An algebraic approach to hyperalgebras

Abstract

In the past 6 decades the theory of hypergroups and other concrete hyperalgebras has fairly developed but there is still no coherent universal-algebra type theory of hyperalgebras. We represent hyperalgebras on a universe A as special universal algebras on the set P*(A) (of all nonvoid subsets of A), define hyperclones on A and for A finite, study the relationship between the hyperclones on A and the inclusion-isotone clones on P* (A). We introduce new notions of subuniverses, congruences and homomorphisms of hyperalgebras. Finally we raise a few natural problems concerning the lattice of inclusion-isotone clones on P*(A); in particular for the boolean case A={0, 1}.