Contributo alle iperstrutture matroidali

Abstract

In this paper we continue the investigation of matroidal hyperstructures, introduced in [8], [9], [15], [17], [22]. In the first section, we define the sub-hypergroupoid [A] generated by a subset A of an hypergroupoid (H, •).  We introduce the notion of defect and optimal defect of [A] and nvestigate their properties.
In the following sections, we define the class of M_λ -hypergroups, attached to an action of a group on a set M . We give necessary, and necessary and sufficient conditions for a M_λ -hypergroupoid to be matroidal. The most significant case is given by the canonical action of the multiplicative group K^∗ of a field K on the set V − {0}, where V is a K-vector space. Moreover, we determine necessary conditions under which the defect of a sub-hypergroupoid [A] of a M_λ -hypergroupoid is optimal; we also give examples of non-commutative matroidal hypergroupoids.
      In the last section, we continue the investigation of finite non-commutative exchange groups.