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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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