By modifying a construction for Hadamard (Menon) difference sets we construct two infinite families of negative Latin square type partial difference sets in groups of the form Z3 × Zp where p is any odd prime. One of these families has the wellknown Paley parameters, which had previously only been constructed in p-groups. This provides new constructions of Hadamard matrices and implies the exis...

Recently, V.Ginzburg introduced the notion of a principal nilpotent pair (= pn-pair) in a semisimple Lie algebra g [Gi99]. It is a double counterpart of the notion of a regular nilpotent element in g. A pair e = (e1, e2) ∈ g × g is called nilpotent, if [e1, e2] = 0 and there exists a pair h = (h1, h2) of semisimple elements such that [h1, h2] = 0, [hi, ej ] = δijej (i, j ∈ {1, 2}). A pn-pair e ...

In this paper first we introduce the notion of weak $Gamma$-(semi)hypergroups, next some classes of equivalence relations which are called good regular and strongly good regular relations are defined.  Then we investigate some properties of this kind of relations on weak $Gamma$-(semi)hypergroups.