Extension theorems for functional equations with bisymmetric operations (Q1601710)

Two operations \(\mu:X^k\to X\) and \(\nu:X^n\to X\) are commuting if \[ {\underset {j}\nu} \bigl[{\underset {i}\mu} [x_{ij}]\bigr] ={\underset {i} \mu}\bigl[ {\underset {j}\nu} [x_{ij}]\bigr] \] (where \({\underset {i}\mu} [x_i]= \mu(x_1,\dots, x_k))\) holds for all \(x_{ij}\in X\), \((i=1,\dots,k,\;j=1, \dots,n)\). If an operation commutes with itself then it is called bisymmetric. Let \(m_1,\dots, m_k:X\times X\to Y\) be binary operations on \(X\). A subset \(K \subset X\) is termed \((m_1, \dots, m_k)\)-convex, if it is closed under these operations. A point \(x\in X\) is said to be \((m_1,\dots, m_k)\)-absorbable with respect to \(K\), if there exists a point \(x_0\in K\) such that \[ m_1(x,x_0), \dots, m_k(x,x_0) \in K \] The set of all \((m_1,\dots, m_k)\) absorbable points of \(K\) is denoted by \(\overline K\). The main result of this paper is the following. Theorem 1. Let \(m_1,\dots, m_k:X\times X\to X\) be pairwise commuting binary operations on \(X\) and let \(K\subset X\) be \((m_1,\dots, m_k)\)-convex. Let, furthermore, \(Y\) be a set and, \(M:Y^k\to Y\) be a bisymmetric operation on \(Y\). Assume that \(f:K\to Y\) satisfies following equation \[ f(x)= M \biggl(f\bigl(m_1 (x,y)\bigr), \dots, f\bigl(m_k(x,y) \bigr)\biggr)\quad (x,y\in K). \] Then there exists a uniquely determined extension \(\overline f:\overline K\to Y\) of \(f\) such that \[ \overline f(x)= M\overline f\bigl(m_1 (x,y)\bigr), \dots,\overline f\bigl(m_k (x,y)\bigr) \quad (x,y\in \overline K). \] As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.