# 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 . Further, in 2009, C. Prabprayak and U. Leerawat  also studied the derivation of BCC-algebra. In 2005, J. Zhan and Y. L. Liut  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  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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Every distributive lattice is a modular lattice.

Definition 2.15. 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. 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. 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. 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. 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

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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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