On exponentiation of \(n\)-ary hyperalgebras (Q2773028)

For a positive integer \(n\), an \(n\)-ary hyperalgebra \(G\) is a pair \((X,p)\) where \(X\) is a set and \(p\) maps \(X^n\) to the set of non-empty subsets of \(X\). It is said to be medial if for any \(n\times n\) matrix\((x_{ij})\) over \(X\) we have \(p(p(x_{11},\dots,x_{1n}),\dots,p(x_{n1},\dots,x_{nn}))\subseteq p(x_1,\dots,x_n)\) whenever \(x_j\in p(x_{1j},\dots, x_{nj})\) for all \(j\). The first theorem shows that if \(G=(X,p)\), \(H=(Y,q)\) and \(G\) is medial, then there is a subhyperalgebra of the direct product \(G^Y\) whose carrier is Hom\((H,G)\). This subhyperalgebra is called the power of \(G\) and \(H\) and is denoted by \(G^H\). Properties of this power operation are studied, and the paper ends by considering links with category theory.