9 research outputs found

    Distributive and trimedial quasigroups of order 243

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    We enumerate three classes of non-medial quasigroups of order 243=35243=3^5 up to isomorphism. There are 1700417004 non-medial trimedial quasigroups of order 243243 (extending the work of Kepka, B\'en\'eteau and Lacaze), 9292 non-medial distributive quasigroups of order 243243 (extending the work of Kepka and N\v{e}mec), and 66 non-medial distributive Mendelsohn quasigroups of order 243243 (extending the work of Donovan, Griggs, McCourt, Opr\v{s}al and Stanovsk\'y). The enumeration technique is based on affine representations over commutative Moufang loops, on properties of automorphism groups of commutative Moufang loops, and on computer calculations with the \texttt{LOOPS} package in \texttt{GAP}

    On Loop Commutators, Quaternionic Automorphic Loops, and Related Topics

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    This dissertation deals with three topics inside loop and quasigroup theory. First, as a continuation of the project started by David Stanovský and Petr Vojtĕchovský, we study the commutator of congruences defined by Freese and McKenzie in order to create a more pleasing, equivalent definition of the commutator inside of loops. Moreover, we show that the commutator can be characterized by the generators of the inner mapping group of the loop. We then translate these results to characterize the commutator of two normal subloops of any loop. Second, we study automorphic loops with the desire to find more examples of small orders. Here we construct a family of automorphic loops, called quaternionic automorphic loops, which have order 2n for n ≥ 3, and prove several theorems about their structure. Although quaternionic automorphic loops are nonassociative, many of their properties are reminiscent of the generalized quaternion groups. Lastly, we find varieties of quasigroups which are isotopic to commutative Moufang loops and prove their full characterization. Moreover, we define a new variety of quasigroups motivated by the semimedial quasigroups and show that they have an affine representation over commutative Moufang loops similar to the semimedial case proven by Kepka

    Associativity conditions for linear quasigroups and equivalence relations on binary trees

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    We characterise the bracketing identities satisfied by linear quasigroups with the help of certain equivalence relations on binary trees that are based on the left and right depths of the leaves modulo some integers. The numbers of equivalence classes of nn-leaf binary trees are variants of the Catalan numbers, and they form the associative spectrum (a kind of measure of non-associativity) of a quasigroup.Comment: 32 page
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