Open Access

Ternary γ-homomorphisms and ternary γ-derivations on ternary semigroups

Journal of Inequalities and Applications20122012:34

https://doi.org/10.1186/1029-242X-2012-34

Received: 6 July 2011

Accepted: 15 February 2012

Published: 15 February 2012

Abstract

In this paper, we introduce the notions of γ-homomorphism and γ-derivation of a ternary semigroup and investigate γ-homomorphism and γ-derivations on ternary semigroup associated with the following functional in-equality |f([xyz]) - f(x) - f(y) - f(z)| ≤ φ(x, y, z) and |f([xxx]) - 3f(x)| ≤ φ(x, x, x), respectively.

2000 MSC: Primary 39B52, Secondary 39B82; 46B99; 17A40.

Keywords

ternary semigroup ternary γ-homomorphism ternary γ-derivation ternary (γ, h)-derivation.

1 Introduction and preliminaries

Ternary algebraic operations were considered in the 19th century by several mathematicians such as Cayley [1] who introduced the notion of "cubic matrix" which in turn was generalized by Kapranov, Gelfand and Zelevinskii et al. [2]. The simplest example of such non-trivial ternary operation is given by the following composition rule:
{ a , b , c } i j k = 1 l , m , n N a n i l b l j m c m k n ( i , j , k = 1 , 2 , , N ) .

Ternary structures and their generalization, the so-called n-ary structures, raise certain hopes in view of their possible applications in physics. Some significant physical applications are described in [3, 4].

In 1940, Ulam [5] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homo-morphisms:

We are given a group G and a metric group G' with metric ρ(·, ·). Given ϵ > 0, does there exist a δ > 0 such that if f : GG' satisfies ρ(f(xy), f(x)f(y)) < δ for all x, y G, then a homomorphism h : GG exists with ρ(f(x), h(x)) < ϵ for all x G?

As mentioned above, when this problem has a solution, we say that the homomorphisms from G1 to G2 are stable. In 1941, Hyers [6] gave a partial solution of Ulams problem for the case of approximate additive mappings under the assumption that G1 and G2 are Banach spaces. In 1978, Rassias [7] generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences. This phenomenon of stability that was introduced by Rassias [7] is called the Hyers-Ulam-Rassias stability. In 1992, a generalization of Rassias theorem was obtained by Găvruta [8].

During the last decades several stability problems of functional equations have been investigated be many mathematicians. A large list of references concerning the stability of functional equations can be found in [915].

In this article, using a sequence of Hyers type, we prove the generalized Hyers-Ulam-Rassias stability of ternary γ-homomorphisms and ternary γ-derivations on commutative ternary semigroups.

In the first section, which have preliminary character, we review some basic definitions and properties related to ternary groups and semigroups (cf. also Rusakov [16]).

Definition 1.1. A nonempty set G with one ternary operation [ ]: G × G × GG is called a ternary groupoid and denoted by (G, [ ]).

We say that (G, [ ]) is a ternary semigroup if the operation [ ] is associative, i.e., if
[ [ x y z ] u v ] = [ x [ y z u ] v ] = [ x y [ z u v ] ]

hold for all x, y, z, u, v G (see [17]). We shall write x3 instead of [xxx].

Definition 1.2. A ternary semigroup (G, [ ]) is a ternary group if for all a, b, c G, there are x, y, z G such that
[ x a b ] = [ a y b ] = [ a b z ] = c .

One can prove (post [18]) that elements x, y, z are uniquely determined. Moreover, according to the suggestion of post [18] one can prove (cf, Dudek et al. [19]) that in the above definition, under the assumption of the associativity, it suffices only to postulate the existence of a solution of [ayb] = c, or equivalently, of [xab] = [abz] = c.

