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The module theory of semisymmetric quasigroups, totally symmetric quasigroups, and triple systems

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Abstract

This paper initiates a unified module theory for four varieties of quasigroups: semisymmetric, semisymmetric idempotent, totally symmetric, and totally symmetric idempotent. These classes correspond, respectively, to extended Mendelsohn triple systems, Mendelsohn triple systems (MTS), extended Steiner triple systems, and Steiner triple systems (STS), which, in turn, correspond to partitions of complete (directed) graphs. Letting \({\mathbf {V}}\) stand for any of the aforementioned quasigroup categories, we determine a ring, \({\mathbb {Z}}{\mathbf {V}}\!{Q}\), such that abelian groups in the slice category \({\mathbf {V}}\!/\!{Q}\) are equivalent to (right) \({\mathbb {Z}}{\mathbf {V}}\!{Q}\)-modules. This ring is a quotient of the group algebra of the so-called universal stabilizer of the \({\mathbf {V}}\)-quasigroup Q. We prove that the universal stabilizer of Q in \({\mathbf {V}}\) is the fundamental group of either the graph from which the triple system sources its blocks, or a closely related space. In each of the four varieties, we provide a free product description of \({\mathbb {Z}}{\mathbf {V}}\!{Q}\), and show that the factorizations are indexed by blocks of the triple system. We also consider the ranks of free group rings that appear in the coproduct for \({\mathbb {Z}}{\mathbf {V}}\!{Q}\) (for the cases \({\mathbf {V}}=\mathbf {MTS}, \mathbf {STS}\)), and we describe these ranks in terms of pentagonal numbers. We establish a relationship between the module theory of MTS and commutative Moufang loops of exponent 3. As an application, we show how the MTS module theory completely accounts for the distributive, nonmedial quasigroups of order 81.

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Notes

  1. This notion of triality should not be conflated with the, perhaps better known, triality inherent to the class of quasigroups known as Moufang loops [16]. Moufang loop triality is related to the, certainly better known, phenomenon of triality in Lie groups and the Dynkin diagram \(D_4\).

  2. Not all medial quasigroups are idempotent, but the representation theory of nonidempotent, medial quasigroups requires extra automorphisms and constants (cf. [50]) we wish to avoid discussing.

  3. At the confluence of universal algebra and category theory, it is important to regard the empty set as a quasigroup (cf. the introduction of [3]).

  4. Since the initial submission of this work, this claim has been further substantiated by the fact that the ambient space for Kanevsky’s construction of a nonassociative CML on a cubic hypersurface is the extension \({\mathbb {Q}}_{(3)}(\omega )\) of the 3-adic rationals by a cubic root of unity.

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Acknowledgements

The author was partially supported by the Fulbright US Scholar Program through the US Department of State’s Bureau of Educational and Cultural Affairs (Award 11590-EZ).

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  1. Department of Mathematics, Howard University, Washington, DC, 20059, USA

    Alex W. Nowak

  2. Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Praha 8, 18675, Prague, Czech Republic

    Alex W. Nowak

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Nowak, A.W. The module theory of semisymmetric quasigroups, totally symmetric quasigroups, and triple systems. J Algebr Comb 56, 565–607 (2022). https://doi.org/10.1007/s10801-022-01124-3

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