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Approximate homomorphisms of ternary semigroups. (English) Zbl 1112.39021
Let $$(G,[\;])$$ be a ternary semigroup, i.e., a nonempty set $$G$$ with an associative ternary operation $$G^3\ni(x,y,z)\mapsto [xyz]\in G$$. If $$(G_1,[\;]_1$$) is another ternary semigroup, then a mapping $$f:G\to G_1$$ is called a ternary homomorphism provided that $$f([xyz])=[f(x)f(y)f(z)]_1$$ for all $$x,y,z\in G$$. The authors consider the stability of ternary homomorphisms, following the classical Ulam’s problem for binary homomorphisms and Hyers’ solution. In the case where the codomain is a Banach space $$X$$ with ternary operation coming from binary addition, the stability is proved. In particular, if $$G$$ is commutative and $$f:G\to X$$ satisfies $\| f([xyz])-f(x)-f(y)-f(z)\| \leq\varepsilon,\qquad x,y,z\in G$ with some $$\varepsilon\geq 0$$, then there exists a unique ternary homomorphism $$T:G\to X$$ such that $\| f(x)-T(x)\| \leq\frac{\varepsilon}{2},\qquad x\in G.$ If $$G$$ is mapped into a normed algebra $$A$$ with multiplicative norm and with ternary multiplication, the superstability is shown. Namely, if $$f:G\to A$$ satisfies $\| f([xyz])-f(x)f(y)f(z)\| \leq \varepsilon,\qquad x,y,z\in G$ then $$f$$ is either bounded or it is a ternary homomorphism.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
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