Triple systems and binary operations

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F14%3A10287285" target="_blank" >RIV/00216208:11320/14:10287285 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1016/j.disc.2014.02.007" target="_blank" >http://dx.doi.org/10.1016/j.disc.2014.02.007</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.disc.2014.02.007" target="_blank" >10.1016/j.disc.2014.02.007</a>

Result language
angličtina
Original language name
Triple systems and binary operations
Original language description
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 thebinary operation induced by HTSs. It turns out that each such operation * satisfies y is an element of {x * (x * y), (x * y) * x} and y is an element of {(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.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2014
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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