notion of a generalized /- derivation for a lattice and investigate some related properties.

[section]2. Preliminaries

We first recall some definitions and results which are essential in the development of this paper.

Definition 2.1.[5] An (algebraic) lattice (L, [conjunction], [disjunction]) is a nonempty set L with two binary operation "[conjunction]" and "[disjunction]"(read "join" and "meet", respectively) on L which satisfy the following condition for all x,y,z [member of] L:

(i) x [conjunction] x = x, x [disjunction] x = x;

(ii) x [conjunction] y = y [conjunction] x, x [disjunction] y = y [disjunction] x;

(iii) x [conjunction] (y [conjunction] z) = (x [conjunction] y) [conjunction] z, x [disjunction] (y [disjunction] z) = (x [disjunction] y) [disjunction] z;

(iv) x = x [conjunction] ( x [disjunction] y) , x = x [disjunction] ( x [conjunction] y).

We often abbreviate L is a lattice to (L, [conjunction], [disjunction]) is an algebraic lattice.

Definition 2.2.[5] A poset (L, [less than or equal to]) is a lattice ordered if and only if for every pair x,y of elements of L both the sup{x,y} and the inf{x, y} exist.

Theorem 2.3.[5] In a lattice ordered set (L, [less than or equal to]) the following statements are equivalent for all x,y [member of] L:

(a) x [less than or equal to] y;

(b)sup{x,y} = y;

(c)inf{x,y} = x.

Definition 2.4.[8] Let L be a lattice. A binary relation "[less than or equal to]"is defined by x [less than or equal to] y if and only if x [conjunction] y = x and x [disjunction] y = y.

Lemma 2.5. Let L be a lattice. Then x [conjunction] y = x if and only if x [disjunction] y = y for all x,y [member of] L.

Proof. Let x,y [member of] L and assume x [conjunction] y = x. Then x [disjunction] y = (x [conjunction] y) [disjunction] y = y. Conversely, let x [disjunction] y = y. So x [conjunction] y = x [conjunction] (x [disjunction] y) = x.

Corollary 2.6. Let L be a lattice. Then x [less than or equal to] y if and only if either x [conjunction] y = x or x [disjunction] y = y.

Lemma 2.7.[8] Let L be a lattice. Define the binary relation "[less than or equal to]" as Definition 2.3. Then ( L, [less than or equal to]) is a poset and for any x, y [member of] L, x [conjunction] y is the inf{x, y} and x [disjunction] y the sup{x, y}.

Theorem 2.8.[5] Let L be a lattice. If we define x [less than or equal to] y if and only if x [conjunction] y = x then (L, [less than or equal to]) is a lattice ordered set.

Definition 2.9.[5] If a lattice L contains a least (greatest) element with respect to [less than or equal to] then this uniquely determined element is called the zero element (one element), denoted by 0 (by 1).

Lemma 2.10.[5] Let L be a lattice. If y [less than or equal to] z, then x [conjunction] y [less than or equal to] x [conjunction] z and x [disjunction] y [less than or equal to] x [disjunction] z for all x, y, z [member of] L.

Definition 2.11.[5] A nonempty subset S of a lattice L is called sublattice of L if S is a lattice with respect to the restriction of [conjunction] and [disjunction] of L onto S.

Definition 2.12.[5] A lattice L is called modular if for any x,y,z [member of] L if x [less than or equal to] z, then x [disjunction] (y [conjunction] z) = (x [disjunction] y) [conjunction] z.

Definition 2.13.[5] A lattice L is called distributive if either of the following condition hold for all x, y, z in L: x [conjunction] ( y [disjunction] z) = (x [conjunction] y) [disjunction] (x [conjunction] z) or x [disjunction] (y [conjunction] z) = (x [disjunction] y) [conjunction] (x [disjunction] z).

Corollary 2.14.[5] Every distributive lattice is a modular lattice.

Definition 2.15.[5] Let f : L - M be a function from a lattice L to a lattice M.

(i) f is called a join-homomorphism if f(x [disjunction] y) = f(x) [disjunction] f(y) for all x,y [member of] L.