In a ternary group, the equation [xxz] = x has a unique solution which is denoted by z = x ̄ and called the skew element to x (cf. Dörnte [20]). As a consequence of results obtained in [20] we have the following theorem:

Theorem 1.3. In any ternary group (G, [ ]) for all x, y, z G, the following identities take place:
[ x x x ̄ ] = [ x x ̄ x ] = [ x ̄ x x ] = x , [ y x x ̄ ] = [ y x ̄ x ] = [ x x ̄ y ] = [ x ̄ x y ] = y , [ x y z ] ¯ = [ z ̄ ȳ x ̄ ] , x ¯ ¯ = x .

Other properties of skew elements are described in [21, 22].

Definition 1.4. A ternary groupoid (G, [ ]) is called σ-commutative, if
[ x 1 x 2 x 3 ] = [ x σ 1 x σ 2 x σ 3 ]
(1)

holds for all x1, x2, x3 G and all σ S3. If (1) holds for all σ S3, then (G, [ ]) is a commutative groupoid. If (1) holds only for σ = (13), i.e., if [x1x2x3] = [x3x2x1], then (G, [ ]) is called semicommutative.

Definition 1.5. An element e G is called a middle identity or a middle neutral element of (G, [ ]), if for all x G we have
[ e x e ] = x .
An element e G satisfying the identity
[ e e x ] = x

is called a left identity or a left neutral element of (G, [ ]). Similarly, we define a right identity. An element which is a left, middle, and right identity is called a ternary identity (or simply identity).

A mapping f : (G, [ ]) → (G, [ ]) is called a ternary homomorphism if
f ( [ x y z ] ) = [ f ( x ) f ( y ) f ( z ) ]

for all x, y, z G.

A mapping f : (G, [ ]) → (G, [ ]) is called a ternary Jordan homomorphism if
f ( [ x x x ] ) = [ f ( x ) f ( x ) f ( x ) ]

for all x G.

In Section 2, we define ternary γ-homomorphism on ternary semigroup and investigate their relations.

2 Ternary γ-homomorphisms on ternary semigroups

Definition 2.1. Let G be a ternary semigroup. Then the maping H : GG is called a ternary γ-homomorphism if there exists a function γ : G → [0, ∞) such that
γ ( H ( [ x y z ] ) ) = γ ( [ H ( x ) H ( y ) H ( z ) ] ) = γ ( H ( x ) ) + γ ( H ( y ) ) + γ ( H ( z ) )

for all x, y, z G.

Theorem 2.2. Let G be a ternary semigroup and φ : G × G × G → [0, ∞) be a function such that
φ ̃ ( x , y , z ) : = 1 3 n = 0 3 - n φ ( x 3 n , y 3 n , z 3 n ) < .
Suppose that H : GG and f : G → [0, ∞) are functions such that
f ( [ x y z ] ) - f ( x ) - f ( y ) - f ( z ) φ ( x , y , z )
(2)
f ( H ( [ x y z ] ) ) - f ( [ H ( x ) H ( y ) H ( z ) ] ) φ ( x , y , z )
(3)
for all x, y, z G. Then there exists a unique function γ : G → [0, ∞) such that
f ( x ) - γ ( x ) φ ̃ ( x , x , x )

and γ(x3) = 3γ(x). If G is commutative and H is a ternary Jordan homomorphism, then mapping H : GG is a ternary γ-homomorphism.

