Multiple-valued hyperstructures

Abstract

An n-ary hyperoperation on A is a map from A/sup n/ into the set P of nonvoid subsets of A. A hyperclone on A is a set of hyperoperations on A containing all projections and closed with respect to a natural composition. Although special hyperalgebras, like hypergroups, hyperrings etc., have been studied for 6 decades there is no universal-algebra type theory for hyperalgebras. We try to close this gap by embedding hyperoperations on A into the set Q of all /spl sube/-isotone operations on P. The very crucial compatible relations are introduced through this embedding. For A finite we search for a general completeness criterion and the related maximal hyperclones via the maximal subclones of Q. For this we determine the position of Q in the lattice of clones on P and initiate the study of such meet-reducible clones. We find all such clones of the form Q/spl cap/Pol /spl rho/ where /spl rho/ is a proper unary relation on P, toe reduce the case of equivalence relations and show that two types of maximal clones on P produce no maximal subclone of Q.