(ii) f is called a meet-homomorphism if f( x [conjunction] y) = f( x) [conjunction] f( y) for all x, y [member of] L.

(iii) f is called a lattice-homomorphism if f are both a join-homomorphism and a meet-homomorphism.

(iv) f is called an order-preserving if x [less than or equal to] y implies f(x) [less than or equal to] f(y) for all x,y [member of] L.

Lemma 2.16.[5] Let f : L [right arrow] M be a function from a lattice L to a lattice M. If f is a join-homomorphism (or a meet-homomorphism , or a lattice-homomorphism), then f is an order-preserving.

Definition 2.17.[5] An ideal is a nonempty subset I of a lattice L with the properties:

(i) if x [less than or equal to] y and y [member of] I, then x [member of] I for all x, y [member of] L;

(ii) if x, y [member of] I, then x [disjunction] y [member of] I.

Definition 2.18.[3] Let L be a lattice and f : L [right arrow] L be a function. A function d : L [right arrow] L is called a f-derivation on L if for any x,y [member of] L, d(x [conjunction] y) = (dx [conjunction] f(y)) [disjunction] (f(x) [conjunction] dy).

Proposition 2.19.[3] Let L be a lattice and d be a f-derivation on L where f : L [right arrow] L is a function. Then the following conditions hold : for any element x, y [member of] L,

(1) dx [less than or equal to] f(x);

(2) dx [conjunction] dy [less than or equal to] d(x [conjunction] y) [less than or equal to] dx [disjunction] dy;

(3) If L has a least element 0, then f(0) = 0 implies d 0 = 0.

[section]3. The generalized f-derivations of lattices

The following definitions introduces the notion of a generalized f-derivation for lattices.

Definition 3.1. Let L be a lattice and f : L [right arrow] L be a function. A function D : L [right arrow] L is called a generalized f-derivation on L if there exists a f-derivation d : L [right arrow] L such that D(x [conjunction] y) = (D(x) [conjunction] f(y)) [disjunction] (f(x) [conjunction] d(y)) for all x, y [member of] L.

We often abbreviate d(x) to dx and Dx to D(x).

Remark. If D = d, then D is a f-derivation.

Now we give some examples and show some properties for a generalized f- derivation in lattices.

Example 3.2. Consider the lattice given by the following diagram of Fig. 1.

[FIGURE 1 OMITTED]

Define, respectively, a function d, a function D and a function / by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then it is easily checked that d is a f-derivation and D is a generalized f- derivation.

Example 3.3. Consider the lattice as shown in Fig. 2.

[FIGURE 2 OMITTED]

Define, respectively, a function d, a function D and a function f by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then it is easily checked that d is a f-derivation and D is a generalized f- derivation.

Proposition 3.4. Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a function. Then the following hold: for any element x, y [member of] L,

(1) dx [less than or equal to] Dx [less than or equal to] f(x);

(2) Dx [conjunction] Dy [less than or equal to] D(x [conjunction] y) [less than or equal to] Dx [disjunction] Dy.

Proof.(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 3.5. Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is an order-preserving. Suppose x,y [member of] L be such that y [less than or equal to] x. If Dx = f(x), then Dy = f(y).

Proof. Since f is an order-preserving, f(y) [less than or equal to] f(x). Thus Dy = D(x [conjunction] y) = (Dx [conjunction] /(y)) [disjunction] (f(xc) [conjunction] dy) = (f(x) [conjunction] f(y)) [disjunction] (f(x) [conjunction] dy) = f(y) [disjunction] dy = f(y).

Proposition 3.6. Let L be a lattice with a least element 0 and D be a generalized f-derivation on L where f : L [right arrow] L is a function. Then

(1) if f(0) = 0, then D0 = 0;

(2) if D0 = 0, then Dx [conjunction] f(0) = 0 for all x [member of] L.

Proof. (1) By Proposition 3.4(1).

(2) Let x [member of] L. It is easily show that d0 = 0. Then

0 = D0 = D(x [conjunction] 0) = (Dx [conjunction] f(0)) [disjunction] (f(x) [conjunction] d0) = Dx [conjunction] f(0).

