# Derivations of MV-Algebras

Authors: N. O. Alshehri
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## 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.


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## Body

## 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(M,⊕,*,0) where ⊕ is a binary operation, * is a unary operation, and 0 is a constant such that the following axioms are satisfied for any a,b∈M: (MV1)(M,⊕,0) is a commutative monoid,(MV2)(a*)*=a,(MV3)0*⊕a=0*,(MV4)(a*⊕b)*⊕b=(b*⊕a)*⊕a.If we define the constant1=0* and the auxiliary operations ⊙,∨, and ∧ by (2.1)a⊙b=(a*⊕b*)*,a∨b=a⊕(b⊙a*),a∧b=a⊙(b⊕a*),
then (M,⊙,1) is a commutative monoid and the structure (M,∨,∧,0,1) is a bounded distributive lattice. Also, we define the binary operation ⊝ by x⊝y=x⊙y*. A subset X of an MV-algebra M is called subalgebra of M if and only if X is closed under the MV-operations defined in M. In any MV-algebras, one can define a partial order ≤ by putting x≤y if and only if x∧y=x for each x,y∈M. If the order relation ≤, defined over M, is total, then we say that M is linearly ordered. For an MV-algebra M, if we define B(M)={x∈M:x⊕x=x}={x∈M:x⊙x=x}. Then, (B(M),⊕,*,0) is both a largest subalgebra of M and a Boolean algebra.An MV-algebraM has the following properties for all x,y,z∈M(1)
x⊕1=1,(2)
x⊕x*=1,(3)
x⊙x*=0,(4)
Ifx⊕y=0, then x=y=0,(5)
Ifx⊙y=1, then x=y=1,(6)
Ifx≤y, then x∨z≤y∨z and x∧z≤y∧z,(7)
Ifx≤y, then x⊕z≤y⊕z and x⊙z≤y⊙z,(8)
x≤y if and only if y*≤x*,(9)
x⊕y=y if and only if x⊙y=x.Theorem 2.2 (see [1]).
The following conditions are equivalent for allx,y∈M(i)
x≤y,(ii)
y⊕x*=1,(iii)
x⊙y*=0.Definition 2.3 (see [1]).
LetM be an MV-algebra and I be a nonempty subset of M. Then, we say that I is an ideal if the following conditions are satisfied: (i)
0∈I,(ii)
x,y∈I imply x⊕y∈I,(iii)
x∈I and y≤x imply y∈I.Proposition 2.4 (see [1]).
LetM be a linearly ordered MV-algebra, then x⊕y=x⊕z and x⊕z≠1 implies that y=z.


