Triple systems and binary operations

Abstract

It is well known that given a Steiner triple system (STS) one can define a binary operation ∗ upon its base set by assigning x∗x=x for all x and x∗y=z, where z is the third point in the block containing the pair {x,y}. The same can be done for Mendelsohn triple systems (MTSs) as well as hybrid triple systems (HTSs), where (x,y) is considered to be ordered. In the case of STSs and MTSs, the operation is a quasigroup, however this is not necessarily the case for HTSs. In this paper we study the binary operation induced by HTSs. It turns out that each such operation ∗ satisfies y∈{x∗(x∗y),(x∗y)∗x}andy∈{(y∗x)∗x,x∗(y∗x)} for all x and y from the base set. We call every binary operation that fulfils this condition hybridly symmetric.Not all idempotent hybridly symmetric operations can be obtained from HTSs. We show that these operations correspond to decompositions of a complete digraph into certain digraphs on three vertices. However, an idempotent hybridly symmetric quasigroup always comes from an HTS. The corresponding HTS is then called a latin HTS (LHTS). The core of this paper is the characterization of LHTSs and the description of their existence spectrum.