Proof. Putting y = z = x in inequality (2), we get
f ( x 3 ) - 3 f ( x ) φ ( x , x , x ) .
By induction, one can show that
3 - n f ( x 3 n ) - f ( x ) 1 3 k = 0 n - 1 3 - k φ x 3 k , x 3 k , x 3 k ,
(4)
for all x G and for all positive integer n, and
3 - n f ( 3 3 n ) - 3 - m f ( x 3 m ) 1 3 k = m n - 1 3 - k φ x 3 k , x 3 k , x 3 k
for all x G and for all nonnegative integers m, n with m < n. Hence, { 3 - n f ( x 3 n ) } is a Cauchy sequence in [0, ∞). Due to the completeness of [0, ∞) we conclude that this sequence is convergent. Now, let
γ ( x ) = lim n 3 - n f ( x 3 n ) , x G .
Hence
γ ( x 3 ) = lim n 3 - n f x 3 n + 1 = 3 lim n 3 - ( n + 1 ) f x 3 n + 1 = 3 γ ( x )
for all x G. If n → ∞ in inequality (4), we obtain
f ( x ) - γ ( x ) φ ̃ ( x , x , x ) .
Next, assume that G is commutative and H : GG is a ternary Jordan homomorphism. Replace x by x 3 n , y by y 3 n and z by z 3 n in inequalities (2) and (3) and divide both sides by 3 n to obtain the following:
3 - n f ( [ x y z ] 3 n ) - 3 - n f ( x 3 n ) - 3 - n f ( y 3 n ) - 3 - n f ( z 3 n ) 3 - n φ ( x 3 n , y 3 n , z 3 n ) ,
and
3 - n f ( ( H [ x y z ] ) 3 n ) - 3 - n f ( [ H ( x ) H ( y ) H ( z ) ] 3 n ) 3 - n φ ( x 3 n , y 3 n , z 3 n ) .
If n tends to infinity. Then
γ ( H [ x y z ] ) = γ ( [ H ( x ) H ( y ) H ( z ) ] ) = γ ( H ( x ) ) + γ ( H ( y ) ) + γ ( H ( z ) ) ,
for all x, y, z G. If γ' is another mapping with the required properties, then
γ ( x ) - γ ( x ) = 1 3 n 3 n γ ( x ) - 3 n γ ( x ) = 1 3 n γ ( x 3 n ) - γ ( x 3 n ) 1 3 n ( γ ( x 3 n ) - f ( x 3 n ) + f ( x 3 n ) - γ ( x 3 n ) ) 2 3 n φ ̃ ( x 3 n , x 3 n , x 3 n ) .

Passing to the limit as n → ∞ we get γ(x) = γ'(x), x G. So γ is unique. Therefore, the mapping H : GG is a unique ternary γ-homomorphism.

Theorem 2.3. Let G be a commutative ternary semigroup and φ : G × G × G → [0, ∞) be a function such that
φ ̃ ( x , y , z ) : = 1 3 n = 0 3 - n φ ( x 3 n , y 3 n , z 3 n ) < .
Suppose that H : GG and f : G → [0, ∞) are functions satisfying (2) and (3). If there exists a mapping T : GG such that T is a ternary Jordan homomorphism and
f ( H ( [ x y z ] ) ) - f ( [ H ( x ) H ( y ) T ( z ) ] ) φ ( x , y , z )
(5)

for all x, y, z G, then the mapping T : GG is a ternary γ-homomorphism.

Proof. By Theorem 2.2, there exists a unique mapping γ : G → [0, ∞) such that
γ ( x ) = lim n 3 - n f ( x 3 n ) , x G ,
and H : GG is a ternary γ-homomorphism. It follows from (5) that
γ ( [ H ( x ) H ( y ) H ( z ) ] ) - γ ( [ H ( x ) H ( y ) T ( z ) ] ) = γ ( H [ x y z ] ) - γ ( [ H ( x ) H ( y ) T ( z ) ] ) = lim n 1 3 n f ( ( H [ x y z ] ) 3 n ) - f ( [ H ( x ) H ( y ) T ( z ) ] 3 n ) lim n 1 3 n φ ( x 3 n , y 3 n , z 3 n ) = 0
for all x, y, z G. So, γ([H(x)H(y)H(z)]) = γ([H(x)H(y)T(z)]) for all x, y, z G. By (2), γ is ternary additive. Hence, γ(H(x)) = γ(T(x)) for all x G. Thus,
γ ( T [ x y z ] ) = γ ( H [ x y z ] ) = γ ( H ( x ) ) + γ ( H ( y ) ) + γ ( H ( z ) ) = γ ( T ( x ) ) + γ ( T ( y ) ) + γ ( T ( z ) ) = γ ( [ T ( x ) T ( y ) T ( z ) ] )

for all x, y, z G. Therefore T is a ternary γ-homomorphism.

