Isomorphism classes of the hypergroups of type U on the right of size five

AbstractBy means of a blend of theoretical arguments and computer algebra techniques, we prove that the number of isomorphism classes of hypergroups of type U on the right of order five, having a scalar (bilateral) identity, is 14751. In this way, we complete the classification of hypergroups of type U on the right of order five, started in our preceding papers [M. De Salvo, D. Freni, G. Lo Faro, A new family of hypergroups and hypergroups of type U on the right of size five, Far East J. Math. Sci. 26(2) (2007) 393–418; M. De Salvo, D. Freni, G. Lo Faro, A new family of hypergroups and hypergroups of type U on the right of size five Part two, Mathematicki Vesnik 60 (2008) 23–45; M. De Salvo, D. Freni, G. Lo Faro, On the hypergroups of type U on the right of size five, with scalar identity (submitted for publication)]. In particular, we obtain that the number of isomorphism classes of such hypergroups is 14865

Some properties of certain Subhypergroups

The structure of the hypergroup is much more complicated than that of the group. Thus there exist various kinds of subhypergroups.This paper deals with some of these subhypergroups and presents certain properties of the closed, invertible and ultra-closed subhypergroups

On the Hypergroups of Type U on the right of size five, with scalar identity

We examine hypergroups of type U on the right of size 5 with bilateral scalar identity.We will prove that these hypergroups are isomorphic to the group Z_5, if there exists a hyperproduct xy of size 1, with x, y not equal to the identity. Moreover, if they are not isomorphic to the group Z_5, then all hyperproducts xy, with x, y different from the identity, always contain the identity and have size greater than or equal to 3