PROPERTIES OF HYPERPRODUCTS AND THE RELATION β IN QUASIHYPERGROUPS

Abstract

Some properties of the complete parts in hyper-groupoids are established. Applying these properties to the case of quasihypergroups H for which H/β*. is a quasigroup, a necessary and sufficient condition for the transitivity of the relation β is proved. Consequently, several classes of quasihypergroups in which β is transitive are obtained (for instance, in any finite quasihypergroup with identity β is a transitive relation). Then, in the case of quasihypergroups having underlying groups, the relation β is completely de-termined.