Matroidal M_\lambda-hypergroupoids: \lambda-linear mappings and automorphisms of M_\lambda-hypergroupoids

In this paper we investigate the matroidal M_λ-hypergroupoids introduced in [D. Freni, Matematiche (Catania) 52 (1997), no. 2, 271–295 (1998); MR1626420 (99g:20125)]. We give necessary and sufficient conditions for an M_λ-hypergroupoid to be matroidal; moreover, we find its cardinality and we analyse the properties of the corresponding homomorphisms. The paper also gives a characterization of the group Λ(G_λ), that is, the group of automorphisms of the hypergroupoid G_λ; this hypergroupoid is obtained from a given group G, an element λ of G, and the action of G on G determined by λ through the same operation of G. If G has finite cardinality we have |Λ(G_λ)|=[G:⟨λ⟩]!|λ|^[G:⟨λ⟩].