International Journal of Mathematics and Mathematical Sciences

Volume 2010, Article ID 312027, 7 pages

http://dx.doi.org/10.1155/2010/312027

## Derivations of MV-Algebras

Department of Mathematics, Faculty of Science (Girl's), King Abdulaziz University, P.O. Box 126238, Jeddah 21352, Saudi Arabia

Received 26 August 2010; Revised 8 November 2010; Accepted 16 December 2010

Academic Editor: Howard Bell

Copyright © 2010 N. O. Alshehri. 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

We introduce the notion of derivation for an MV-algebra and discuss some related properties. Using the notion of an isotone derivation, we give some characterizations of a derivation of an MV-algebra. Moreover, we define an additive derivation of an MV-algebra and investigate some of its properties. Also, we prove that an additive derivation of a linearly ordered MV-algebral is an isotone.

#### 1. Introduction

In his classical paper [1], Chang invented the notion of MV-algebra in order to provide an algebraic proof of the completeness theorem of infinite valued Lukasiewicz propositional calculus. Recently, the algebraic theory of MV-algebras is intensively studied, see [2–5].

The notion of derivation, introduced from the analytic theory, is helpful to the research of structure and property in algebraic system. Several authors [6–9] studied derivations in rings and near rings. Jun and Xin [10] applied the notion of derivation in ring and near-ring theory to *BCI*-algebras. In [11], Szász introduced the concept of derivation for lattices and investigated some of its properties, for more details, the reader is referred to [9, 12–19].

In this paper, we apply the notion of derivation in ring and near-ring theory to MV-algebras and investigate some of its properties. Using the notion of an isotone derivation, we characterize a derivation of MV-algebra. We introduce a new concept, called an additive derivation of MV-algebras, and then we investigate several properties. Finally, we prove that an additive derivation of a linearly ordered MV-algebra is an isotone.

#### 2. Preliminaries

*Definition 2.1 (see [5]). *An MV-algebra is a structure where is a binary operation, is a unary operation, and 0 is a constant such that the following axioms are satisfied for any : (MV1) is a commutative monoid,(MV2) ,(MV3) ,(MV4) .

If we define the constant and the auxiliary operations , and by then is a commutative monoid and the structure is a bounded distributive lattice. Also, we define the binary operation by . A subset of an MV-algebra is called subalgebra of if and only if is closed under the MV-operations defined in . In any MV-algebras, one can define a partial order ≤ by putting if and only if for each . If the order relation ≤, defined over , is total, then we say that is linearly ordered. For an MV-algebra , if we define . Then, is both a largest subalgebra of and a Boolean algebra.

An MV-algebra has the following properties for all (1),(2),(3), (4)If , then ,(5)If , then ,(6)If , then and ,(7)If , then and ,(8) if and only if ,(9) if and only if .

Theorem 2.2 (see [1]). *The following conditions are equivalent for all *(i)*,
*(ii)*,
*(iii)*. *

*Definition 2.3 (see [1]). *Let be an MV-algebra and be a nonempty subset of . Then, we say that is an ideal if the following conditions are satisfied: (i),
(ii) imply ,(iii) and imply .

Proposition 2.4 (see [1]). *Let be a linearly ordered MV-algebra, then and implies that .*

#### 3. Derivations of MV-Algebras

*Definition 3.1. *Let be an MV-algebra, and let be a function. We call a derivation of , if it satisfies the following condition for all

We often abbreviate to .

*Example 3.2. *Let . Consider Tables 1 and 2.

Then is an MV-algebra. Define a map by

Since and is not derivation.

*Example 3.3. *Let . Consider Tables 3 and 4.

Then, is an MV-algebra. Define a map by

Then, it is easily checked that is a derivation of .

Proposition 3.4. *Let be an MV-algebra, and let be a derivation on . Then, the following hold for every : *(i)*,
*(ii)*,
*(iii)*,
*(iv)*,*(v)*If is an ideal of an MV-algebra , then .*

*Proof. *(i) .

Putting , we get .

(ii) Let , then
and so (ii) follows from (4). (iii)It is clear.(iv)Let , from (ii), we have
from Theorem 2.2 we get . (v)Let , then for some . Since , thus and so .

Proposition 3.5. *Let be a derivation of an MV-algebra , and let . If . Then, the following hold: *(i)*,
*(ii)*,
*(iii)*. *

*Proof. *(i) Let , then Theorem 2.2 implies that , and so .

(ii)From (i), we get
and by (4), we have . Therefore, .(iii)If , then , thus , also , and so . Hence, .

Proposition 3.6. *Let be an MV-algebra, and let be a derivation on . Then, the following hold: *(i)*,
*(ii)* if and only if is the identity on .*

*Proof. *(i) It follows directly from Proposition 3.5(iii).

(ii)It is sufficient to show that if , then is the identity on .

Assume that , from Proposition 3.4(ii), we have , which implies that . Therefore, .

*Definition 3.7. *Let be an MV-algebra and be a derivation on . If implies for all , is called an isotone derivation.

*Example 3.8. *Let be an MV-algebra as in Example 3.3. It is easily checked that is an isotone derivation of .

Proposition 3.9. *Let be an MV-algebra, and let be aderivation of . If for all , then the following hold:*(i)*,*(ii)*, *(iii)*If is an isotone derivation of , then is zero.*

*Proof. *(i) It follows by putting .

(ii)It follows from Proposition 3.6(i).(iii)Since is an isotone, hence for all . By (i), we have , and so is zero.

*Definition 3.10. *Let be an MV-algebra, and let be a derivation on . If for all is called an additive derivation.

*Example 3.11. *Let be an MV-algebra as in Example 3.3. It is easily checked that is an additive derivation of .

Theorem 3.12. *Let M be an MV-algebra, and let be a nonzero additive derivation of . Then, .*

*Proof. *Let , thus for some . Then,

Therefore , this complete the proof.

Theorem 3.13. *Let be an additive derivation of a linearly ordered MV-algebra M. Then, either or .*

*Proof. *Let be an additive derivation of a linearly ordered MV-algebra . Hence,
also,
for all . If , then Proposition 2.4 implies that . Putting , we get that . Therefore,
for all , and so is zero.

Proposition 3.14. *Let be a linearly ordered MV-algebra, and let additive derivations of . Define for all . If , then or .*

*Proof. *Let , , and suppose that . Then,
thus . Similarly, we can prove that .

Proposition 3.15. *Let be a linearly ordered MV-algebra, and let be a nonzero additive derivation of . Then,
*

*Proof. *From Proposition 3.4(iii) and Theorem 3.13, we get that ; applying (9), we have . Thus,

Theorem 3.16. *Every nonzero additive derivation of a linearly ordered MV-algebra is an isotone derivation.*

*Proof. *Assume that is an additive derivation of , and . If , then , hence
and so, , from (8), we have . Otherwise, , again by (8) . Since , we get .

Theorem 3.17. *Let be a linearly ordered MV-algebra, and let be a nonzero additive deriviation of . Then, is an ideal of .*

*Proof. *From Proposition 3.4(i), we get that . Let ; this implies that . And so .

Now, let and . Using Theorem 3.16, we have that , and so .

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