Corollary 2.4. Let G be a ternary group with identity element e and φ : G5 → [0, ∞) be a function such that
φ ̃ ( x , y , u . v . w ) : = 1 3 n = 0 3 - n φ ( x 3 n , y 3 n , u 3 n , v 3 n , w 3 n ) < .
Suppose that H : GG and f : G → [0, ∞) are functions such that f(e) = 0, H(e) = e and
f ( [ x y H ( [ u v w ] ) ] ) - f ( x ) - f ( y ) - f ( [ H ( u ) H ( v ) H ( w ) ] )
(6)
φ ( x , y , H ( u ) , v , w )
(7)
for all x, y, u, v, w G. Then there exists a unique function γ : G → [0, ∞) such that
f ( x ) - γ ( x ) φ ̃ ( x , x , x , e , e )

and γ(x3) = 3γ(x). If G is commutative and H is a ternary Jordan homomorphism, then the mapping H : GG is a ternary γ-homomorphism.

Proof. Letting v = w = e in (6), we get
f ( [ x y H ( u ) ] ) - f ( x ) - f ( y ) - f ( H ( u ) ) φ ( x , y , H ( u ) , e , e )
and by putting x = y = e in (6) we get
f ( [ H ( [ u v w ] ) ] ) - f ( [ H ( u ) H ( v ) H ( w ) ] ) φ ( e , e , H ( u ) , v , w ) .

The rest of the proof are similar to the proof of Theorem 2.2.

In next section, firstly we define ternary γ-derivation on ternary semigroup and investigate ternary γ-derivations on ternary semigroups with the following functional inequality |f([xxx]) - 3f(x)| ≤ φ(x, x, x).

3 Ternary γ-derivations on ternary semigroups

Definition 3.1. Let G be a ternary semigroup. Then the map D : GG is called a ternary γ-derivation if there exists a function γ : G → [0, ∞) such that
γ ( D ( [ x y z ] ) ) = γ ( [ D ( x ) y z ] ) + γ ( [ x D ( y ) z ] ) + γ ( [ x y D ( z ) ] )

for all x, y, z G.

Theorem 3.2. Let G be a ternary semigroup and φ : G × G × G → [0, ∞) be a function such that
φ ̃ ( x , y , z ) : = 1 3 n = 0 3 - n φ ( x 3 n , y 3 n , z 3 n ) < .
Suppose that f : G → [0, ∞) is a function such that
f ( x 3 ) - 3 f ( x ) φ ( x , x , x )
(8)
f ( D ( [ x y z ] ) ) - f ( [ D ( x ) y z ] ) - f ( [ x D ( y ) z ] ) - f ( [ x y D ( z ) ] ) φ ( x , y , z )
(9)
for all x, y, z G and mapping D : GG. Then there exists a unique function γ : G → [0, ∞) such that
f ( x ) - γ ( x ) φ ̃ ( x , x , x )

and γ (x3) = 3γ(x). If G is commutative and D is a ternary Jordan homomorphism, then mapping D : GG is a ternary γ-derivation.

