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. PreliminariesDefinition 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 constant 1=0* and the auxiliary operations ⊙,∨, and ∧ by 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-algebra M has the following properties for all x,y,z∈M
x⊕1=1,
x⊕x*=1,
x⊙x*=0,
If x⊕y=0, then x=y=0,
If x⊙y=1, then x=y=1,
If x≤y, then x∨z≤y∨z and x∧z≤y∧z,
If x≤y, then x⊕z≤y⊕z and x⊙z≤y⊙z,
x≤y if and only if y*≤x*,
x⊕y=y if and only if x⊙y=x.
Theorem 2.2 (see [1]).
The following conditions are equivalent for all x,y∈M
x≤y,
y⊕x*=1,
x⊙y*=0.
Definition 2.3 (see [1]).
Let M 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:
0∈I,
x,y∈I imply x⊕y∈I,
x∈I and y≤x imply y∈I.
Proposition 2.4 (see [1]).
Let M be a linearly ordered MV-algebra, then x⊕y=x⊕z and x⊕z≠1 implies that y=z.
3. Derivations of MV-AlgebrasDefinition 3.1.
Let M 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∈Md(x⊙y)=(dx⊙y)⊕(x⊙dy).
We often abbreviate d(x) to dx.
Example 3.2.
Let M={0,a,b,1}. Consider Tables 1 and 2.
Then (M,⊕,*,0) is an MV-algebra. Define a map d:M→M by
dx={0ifx=0,a,1,aifx=b.
Since d(a⊙b)=0 and (da⊙b)⊕(a⊙db)=(0⊙b)⊕(a⊙a)=0⊕a=a,d is not derivation.
⊕
0
a
b
1
0
0
a
b
1
a
a
a
1
1
b
b
1
b
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1
*
0
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a
0
Example 3.3.
Let M={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
dx={0ifx=0,x1,x3,x2ifx=x2,x4,1.
Then, it is easily checked that d is a derivation of M.
⊕
0
x1
x2
x3
x4
1
0
0
x1
x2
x3
x4
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x1
x1
x3
x4
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1
1
x2
x2
x4
x2
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x4
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x3
x3
x3
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x3
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x4
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x4
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1
1
1
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0
x1
x2
x3
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x4
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x1
0
Proposition 3.4.
Let M be an MV-algebra, and let d be a derivation on M. Then, the following hold for every x∈M:
d0=0,
dx⊙x*=x⊙dx*=0,
dx=dx⊕(x⊙d1),
dx≤x,
If I is an ideal of an MV-algebra M, then d(I)⊆I.
Proof.
(i) d0=d(x⊙0)=(dx⊙0)⊕(x⊙d0)=x⊙d0.
Putting x=0, we get d0=0.
Let x∈M, then
0=d0=d(x⊙x*)=(dx⊙x*)⊕(x⊙dx*),
and so (ii) follows from (4).
It is clear.
Let x∈M, from (ii), we have
1=0*=(dx⊙x*)*=(dx)*⊕x,
from Theorem 2.2 we get dx≤x.
Let y∈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.
Let d be a derivation of an MV-algebra M, and let x,y∈M. If x≤y. Then, the following hold:
d(x⊙y*)=0,
dy*≤x*,
dx⊙dy*=0.
Proof.
(i) Let x≤y, then Theorem 2.2 implies that x⊙y*=0, and so d(x⊙y*)=d0=0.
From (i), we get
0=d(x⊙y*)=(dx⊙y*)⊕(x⊙dy*),
and by (4), we have x⊙dy*=0. Therefore, dy*≤x*.
If x≤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.
Let M be an MV-algebra, and let d be a derivation on M. Then, the following hold:
dx⊙dx*=0,
dx*=(dx)* if and only if d is the identity on M.
Proof.
(i) It follows directly from Proposition 3.5(iii).
It is sufficient to show that if dx*=(dx)*, then d is the identity on M.
Assume that dx*=(dx), from Proposition 3.4(ii), we have x⊙(dx)*=0, which implies that x≤dx. Therefore, dx=x.
Definition 3.7.
Let M 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.
Let M be an MV-algebra as in Example 3.3. It is easily checked that d is an isotone derivation of M.
Proposition 3.9.
Let M be an MV-algebra, and let d be aderivation of M. If dx*=dx for all x∈M, then the following hold:
d1=0,
dx⊙dx=0,
If d is an isotone derivation of M, then d is zero.
Proof.
(i) It follows by putting x=0.
It follows from Proposition 3.6(i).
Since d is an isotone, hence dx≤d1 for all x∈M. By (i), we have dx≤0, and so d is zero.
Definition 3.10.
Let M 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.
Let M 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 let d be a nonzero additive derivation of M. Then, d(B(M))⊆B(M).
Proof.
Let y∈d(B(M)), thus y=d(x) for some x∈B(M). Then,
y⊕y=dx⊕dx=d(x⊕x)=dx=y.
Therefore y∈B(M), this complete the proof.
Theorem 3.13.
Let d be an additive derivation of a linearly ordered MV-algebra M. Then, either d=0 or d1=1.
Proof.
Let d be an additive derivation of a linearly ordered MV-algebra M. Hence,
d1=d(x⊕x*)=dx⊕dx*,
also,
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,
0=d1=dx⊕d1=dx,
for all x∈M, and so d is zero.
Proposition 3.14.
Let M 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.
Let d1d2=0, x∈M, and suppose that d2≠0. Then,
0=d1d2x=d1(d2x⊕(x⊙d21))=d1d2x⊕d1x=d1x,
thus d1=0. Similarly, we can prove that d2=0.
Proposition 3.15.
Let M be a linearly ordered MV-algebra, and let d be a nonzero additive derivation of M. Then,
d(x⊙x)=x⊕x,∀x∈M.
Proof.
From Proposition 3.4(iii) and Theorem 3.13, we get that dx=dx⊕x; applying (9), we have dx⊙x=x. Thus,
d(x⊕x)=(dx⊙x)⊕(dx⊙x)=x⊕x.
Theorem 3.16.
Every nonzero additive derivation of a linearly ordered MV-algebra M is an isotone derivation.
Proof.
Assume that d is an additive derivation of M, and x,y∈M. If x≤y, then x*⊕y=1, hence
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.
Let M 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 Proposition 3.4(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, let x∈d-1(0) and y≤x. Using Theorem 3.16, we have that dy≤dx, and so dy=0.
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