Hypergroupoids and combinatorial structures

Let (H,∘) be a finite hypergroupoid, i.e., H is a finite set and with any a,b∈H is associated a non-empty subset a∘b⊂H . Given a finite hypergroupoid, one may consider a block design (H,B) , B={a∘b; a,b∈H} , where multiplicity is taken into account. Conversely, if (H,B) is a block design, there exist many hypergroupoids whose associated design is (H,B) . A t -(v,k,λ) design is a block design (H,B) such that (i) |H|=v ; (ii) |B|=k for any B∈B ; and (iii) any subset of H consisting of t elements appears as a subset of exactly λ blocks. If the design associated with a hypergroupoid is a 2 -(v,k,λ) design, it holds that λ(v−1)=vk(k−1) . A hypergroupoid (H,∘) is called a 2 -(v,k) hypergroupoid if it is associated with a 2 -(v,k,λ) design. In this paper the author studies conditions on v and k in order for a 2 -(v,k) hypergroupoid to exist. In that case, such a hypergroupoid is called a (k,d) -hypergroupoid, where d=λ−k(k−1) . Some extremal cases are studied in more detail. For example, (k,1) -hypergroupoids are studied in connection with a Steiner system of projective planes. A necessary and sufficient condition is obtained for a (k,k−1) -hypergroupoid to be a hypergroup. Finally, the author obtains a classification of quasigroups of order three.