We say that is a ternary semigroup if the operation is associative, that is, if holds for all (see [2]).
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 say that ((G, [cdot, cdot, cdot])) is a ternary semigroup if the operation ([cdot, cdot, cdot]) is associative, i.e., if bigl[[x, y, z], u, vbigr] = bigl[x, [y, z, u], vbigr] = bigl[x, y, [z, u, v]bigr] for all (x, y, z, u, v in G) (see [29]).
Let A be a ternary algebra.
Corollary 4.7 Let A be a ternary Banach algebra.
Throughout this article, for a ternary Banach algebra (or C*-ternary ring) A, we denote A × A × ⋯ × A ⏞ n - t i m e s by A n. Definition 4.1 Let A be a ternary Banach algebra or C*-ternary ring.
Example 1.2 Let ( X, ∥ ⋅ ∥ ) be a ternary normed (Banach) algebra.
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 G → → G and f : G → [0, ∞) are functions satisfying (2) and (3).
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.
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.