Proof. By induction in (8), one can show that
3 - n f ( x 3 n ) - f ( x ) 1 3 k = 0 n - 1 3 - k φ x 3 k , x 3 k , x 3 k ,
(10)
for all x G and for all positive integer n, and
3 - n f ( 3 3 n ) - 3 - m f ( x 3 m ) 1 3 k = m n - 1 3 - k φ x 3 k , x 3 k , x 3 k
for all x G and for all nonnegative integers m, n with m < n. Hence, { 3 - n f ( x 3 n ) } is a Cauchy sequence in [0, ∞). Due to the completeness of [0, ∞) we conclude that this sequence is convergent. Set now
γ ( x ) = lim n 3 - n f ( x 3 n ) , x G .
Hence
γ ( x 3 ) = lim n 3 - n f x 3 n + 1 = 3 lim n 3 - ( n + 1 ) f x 3 n + 1 = 3 γ ( x )
for all x G. If n → ∞ in inequality (10), we obtain
f ( x ) - γ ( x ) φ ̃ ( x , x , x ) .
Next, assume that G is commutative and D : GG is a ternary Jordan homomorphism. Replace x by x 3 n , y by y 3 n and z by z 3 n in inequality (9) and divide both sides by 3 n , we have
3 - n f ( D ( [ x y z ] ) 3 n ) - 3 - n f ( [ D ( x ) y z ] 3 n ) - 3 - n f ( [ x D ( y ) z ] 3 n ) - 3 - n f ( [ x y D ( z ) ] 3 n ) 3 - n φ ( x 3 n , y 3 n , z 3 n ) .
If n tends to infinity. Then
γ ( D ( [ x y z ] ) ) = γ ( [ D ( x ) y z ] ) + γ ( [ x D ( y ) z ] ) + γ ( [ x y D ( z ) ] )
for all x, y, z G. If γ' is another mapping with the required properties, then
γ ( x ) - γ ( x ) = 1 3 n 3 n γ ( x ) - 3 n γ ( x ) = 1 3 n γ ( x 3 n ) - γ ( x 3 n ) 1 3 n ( γ ( x 3 n ) - f ( x 3 n ) + f ( x 3 n ) - γ ( x 3 n ) ) 2 3 n φ ̃ ( x 3 n , x 3 n , x 3 n ) .

Passing to the limit as n → ∞ we get γ(x) = γ'(x), x G. This proves the uniqueness of γ. Thus, the mapping D : GG is a unique ternary γ-derivation.

Corollary 3.3. Let G be a ternary semigroup, and ϵ > 0. Suppose that f : G → [0, ∞) is a function such that
f ( x 3 ) - 3 f ( x ) ε ,
f ( D ( [ x y z ] ) ) - f ( [ D ( x ) y z ] ) - f ( [ x D ( y ) z ] ) - f ( [ x y D ( z ) ] ) ε
for all x, y, z G and mapping D : GG. Then there exists a unique function γ : G → [0, ∞) such that
f ( x ) - γ ( x ) 1 2 ε

and γ(x3) = 3γ(x). If G is commutative and D is a ternary Jordan homomorphism, then mapping D : GG is a ternary γ-derivation.

Theorem 3.4. Let G be a commutative ternary semigroup and φ : G × G × G → [0, ∞) be a function such that
φ ̃ ( x , y , z ) : = 1 3 n = 0 3 - n φ ( x 3 n , y 3 n , z 3 n ) < .
Suppose that D : GG is a ternary Jordan homomorphism and f : G → [0, ∞) is a function such that
f ( x 3 n ) = 3 n f ( x )
f ( D ( [ x y z ] ) ) - f ( [ D ( x ) y z ] ) - f ( [ x D ( y ) z ] ) - f ( [ x y D ( z ) ] ) φ ( x , y , z )
(11)

for all x, y, z G and for all positive integer n. Then the mapping D : GG is a ternary f-derivation.

Proof. Since G is commutative and D : GG is ternary Jordan homomorphism. Replace x by x 3 n , y by y 3 n and z by z 3 n in inequality (11) and divide both sides by 3 n to obtain the following:
3 - n f ( D ( [ x y z ] ) 3 n ) - 3 - n f ( [ D ( x ) y z ] 3 n ) - 3 - n f ( [ x D ( y ) z ] 3 n ) - 3 - n f ( [ x y D ( z ) ] 3 n ) 3 - n φ ( x 3 n , y 3 n , z 3 n ) .
If n tends to infinity. Then
f ( D ( [ x y z ] ) ) = f ( [ D ( x ) y z ] ) + f ( [ x D ( y ) z ] ) + f ( [ x y D ( z ) ] )

for all x, y, z G. Thus, the mapping D : GG is a ternary f-derivation.

