Extension theorems for functional equations with bisymmetric operations

Summary.

In this paper we deal with the extension of the following functional equation¶¶$f (x) = M \bigl(f (m_{1}(x, y)), \dots, f (m_{k}(x, y))\bigr) \qquad (x, y \in K)$, (*)¶ where M is a k-variable operation on the image space Y, m ₁,..., m _k are binary operations on X, $K \subset X$ is closed under the operations m ₁,..., m _k , and $f : K \rightarrow Y$ is considered as an unknown function.¶ The main result of this paper states that if the operations m ₁,..., m _k , M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m ₁,..., m _k )-affine hull K ^* of K, such that (*) holds over K ^*. (The set K ^* is defined as the smallest subset of X that contains K and is (m ₁,..., m _k )-affine, i.e., if $x \in X$, and there exists $y \in K^*$ such that $m_{1}(x, y), \ldots, m_{k}(x, y) \in K^*$ then $x \in K^*$). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.