Strongly transitive geometric spaces: Applications to hypergroups and semigroups theory

In this paper we determine a family P,(H) of subsets of a hypergroup H such that the geometric space (H, P,(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, (H). In this case a n-tuple (x(1), x(2),..., x(n)) is an element of H-n exists such that D(H) = B(x(1), x(2),...,x(n)) = {x is an element of H\There Exists sigma is an element of a E S-n, x is an element of Pi(i=1)(n) (sigma(i))}. Moreover, in the last section, we prove that in every semigroup the transitive closure gamma* of the relation gamma is the smallest congruence such that G/(gamma*). is a commutative semigroup. We determine a necessary and sufficient condition such that the geometric space (G, P,(G)) of a 0-simple semigroup is strongly transitive. Finally, we prove that if G is a simple semigroup, then the space (G, Psigma (G)). is strongly transitive and the relation gamma of G is transitive.