Groups, triality, and hyperquasigroups

Smith, Jonathan D. H.

doi:10.1016/j.jpaa.2011.08.014

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Cited by 8 publications

(1 citation statement)

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“…We consider the action of S 3 on the quasigroup operations. Let σ and τ represent (12) and (23), respectively [14]. We write Let g ∈ S 3 .…”

Section: Mirror Symmetrymentioning

confidence: 99%

On free quasigroups and quasigroup representations

Wang

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“…We consider the action of S 3 on the quasigroup operations. Let σ and τ represent (12) and (23), respectively [14]. We write Let g ∈ S 3 .…”

Section: Mirror Symmetrymentioning

confidence: 99%

On free quasigroups and quasigroup representations

Wang

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Isomorphism invariants for linear quasigroups

Smith

Wang

2019

São Paulo J. Math. Sci.

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For a unital ring S, an S-linear quasigroup is a unital S-module, with automorphisms ρ and λ giving a (nonassociative) multiplication x • y = x ρ + y λ. If S is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional S-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for Z-linear quasigroups. We consider the extent to which ordinary characters classify Z-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic Z-linear quasigroups with the same ordinary character. For a subclass of Z-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on Z-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy.

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Stochastic tensors and approximate symmetry

Lee

Smith

2017

Discrete Mathematics

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Groups, triality, and hyperquasigroups

Cited by 8 publications

References 11 publications

On free quasigroups and quasigroup representations

On free quasigroups and quasigroup representations

Isomorphism invariants for linear quasigroups

Stochastic tensors and approximate symmetry

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