4 Ternary (γ, h)-derivations on ternary semigroups

In this section, we introduce concept ternary (γ, h)-derivations on ternary semigroups and investigate ternary (γ, h)-derivations on ternary semigroups with the following functional inequality |f([xxx]) - 3f(x)| < φ(x, x, x).

Definition 4.1. Let G be a ternary semigroup. Then the maping D : GG is called ternary (γ, h)-derivation if there exists mappings h : GG and γ : G → [0, ∞) such that
γ ( D ( [ x y z ] ) ) = γ ( [ D ( x ) h ( y ) h ( z ) ] ) + γ ( [ h ( x ) D ( y ) h ( z ) ] ) + γ ( [ h ( x ) h ( y ) D ( z ) ] )

for all x, y, z G.

Theorem 4.2. Let G be a ternary semigroup, and let φ : G × G × G → [0, ∞) be a function such that
φ ̃ ( x , y , z ) : = 1 3 n = 0 3 - n φ ( x 3 n , y 3 n , z 3 n ) < .
Suppose that D, h : GG and f : G → [0, ∞) are functions such that
f ( x 3 ) - 3 f ( x ) φ ( x , x , x )
(12)
f ( D ( [ x y z ] ) ) - f ( [ D ( x ) h ( y ) h ( z ) ] ) - f ( [ h ( x ) D ( y ) h ( z ) ] )
(13)
- f ( [ h ( x ) h ( y ) D ( z ) ] ) φ ( x , y , z )
(14)
for all x, y, z G. Then there exist a unique function γ : G → [0, ∞) such that
f ( x ) - γ ( x ) φ ̃ ( x , x , x )

and γ(x3) = 3γ(x). If G is commutative and D, h are ternary homomorphisms, then mapping D : GG is a ternary (γ, h)-derivation.

Proof. By a similar method to the proof of Theorem 3.2 we obtain
γ ( x ) = lim n 3 - n f ( x 3 n ) , x G .
Such that
f ( x ) - γ ( x ) φ ̃ ( x , x , x )
and
γ ( x 3 ) = 3 γ ( x )

for all x G.

Now suppose that G is commutative and D, h : GG are ternary homomorphism. Replace x by x 3 n , y by y 3 n and z by z 3 n in inequality (13) and divide both sides by 3 n to obtain the following:
3 - n f ( D ( [ x y z ] ) 3 n ) - 3 - n f ( [ D ( x ) h ( y ) h ( z ) ] 3 n ) - 3 - n f ( [ h ( x ) D ( y ) h ( z ) ] 3 n ) - 3 - n f ( [ h ( x ) h ( y ) D ( z ) ] 3 n ) 3 - n φ ( x 3 n , y 3 n , z 3 n ) .
Let n tend to infinity. Then
γ ( D ( [ x y z ] ) ) = γ ( [ D ( x ) h ( y ) h ( z ) ] ) + γ ( [ h ( x ) D ( y ) h ( z ) ] ) + γ ( [ h ( x ) h ( y ) D ( z ) ] )

for all x, y, z G.

If in Theorem 4.2 replace inequality 12 by equation f ( x 3 n ) = 3 n f ( x ) to obtain the following Theorem.

Theorem 4.3. Let G be a commutative ternary semigroup and φ : G × G × G → [0, ∞) be a function such that
φ ̃ ( x , y , z ) : = 1 3 n = 0 3 - n φ ( x 3 n , y 3 n , z 3 n ) < .
Suppose that D, h : GG are ternary Jordan homomorphism and f : G → [0, ∞) is a function such that
f ( x 3 n ) = 3 n f ( x )
f ( D ( [ x y z ] ) ) - f ( [ D ( x ) h ( y ) h ( z ) ] ) - f ( [ h ( x ) D ( y ) h ( z ) ] ) - f ( [ h ( x ) h ( y ) D ( z ) ] ) φ ( x , y , z )

for all x, y, z G and for all positive integer n. Then the mapping D : GG is a ternary (f, h)-derivation.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Yazd University

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© Dehghanian and Modarres; licensee Springer. 2012

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