## 3. Derivations of MV-Algebras

Definition 3.1.
LetM be an MV-algebra, and let d:M→M be a function. We call d a derivation of M, if it satisfies the following condition for all x,y∈M(3.1)d(x⊙y)=(dx⊙y)⊕(x⊙dy).
We often abbreviated(x) to dx.Example 3.2.
LetM={0,a,b,1}. Consider Tables 1 and 2.
Then(M,⊕,*,0) is an MV-algebra. Define a map d:M→M by
(3.2)dx={0ifx=0,a,1,aifx=b.
Sinced(a⊙b)=0 and (da⊙b)⊕(a⊙db)=(0⊙b)⊕(a⊙a)=0⊕a=a,d is not derivation.Table 1⊕0ab100ab1aaa11bb1b111111Table 2*0ab11ba0Example 3.3.
LetM={0,x1,x2,x3,x4,1}. Consider Tables 3 and 4.
Then,(M,⊕,*,0) is an MV-algebra. Define a map d:M→M by
(3.3)dx={0ifx=0,x1,x3,x2ifx=x2,x4,1.
Then, it is easily checked thatd is a derivation of M.Table 3⊕0x1x2x3x4100x1x2x3x41x1x1x3x4x311x2x2x4x21x41x3x3x31x311x4x41x41111111111Table 4*0x1x2x3x411x4x3x2x10Proposition 3.4.
LetM be an MV-algebra, and let d be a derivation on M. Then, the following hold for every x∈M: (i)
d0=0,(ii)
dx⊙x*=x⊙dx*=0,(iii)
dx=dx⊕(x⊙d1),(iv)
dx≤x,(v)
IfI is an ideal of an MV-algebra M, then d(I)⊆I.Proof.
(i)d0=d(x⊙0)=(dx⊙0)⊕(x⊙d0)=x⊙d0.
Puttingx=0, we get d0=0.
(ii)
Letx∈M, then
(3.4)0=d0=d(x⊙x*)=(dx⊙x*)⊕(x⊙dx*),
and so (ii) follows from (4).(iii)
It is clear.(iv)
Letx∈M, from (ii), we have
(3.5)1=0*=(dx⊙x*)*=(dx)*⊕x,
from Theorem 2.2 we get dx≤x.(v)
Lety∈d(I), then y=d(x) for some x∈I. Since y=d(x)≤x∈I, thus y∈I and so d(I)⊆I.Proposition 3.5.
Letd be a derivation of an MV-algebra M, and let x,y∈M. If x≤y. Then, the following hold: (i)
d(x⊙y*)=0,(ii)
dy*≤x*,(iii)
dx⊙dy*=0.Proof.
(i) Letx≤y, then Theorem 2.2 implies that x⊙y*=0, and so d(x⊙y*)=d0=0.
(ii)
From (i), we get(3.6)0=d(x⊙y*)=(dx⊙y*)⊕(x⊙dy*),
and by (4), we have x⊙dy*=0. Therefore, dy*≤x*.(iii)
Ifx≤y, then dx≤y, thus dx⊙dy*≤y⊙dy*, also dy*≤y*, and so y⊙dy*≤y⊙y*=0. Hence, dx⊙dy*=0.Proposition 3.6.
LetM be an MV-algebra, and let d be a derivation on M. Then, the following hold: (i)
dx⊙dx*=0,(ii)
dx*=(dx)* if and only if d is the identity on M.Proof.
(i) It follows directly from Proposition3.5(iii).
(ii)
It is sufficient to show that ifdx*=(dx)*, then d is the identity on M.
Assume thatdx*=(dx), from Proposition 3.4(ii), we have x⊙(dx)*=0, which implies that x≤dx. Therefore, dx=x.Definition 3.7.
LetM be an MV-algebra and d be a derivation on M. If x≤y implies dx≤dy for all x,y∈M, d is called an isotone derivation.Example 3.8.
LetM be an MV-algebra as in Example 3.3. It is easily checked that d is an isotone derivation of M.Proposition 3.9.
LetM be an MV-algebra, and let d be aderivation of M. If dx*=dx for all x∈M, then the following hold:(i)
d1=0,(ii)
dx⊙dx=0,(iii)
Ifd is an isotone derivation of M, then d is zero.Proof.
(i) It follows by puttingx=0.
(ii)
It follows from Proposition3.6(i).(iii)
Sinced is an isotone, hence dx≤d1 for all x∈M. By (i), we have dx≤0, and so d is zero.Definition 3.10.
LetM be an MV-algebra, and let d be a derivation on M. If d(x⊕y)=dx⊕dy for all x,y∈M,d is called an additive derivation.Example 3.11.
LetM be an MV-algebra as in Example 3.3. It is easily checked that d is an additive derivation of M.Theorem 3.12.
Let M be an MV-algebra, and letd be a nonzero additive derivation of M. Then, d(B(M))⊆B(M).Proof.
Lety∈d(B(M)), thus y=d(x) for some x∈B(M). Then,
(3.7)y⊕y=dx⊕dx=d(x⊕x)=dx=y.
Thereforey∈B(M), this complete the proof.Theorem 3.13.
Letd be an additive derivation of a linearly ordered MV-algebra M. Then, either d=0 or d1=1.Proof.
Letd be an additive derivation of a linearly ordered MV-algebra M. Hence,
(3.8)d1=d(x⊕x*)=dx⊕dx*,
also,
(3.9)d1=d(x⊕1)=dx⊕d1,
for all x∈M. If d1≠1, then Proposition 2.4 implies that dx*=d1. Putting x=1, we get that d1=0. Therefore,
(3.10)0=d1=dx⊕d1=dx,
for all x∈M, and so d is zero.Proposition 3.14.
LetM be a linearly ordered MV-algebra, and let d1,d2 additive derivations of M. Define d1d2(x)=d1(d2x) for all x∈M. If d1d2=0, then d1=0 or d2=0.Proof.
Letd1d2=0, x∈M, and suppose that d2≠0. Then,
(3.11)0=d1d2x=d1(d2x⊕(x⊙d21))=d1d2x⊕d1x=d1x,
thus d1=0. Similarly, we can prove that d2=0.Proposition 3.15.
LetM be a linearly ordered MV-algebra, and let d be a nonzero additive derivation of M. Then,
(3.12)d(x⊙x)=x⊕x,∀x∈M.Proof.
From Proposition3.4(iii) and Theorem 3.13, we get that dx=dx⊕x; applying (9), we have dx⊙x=x. Thus,
(3.13)d(x⊕x)=(dx⊙x)⊕(dx⊙x)=x⊕x.Theorem 3.16.
Every nonzero additive derivation of a linearly ordered MV-algebraM is an isotone derivation.Proof.
Assume thatd is an additive derivation of M, and x,y∈M. If x≤y, then x*⊕y=1, hence
(3.14)1=d1=d(x*⊕y)=dx*⊕dy,
and so, (dy)*≤dx*, from (8), we have (dx*)*≤dy. Otherwise, dx*≤x*, again by (8) x≤(dx*)*. Since dx≤x, we get dx≤dy.Theorem 3.17.
LetM be a linearly ordered MV-algebra, and let d be a nonzero additive deriviation of M. Then, d-1(0)={x∈M∣dx=0} is an ideal of M.Proof.
From Proposition3.4(i), we get that 0∈d-1(0). Let x,y∈d-1(0); this implies that d(x⊕y)=0. And so x⊕y∈d-1(0).
Now, letx∈d-1(0) and y≤x. Using Theorem 3.16, we have that dy≤dx, and so dy=0.



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