Strongly Transitive Geometric Spaces: Applications to Hypergroups and Semigroups Theory

In this paper we determine a family P_\sigma(H) of subsets of a hypergroup H such that the geometric space (H, P_\sigma(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_\sigma(H). In this case a n-tuple (x_1, x_2,...,x_n)\in H^n exists such that D(H) = B(x_1, x_2,...,x_n) = {x\inH | \exist \sigma \in S_n, x\in x_\sigma(1)...x_sigma(n)}. 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_\sigma(G)) of a 0-simple semigroup is strongly transitive. Finally, we prove that if G is a simple semigroup, then the space (G, P_\sigma(G)) is strongly transitive and the relation \gamma of G is transitive.