On exponentiation of universal hyperalgebras (Q1866839)

The paper is a continuation of the author's earlier work devoted to a problem of exponentiation of relational systems or universal algebras, respectively. Here a binary operation of exponentiation, which to a pair of universal hyperalgebras of a given type assigns their power, is introduced and studied. Universal hyperalgebras lie between relational systems and universal algebras. More precisely, a universal hyperalgebra is a pair \(\langle X,(p_{\lambda };\lambda \in \Omega)\rangle \) where \(X\) is a set, \(p_{\lambda }\) is an \(n_{\lambda }\)-ary hyperoperation on \(X\) (i.e. the map \(p_{\lambda }:X^{n_{\lambda }}\rightarrow \exp X\backslash \{\emptyset \})\) for each \(\lambda \in \Omega\), and \(\Omega \) is a class. The result of exponentiation is a universal hyperalgebra of the same type carried by the corresponding set of homomorphisms. Sufficient conditions for the existence of such a power and for a decent behavior of the exponentiation are given. As a consequence, a Cartesian closed subcategory of the category of universal hyperalgebras with homomorphisms as morphisms is discovered.