The following result is immediately from Proposition 3.7(2).

Corollary 3.7. Let L be a lattice with a least element 0 and D be a generalized f-derivation on L where f : L [right arrow] L is a function such that D0 = 0. Then we have,

(1) Dx [less than or equal to] f(0) if and only if Dx = 0 for all x [member of] L;

(2) f(0) [less than or equal to] Dx for all x [member of] L if and only if f(0) = 0;

(3) if f(0) [not equal to] 0 and there exist x [member of] L such that Dx [not equal to] 0, then (L, [less than or equal to]) is not a chain.

Proposition 3.8. Let L be a lattice with a greatest element 1 and D be a generalized f-derivation on L where f : L [right arrow] L is a function. Then

(1) if D1 = 1, then f (1) = 1;

(2) if f(1) = 1, then Dx = (D1 [conjunction] f(x)) [disjunction] dx for all x [member of] L.

Proof. (1) By Proposition 3.4(1).

(2) Note that Dx = D(1 [conjunction] x) = (D1 [conjunction] f(x)) [disjunction] (f(1) [conjunction] dx) = (D1 [conjunction] f(x)) [disjunction] (1 [conjunction] dx) = (D1 [conjunction] f(x)) [disjunction] dx.

Corollary 3.9. Let L be a lattice with a greatest element 1 and D be a generalized f-derivation on L where f : L [right arrow] L is a function such that f(1) = 1. Then we have, for all x [member of] L,

(1) D1 [less than or equal to] f(x) if and only if D1 [less than or equal to] Dx;

(2) if D1 [less than or equal to] f(x) and D is an order-preserving, then Dx = D1;

(3) f(x) [less than or equal to] D1 if and only if Dx = f(x);

(4) D1 = 1 if and only if Dx = f(x).

Proposition 3.10. Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a join-homomorphism. Then D = f if and only if D(x [disjunction] y) = (Dx [disjunction] f(y)) [conjunction] (f(x) [disjunction] Dy) for all x, y [member of] L.

Proof. ([??]) Let x,y [member of] L. Then D(x [conjunction] y) = f(x [conjunction] y) = f(x [conjunction] y) = (f(x) [conjunction] (f(y)) [disjunction] (f(x) [conjunction] f(y)) = (Dx [conjunction] f(y) [disjunction] (f(x) [conjunction] Dy).

([??]) Assume that D(x [conjunction] f(y)) = (Dx [conjunction] f(y)) [disjunction] (f(x) [conjunction] Dy). By putting y = x in the assumption, we get Dx = f(x) for all x [member of] L.

Proposition 3.11. Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is an order-preserving. Then Dx = (D(x [disjunction] y) [conjunction] f(x)) [disjunction] dx for all x, y [member of] L.

Proof. Let x, y [member of] L. Then dx [less than or equal to] f(x) [less than or equal to] f(x [conjunction] y). So Dx = D((x [conjunction] y) [disjunction] x) = (D(x [conjunction] y) [disjunction] f(x)) [conjunction] (f(x [conjunction] y) [disjunction[ dx) = (D(x [conjunction] y) [disjunction] f(x)) [conjunction] dx.

Proposition 3.12. Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a function. If D is an order-preserving, then Dx = D(x [disjunction] y) [conjunction] f(x) for all x [member of] L.

Proof. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 3.1.3 Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a function. Then the following conditions are equivalent:

(1) D is an order-preserving;

(2) Dx [disjunction] Dy [less than or equal to] D(x [disjunction] y) for all x,y [member of] L;

(3) D(x [conjunction] y) = Dx [conjunction] Dy for all x,y [member of] L.

Proof. (1) [??] (2) Let x,y [member of] L. Then Dx [less than or equal to] D(x [disjunction] y) and Dy [less than or equal to] D(x [disjunction] y). Therefore Dx [disjunction] Dy [less than or equal to] D(x [disjunction] y).

(2) [??] (1): Assume that (2) holds. Let x, y [member of] L be such that x [less than or equal to] y. Then Dy = D(x [disjunction] y) [greater than or equal to] Dx [disjunction] Dy but we have Dy [less than or equal to] Dx [disjunction] Dy. So Dy = Dx [disjunction] Dy, it follow that Dx [less than or equal to] Dy.

