Domenico Freni

A NEW CHARACTERIZATION OF THE DERIVED HYPERGROUP VIA STRONGLY REGULAR EQUIVALENCES

ABSTRACT A new strongly regular equivalence is defined and a new characterization of the derived hypergroup of a hypergroup is determined.

Strongly Transitive Geometric Spaces: Applications to Hypergroups and Semigroups Theory

Abstract In this paper we determine a family P σ(H) of subsets of a hypergroup H such that the geometric space (H, P σ(H)) is strongly transitive and we use this fact to characterize the hypergroups such that the derived hypergroup D(H) of H coincides with an element of P σ(H). In this case a n-tuple (x 1, x 2,…, x n ) ∈ H n exists such that . Moreover, in the last section, we prove that in every semigroup the transitive closure γ* of the relation γ is the smallest congruence such that is a commutative semigroup. We determine a necessary and sufficient condition such that the geometric space (G, P σ(G)) of a 0-simple semigroup is strongly transitive. Finally, we prove that if G is a simple semigroup, then the space (G, P σ(G)) is strongly transitive and the relation γ of G is transitive.

Existence of proper semihypergroups of type U on the right

We generalize the classical definition of hypergroups of type U on the right to semihypergroups, and we prove some properties of their subsemihypergroups and subhypergroups. In particular, we obtain that a finite proper semihypergroup of type U on the right can exist only if its order is at least 6. We prove that one such semihypergroup of order 6 actually exists. Moreover, we show that there exists a hypergroup of type U on the right of cardinality 9 containing a proper non-trivial subsemihypergroup. In this way, we solve a problem left open in [D. Freni, Sur les hypergroupes de type U et sous-hypergroupes engendres par un sous-ensemble, Riv. Mat. Univ. Parma 13 (1987) 29-41].

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

By 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.