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 (f (m₁(x, y)), ..., f (m_k(x, y)))        (x, y ? K) 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, m₁,..., m_k are binary operations on X, K ì X K subset X is closed under the operations m₁,..., m_k, and f : K ? Y f : K rightarrow Y is considered as an unknown function.¶ The main result of this paper states that if the operations m₁,..., m_k, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m₁,..., 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 (m₁,..., m_k)-affine, i.e., if x ? X x in X , and there exists y ? K^* y in K^* such that m₁(x, y), ?, m_k(x, y) ? K^* m_{1}(x, y), ldots, m_{k}(x, y) in K^* then x ? K^* x in K^* ). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.