International Journal of Mathematics and Mathematical Sciences

Volume 2013 (2013), Article ID 451212, 4 pages

http://dx.doi.org/10.1155/2013/451212

Research Article

Generalized Derivations of BCC-Algebras

S. M. Bawazeer,¹ N. O. Alshehri,² and Rawia Saleh Babusail³

¹Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

²Department of Mathematics, Faculty of Science for Girls, King Abdulaziz University, Jeddah, Saudi Arabia

³Faculty of Science for Girls, King Abdulaziz University, Saudi Arabia

Received 29 April 2013; Revised 2 August 2013; Accepted 27 August 2013

Academic Editor: Brigitte Forster-Heinlein

Copyright © 2013 S. M. Bawazeer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The notion of generalized derivations of BCC-algebras is introduced, and some related properties are investigated. Also, we consider regular generalized derivations and the D-invariant on ideals of BCC-algebras. We also characterized KerD by generalized derivations.

1. Introduction

Imai and Iséki [1] defined a class of algebras of type called -algebras which generalizes on one hand the notion of algebra of sets with the set subtraction as the only fundamental nonnullary operation on the other hand the notion of implication algebra. The class of all -algebras is a quasivariety. Iséki posed an interesting problem (solved by Wroński [2]) whether the class of -algebras is a variety. In connection with this problem, Komori [3] introduced a notion of -algebras, and Dudek [4] redefined the notion of -algebras by using a dual form of the ordinary definition in the sense of Komori. Dudek and Zhang [5] introduced a new notion of ideals in -algebras and described connections between such ideals and congruences. On the other hand, Jun and Xin [6] applied the notion of derivations in ring and near-ring theories to -algebras, and they also introduced a new concept called a regular derivation in -algebras and investigated related properties. They defined a -derivation ideal and gave conditions for an ideal to be -derivation. Prabayak and Leerawat [7] introduced the notion of derivation in -algebras. In this paper, the notion of generalized derivations of -algebras is introduced, and some related properties are investigated. Also, we consider regular generalized derivations and the -invariant on ideals of -algebras. We also characterized Ker by generalized derivations.

2. Preliminaries

We review some definitions and properties that will be useful in our results.

Definition 1. Let be a set with a binary operation “” and a constant . Then is called a -algebra if the following axioms are satisfied for all :

(i), (ii), (iii), (iv), (v) and .

Define a binary relation on by letting if and only if . Then is a partially ordered set.

In any -algebra for all , the following properties hold:

(1), (2) implies ,(3) implies .

Any -algebra is a -algebra, but there are -algebras which are not -algebras. Note that a -algebra is a -algebra if and only if satisfies

(4), or(5).

Definition 2. A nonempty subset of a -algebra is called subalgebra of if whenever . For a -algebra , we denote for all .

Remark 3. Let be a -algebra and ; then, consider

(1), (2).

Definition 4. A -algebra is said to be commutative if and only if satisfies for all , ; that is, .

Definition 5. Let be a -algebra and . is called an ideal of if it satisfies the following conditions:

(i), (ii) and imply .

Definition 6. Let be a -algebra. A map is a left-right derivation (briefly,-derivation) of , if it satisfies the identity

If satisfies the identity

then is a right-left derivation (briefly, -derivation) of . Moreover, if is both -derivation and -derivation, then is a derivation of .

Definition 7. A self map of a -algebra is said to be regular if

Corollary 8. A derivation of a -algebra is regular.

Definition 9. Let be a derivation of a -algebra . An ideal of is said to be -invariant if , where .

Proposition 10. Let be a -algebra with partial order , and let be a derivation of . Then the following hold for all :

(i),(ii), (iii), (iv), (v) ,(vi) is a subalgebra of .

3. Generalized Derivations of BCC-Algebras

Definition 11. Let be a -algebra. A mapping is called a generalized -derivation if there exists an -derivation such that for all ; if there exists an -derivation such that for all , the mapping is called a generalized -derivation. Moreover, if is both a generalized - and -derivation, we say that is a generalized derivation.

