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 (MTS), directed triple systems (DTS) as well as hybrid triple systems (HTS), where (x, y) is considered to be ordered. In the case of STSs and MTSs the operation yields a quasigroup, however this is not necessarily the case for DTSs and HTSs. A DTS or an HTS which induces a quasigroup is said to be Latin. In this paper we study Latin DTSs and Latin HTSs which admit a cyclic or a 1-rotational automorphism. We prove the existence spectra for these systems as well as the existence spectra for their pure variants. As a side result we also obtain the existence spectra of pure cyclic and pure 1-rotational MTSs.
This work was supported by grant SVV-2013-267317.
Acknowledgement
The author would like to thank the reviewers for their comments which helped to improve the paper.
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