Contributo alle iperstrutture matroidali

• Main Author: Domenico Freni
• Format: Article
• Language: English
• Published: Università degli Studi di Catania 1997-11-01
• Series: Le Matematiche
• Online Access: http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/408

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