Combinatorial structures in loops I. Elements of the decomposition theory

Abstract

Difference sets have been extensively studied in groups, principally in Abelian groups. Here we extend the notion of a difference set to loops. This entails considering the class of 〈 υ, k〉 systems and the special subclasses of 〈 υ, k, λ〉 principal block partial designs (PBPDs) and 〈 υ, k, λ〉 designs. By means of a certain permutation matrix decomposition of the incidence matrices of a system and its complement, we can isomorphically identify an abstract 〈 υ, k〉 system with a corresponding system in a loop. Special properties of this decomposition correspond to special algebraic properties of the loop. Here we investigate the situation when some or all of the elements of the loop are right inversive. We identify certain classes of 〈 υ, k, λ〉 designs, including skew-Hadamard designs and finite projective planes, with designs and difference sets in right inverse property loops and prove a universal existence theorem for 〈 υ, k, λ〉 PBPDs and corresponding difference sets in such loops.