(1) [right arrow] (3): Let x,y [member of] L. Then D(x [conjunction] y) [less than or equal to] Dx and D(x [conjunction] y) [less than or equal to] Dy. Therefore D(x [conjunction] y) [less than or equal to] Dx [conjunction] Dy. By Proposition 3.4(2), we have D(x [conjunction] y) [greater than equal to] Dx [conjunction] Dy. Hence D(x [conjunction] y) = Dx [conjunction] Dy.

(3) [right arrow] (1): Assume that (3) holds. Let x, y [member of] L be such that x [less than or equal to] y. Then Dx = D(x [conjunction] y) = Dx [conjunction] Dy, it follow that Dx [less than ot equal to] Dy.

Theorem 3.14. Let L be a lattice with a greatest element 1 and D be a generalized f-derivation on L where f : L [right arrow] L is a meet-homomorphism such that f(1) = 1. Then the following conditions are equivalent:

(1) D is an order-preserving;

(2) Dx = f(x) [conjunction] D1 for all x [member of] L;

(3) D(x [conjunction] y) = Dx [conjunction] Dy for all x,y [member of] L;

(4) Dx [disjunction] Dy [less than or equal to] D(x [disjunction] y) for all x, y [member of] L.

Proof. By Theorem 3.13, we get the conditions (1) and (4) are equivalent.

(1) [??] (2): Assume that (1) holds. Let x [member of] L. Since x [less than or equal to] 1, Dx [less than or equal to] D1. We have Dx [less than or equal to] f(x). So we get Dx [less than or equal to] f(x) [conjunction] D1. By Proposition 3.8(2), we have Dx = dx [disjunction] (f(x) [conjunction] D1). Thus Dx = f(x) [conjunction] D1.

(2) [??] (3): Assume that (2) holds. Then Dx [conjunction] Dy = (f(x) [conjunction] D1) [conjunction] (f(y) [conjunction] D1) = f(x [conjunction] y) [conjunction] D1 = D(x [conjunction] y).

(3) [??] (1): Assume that (3) holds. Let x, y [member of] L such that x [less than or equal to] y. By (3), we get Dx = D(x [conjunction] y) = Dx [conjunction] Dy, it follows that Dx [less than or equal to] Dy.

Theorem 3.15. Let L be a distributive lattice and D be a generalized f- derivation on L where f : L [right arrow] L is a join-homomorphism. Then the following conditions are equivalent:

(1) D is an order-preserving;

(2) D(x [conjunction] y) = Dx [conjunction] Dy for all x,y [member of] L;

(3) D(x [disjunction] y) = Dx [disjunction] Dy for all x,y [member of] L.

Proof. By Theorem 3.13, we get the conditions (1) and (2) are equivalent.

(1)[right arrow](3): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3) [??](1): Assume that (3) holds and let x, y [member of] L be such that x [less than or equal to] y. Then Dy = D(x[disjunction]y) = Dx [disjunction] Dy by (3). It follows that Dx [less than or equal to] Dy, this shows that D is an order-preseving.

Theorem 3.16. Let L be a modular lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a join-homomorphism. If there exist a [member of] L such that Da = f(a), then D is an order-preserving implies D(x [disjunction] a) = Dx [disjunction] Da for all x [member of] L.

Proof. Let x [member of] L. Suppose that there exist a [member of] L such that Da = f(a) and D is an order-preserving. Then Da [less than or equal to] D(x [disjunction] a). By Proposition 3.12, we get Dx = D(x [disjunction] y) [conjunction] f(x). So Dx [disjunction] Da = (D(x [disjunction] a) [conjunction] f(x)) [disjunction] Da = D(x [disjunction] a) [conjunction] (Da [disjunction] f(x)) = D(x [disjunction] a) [conjunction] (f(a) [disjunction] f(x)) = D(x [disjunction] a) [conjunction] f(x [disjunction] a) = D(x [disjunction] a).

Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a function. Denote [Fix.sub.D](L) = {x [member of]|Dx = f(x)}.

In the following results, we assume that [Fix.sub.D](L) is a nonempty proper subset of L.

Theorem 3.17. Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a lattice-homomorphism. If D is an order-preserving, then [Fix.sub.D](L) is a sublattice of L.

Proof. Let x,y [member of] [Fix.sub.D](L). Then Dx = f(x) and Dy = f(y). Then f(x [conjunction] y) = f(x) [conjunction] f(y) = Dx [conjunction] Dy [less than or equal to] D(x [conjunction] y). So D(x [conjunction] y) = f(x [conjunction] y), that is x [conjunction] y [member of] [Fix.sub.D](L). Moreover, we get f(x [disjunction] y) = f(x) [disjunction] d(y) = Dx [disjunction] Dy [less than or equal to] D(x [disjunction] y) by Theorem 3.13. So D(x [disjunction] y) = f(x [disjunction] y), this shows that x [disjunction] y [member of] [Fix.sub.D](L).

Theorem 3.18. Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a lattice-homomorphism. If D is an order-serving, then [Fix.sub.D](L) is an ideal of L.

Proof. The proof is by Proposition 3.5 and Theorem 3.17.

Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a function. Denote kerD = {x [member of] L|Dx = 0}.

In the following results, we assume that kerD is a nonempty proper subset of L.

Theorem 3.19. Let L be a distributive lattice and D be a generalized f- derivation on L where f : L [right arrow] L is a lattice-homomorphism. If D is an order- preserving, then kerD is a sublattice of L.

Proof. The proof is by Theorem 3.15.

Definition 3.20. Let L be a lattice and f : L [right arrow] L be a function. A nonempty subset I of L is said to be a f-invariant if f(I) [subset or equal to] I where denote f(I) = {y [member of] L|y = f(x) for some x [member of] I}.

Theorem 3.21. Let L be a lattice and D be a generalized f-derivation on L where f : L [right arrow] L is a function. Let I be an ideal of L such that I is a f- invariant. Then I is a D-invariant.

Proof. Let y [member of] DI. Then there exist x [member of] I such that y = Dx. Since I is a f-invariant, f(x) [member of] I. We have y = Dx [less than or equal to] f(x). Since I is an ideal and f(x) [member of] I, y [member of] I. Thus dI [subset or equal to] I.

Acknowledgements

The author is greatly indebted to the referee for several useful suggestions and valuable comments which led to improvement the exposition. Moreover, this work was supported by a grant from Kasetsart University.

References

[1] Hamza, A. S. A. and N. O. Al-Shehri, Some results on derivations of BCI- algebras, Coden Jnsmac, 46(2006), 13-19.

[2] Hamza, A. S. A. and N. O. Al-Shehri, On left derivations of BCI-algebras, Soochow Journal of Mathematics, 33(2007), No. 6, 435-444.

[3] Harmaitree, S. and U. Leerawat, On /-derivations in Lattices, To appear in the Far East Journal of Mathematical Sciences (FJMS).

[4] Y. B. Jun and X. L. Xin, On derivations of BCI-algebras, Information Sciences, 159(2004), 167-176.

[5] R. Lidl and G. Pilz, Applied Abstract Algebra, Springer-Verlag New York Inc, U.S.A, 1984.

[6] N. O. Alshehri, Generalized Derivation of Lattices, Int. J. Contemp. Math. Science, 5(2010), No. 13, 629-640.

[7] Prabprayak, C. and U. Leerawat, On derivations of BCC-algebras, Kasetsart Journal, 53(2009), 398-401.

[8] L. X. Xin, T. Y. Li and J. H. Lu, On derivations of lattice, Information Sciences, 178(2008), 307-316.

[9] J. Zhan, and Y. L. Liut, On f-derivations of BCI-algebras, International Journal of Mathematics and Mathematical Sciences, 11(2005), 1675-1684.

Sureeporn Harmaitreet ([dagger]) and Utsanee Leerawat ([double dagger])

Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok,

Thailand

E-mail: fsciutl@ku.ac.th
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