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- W2045589413 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 } 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 . 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." @default.
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- W2045589413 date "2014-06-01" @default.
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- W2045589413 title "Triple systems and binary operations" @default.
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- W2045589413 doi "https://doi.org/10.1016/j.disc.2014.02.007" @default.
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