Ternary γ-homomorphisms and ternary γ-derivations on ternary semigroups
© Dehghanian and Modarres; licensee Springer. 2012
Received: 6 July 2011
Accepted: 15 February 2012
Published: 15 February 2012
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.
Keywordsternary semigroup ternary γ-homomorphism ternary γ-derivation ternary (γ, h)-derivation.
1 Introduction and preliminaries
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  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 : G → G' satisfies ρ(f(xy), f(x)f(y)) < δ for all x, y ∈ G, then a homomorphism h : G → G 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  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  generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences. This phenomenon of stability that was introduced by Rassias  is called the Hyers-Ulam-Rassias stability. In 1992, a generalization of Rassias theorem was obtained by Găvruta .
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 [9–15].
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 ).
Definition 1.1. A nonempty set G with one ternary operation [ ]: G × G × G → G is called a ternary groupoid and denoted by (G, [ ]).
hold for all x, y, z, u, v ∈ G (see ). We shall write x3 instead of [xxx].
One can prove (post ) that elements x, y, z are uniquely determined. Moreover, according to the suggestion of post  one can prove (cf, Dudek et al. ) 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 and called the skew element to x (cf. Dörnte ). As a consequence of results obtained in  we have the following theorem:
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.
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).
for all x, y, z ∈ G.
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
for all x, y, z ∈ G.
and γ(x3) = 3γ(x). If G is commutative and H is a ternary Jordan homomorphism, then mapping H : G → G is a ternary γ-homomorphism.
Passing to the limit as n → ∞ we get γ(x) = γ'(x), x ∈ G. So γ is unique. Therefore, the mapping H : G → G is a unique ternary γ-homomorphism.
for all x, y, z ∈ G, then the mapping T : G → G is a ternary γ-homomorphism.
for all x, y, z ∈ G. Therefore T is a ternary γ-homomorphism.
and γ(x3) = 3γ(x). If G is commutative and H is a ternary Jordan homomorphism, then the mapping H : G → G is a ternary γ-homomorphism.
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
for all x, y, z ∈ G.
and γ (x3) = 3γ(x). If G is commutative and D is a ternary Jordan homomorphism, then mapping D : G → G is a ternary γ-derivation.
Passing to the limit as n → ∞ we get γ(x) = γ'(x), x ∈ G. This proves the uniqueness of γ. Thus, the mapping D : G → G is a unique ternary γ-derivation.
and γ(x3) = 3γ(x). If G is commutative and D is a ternary Jordan homomorphism, then mapping D : G → G is a ternary γ-derivation.
for all x, y, z ∈ G and for all positive integer n. Then the mapping D : G → G is a ternary f-derivation.
for all x, y, z ∈ G. Thus, the mapping D : G → G 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).
for all x, y, z ∈ G.
and γ(x3) = 3γ(x). If G is commutative and D, h are ternary homomorphisms, then mapping D : G → G is a ternary (γ, h)-derivation.
for all x ∈ G.
for all x, y, z ∈ G.
If in Theorem 4.2 replace inequality 12 by equation to obtain the following Theorem.
for all x, y, z ∈ G and for all positive integer n. Then the mapping D : G → G is a ternary (f, h)-derivation.
- Cayley A: On the 34 concomitants of the ternary cubic. Am J Math 1881, 4: 1–15. 10.2307/2369145MATHMathSciNetView ArticleGoogle Scholar
- Kapranov M, Gelfand IM, Zelevinskii A: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Berlin; 1994.Google Scholar
- Kerner R: Ternary algebraic structures and their applications in physics. Pierre et Marie Curie University, Paris; 2000.http://arxiv.org/list/math-ph/0011 Ternary algebraic structures and their applications in physics, Proc. BTLP, 23rd International Conference on Group Theoretical Methods in Physics, Dubna, Russia (2000);Google Scholar
- Kerner R: The cubic chessboard: geometry and physics. Class Quantum Gravity 1997, 14: A203-A225. 10.1088/0264-9381/14/1A/017MATHMathSciNetView ArticleGoogle Scholar
- Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.Google Scholar
- Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
- Rassias ThM: On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1MATHView ArticleGoogle Scholar
- Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl 1994, 184: 431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleGoogle Scholar
- Park C, Cho Y-S, Han M-H: Functional inequalities associated with Jordan-von Neumann-type additive functional equations. J Inequal Appl 2007., 13: 2007(Article ID 41820)Google Scholar
- Park W-G, Bae J-H: Approximate behavior of bi-quadratic mappings in quasinormed spaces. J Inequal Appl 2010., 8: 2010(Article ID 472721)Google Scholar
- Park C: Isomorphisms between C*-ternary algebras. J Math Anal Appl 2007, 327: 101–115. 10.1016/j.jmaa.2006.04.010MATHMathSciNetView ArticleGoogle Scholar
- Bae J-H, Park W-G: On a cubic equation and a Jensen-quadratic equation. Abst Appl Anal 2007., 10: 2007(Article ID 45179)Google Scholar
- Gordji Eshaghi M, Karimi T, Gharetapeh Kaboli S: Approximately n -Jordan homomorphisms on Banach algebras. J Inequal Appl 2009., 8: 2009(Ar-ticle ID 870843)Google Scholar
- Kim H-M, Kang S-Y, Chang I-S: On functional inequalities originating from module Jordan left derivations. J Inequal Appl 2008., 9: 2008(Article ID 278505)Google Scholar
- Hyers D-H, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View ArticleGoogle Scholar
- Rusakov S-A: Some Applications of n -ary Group Theory. Belaruskaya navuka, Minsk; 1998.Google Scholar
- Bazunova N, Borowiec A, Kerner R: Universal diferential calculus on ternary algebras. Lett Math Phys 2004, 67(3):195–206.MATHMathSciNetView ArticleGoogle Scholar
- Post E-L: Polyadic groups. Trans Am Math Soc 1940, 48: 208–350. 10.1090/S0002-9947-1940-0002894-7MathSciNetView ArticleGoogle Scholar
- Dudek W-A, Glazek K, Gleichgewicht B: A Note on the Axioms of n -Groups. In Coll Math Soc J Bolyai 29. Universal Algebra, Esztergom, Hungary; 1977:195–202.Google Scholar
- Dörnte W: Unterschungen ber einen verallgemeinerten Gruppenbegriff. Math Z 1929, 29: 1–19. 10.1007/BF01180515MathSciNetView ArticleGoogle Scholar
- Dudek W-A: Autodistributive n -groups. Annales Sci Math Polonae Comment Math 1993, 23: 1–11.Google Scholar
- Dudek I, Dudek W-A: On skew elements in n -groups. Demon Math 1981, 14: 827–833.MATHMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.