Example 12. Let be a -algebra with Cayley table (see Table 1).

Define a map by

Then is a derivation of . Now we define a map by

It is easy to verify that is a generalized -derivation of .

Table 1

*
0
1
2
3

0
0
0
0
0

1
1
0
1
0

2
2
2
0
0

3
3
3
1
0

Proposition 13. A generalized derivation of a -algebra is regular.

Proof. Directly from Corollary 8, we have that .

Proposition 14. Let be a self-map of a -algebra . Then,

(i)if is a generalized -derivation of , then for all ;(ii)if is a generalized -derivation of , then for all .

Proof. (i) If is a generalized -derivation of , then there exists an -derivation such that for all . Hence, we get

(ii) If is a generalized -derivation of , then there exists an -derivation such that for all . Hence, we get

Proposition 15. Let be a -algebra with partial order , and let be a generalized derivation of . Then the following hold for all :

(1), (2), (3), (4),(5),(6),(7),(8).

Proof. Consider (1) .

  From Proposition 10,   .

(2) .

(3) .

(4) .

(5) .

(6) .

(7) .

(8) From Proposition 10, we have

Proposition 16. Let be a -algebra. Then for , where are generalized derivations of .

Proof. Trivial.

Definition 17. Let be a generalized derivation of a -algebra . An ideal of is said to be -invariant if , where

Theorem 18. Let be a generalized derivation of a -algebra . Then every ideal of is -invariant.

Proof. Let be an ideal of a -algebra . Let . Then for some . It follows that , which implies that . Thus . Hence, is -invariant.

Definition 19. Let be a -algebra and let be a generalized derivation. Define a Ker by .

Theorem 20. Let be a -algebra and let be generalized derivation. If and , then .

Proof. Let . Then, we get , and so

Hence, we have .

Theorem 21. Let be a commutative -algebra and be a generalized derivation. If and , then .

Proof. Let and . Then we get and , and so

Hence we have .

Theorem 22. Let be a -algebra and let be a generalized derivation. If , one has for all .

Proof. Let . Then, . Thus, we have

Hence, .

Theorem 23. Let be a -algebra and let be a generalized derivation.Then is a subalgebra of .

Proof. Directly from Theorem 22.

Definition 24. Let be a -algebra and let be a generalized derivation on . Denote .

Proposition 25. Let be a -algebra and let be a generalized derivation on . If , then .

Proof. Since , then . From Proposition 15, we have that implies that . Hence, .

Proposition 26. Let be a -algebra and let be a generalized derivation on . Then is a subalgebra of .

Proof. If , we get and , and so . Hence, .

Proposition 27. Let be a -algebra and let be a generalized derivation on . If , we have .

Proof. Let , Then and .

From Proposition 26, we have , and so . Hence, we have

Hence, .

References

Y. Imai and K. Iséki, “On axiom systems of propositional calculi. XIV,” Proceedings of the Japan Academy, vol. 42, pp. 19–22, 1966.

A. Wroński, “BCK-algebras do not form a variety,” Mathematica Japonica, vol. 28, no. 2, pp. 211–213, 1983.

Y. Komori, “The class of BCC-algebras is not a variety,” Mathematica Japonica, vol. 29, no. 3, pp. 391–394, 1984.

W. A. Dudek, “The number of subalgebras of finite BCC-algebras,” Bulletin of the Institute of Mathematics, vol. 20, no. 2, pp. 129–135, 1992.

W. A. Dudek and X. Zhang, “On ideals and congruences in BCC-algebras,” Czechoslovak Mathematical Journal, vol. 48, no. 123, pp. 21–29, 1998.

Y. B. Jun and X. L. Xin, “On derivations of BCI-algebras,” Information Sciences, vol. 159, no. 3-4, pp. 167–176, 2004.

C. Prabayak and U. Leerawat, “On dervations of BCC-algebra,” Kasetsart Journal, vol. 43, no. 398, 401 pages, 2009.