This thesis follows up the results of article A. Drápal a I. M. Wanless, On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups. This paper dealt with the density of maximally non-associative quasigroups of a certain cons- truction. However, certain cases had to be neglected in the calculations due to restrictive conditions. The examination of these cases is the subject of this work. It turned out that the asymptotic behavior in the general case as in the article differs from the beha- vior in cases examined in our work. In addition to the calculations themselves, the work contains a theoretical introduction with an explanation of the constructions used in the previous article, as well as our own theory necessary for our calculations. In addition, we experimentally verified our results. 1

It is well known that for any 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. The quasigroups associated with STSs and MTSs satisfy the flexible law x · (y · x) = (x · y) · x but those associated with Latin DTSs and Latin HTSs need not. A DTS or an HTS is said to be pure if when considered as a twofold triple system it contains no repeated blocks. This thesis focuses on the study of Latin DTSs and Latin HTSs, in particular it aims to examine flexibility, purity and other related properties in these systems. Latin DTSs and Latin HTSs which admit a cyclic or a rotational automorphism are also studied. The existence spectra of these systems are proved and enumeration results are presented. A smaller part of the thesis is then devoted to examining the size of the centre of a Steiner loop and the connection to...