Triple systems and binary operations

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 } and y ∈ { ( 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. l 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.