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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On groupoids defined by commutators


Authors: Ki Hang Kim and Fred W. Roush
Journal: Proc. Amer. Math. Soc. 71 (1978), 15-18
MSC: Primary 20L10
DOI: https://doi.org/10.1090/S0002-9939-1978-0495827-2
MathSciNet review: 495827
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Abstract: We study matrices R, L which count the numbers of solutions of $ ix = j$ and $ xi = j$. For slight generalizations of R, L, the relation $ RL = LR$ characterizes associativity of a groupoid. For groupoids defined by group commutators $ xy{x^{ - 1}}{y^{ - 1}}$ the relation $ RL = LR$ is valid. In addition one can study analogues of Green's relations. Any $ \mathcal{I}$-class contains at most four $ \mathcal{H}$-classes in a commutator groupoid.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0495827-2
Keywords: Brandt groupoid, commutator groupoid, Green's relation, Lie algebra, nilpotent, strongly connected graph
Article copyright: © Copyright 1978 American Mathematical Society