Lyndon's groupoid generates a small almost Cross variety

Citation data:

Algebra Universalis, ISSN: 0002-5240, Vol: 60, Issue: 2, Page: 239-246

Publication Year:
2009
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Repository URL:
http://works.bepress.com/edmond-lee/33; https://nsuworks.nova.edu/math_facarticles/154
DOI:
10.1007/s00012-009-2120-6
Author(s):
Lee, Edmond W. H.
Publisher(s):
Springer Nature; NSUWorks
Tags:
Mathematics; Lyndon's Groupoid; Groupoids; Varieties; Finitely Based; Finitely Generated; Small; Cross; Almost Cross
article description
Lyndon's groupoid of order seven is the first published example of a non-finitely based finite algebra. The main objective of the present article is to investigate the variety L generated by this groupoid and its subvarieties. It is shown that the subvarieties of L form a chain of order five, all elements of which except L are Cross varieties. It follows that the variety L is also generated by a groupoid of order six and that any groupoid with five or fewer elements does not generate L. Consequently, Lyndon's example of a non-finitely based finite algebra could have been of order six instead of seven. It is also shown that, with respect to some important properties, Lyndon's groupoid contrasts greatly with several well-known non-finitely based finite groupoids that were discovered shortly after its publication. © 2009 Birkhäuser Verlag, Basel.