Mathematics Faculty Articles
Lyndon's Groupoid Generates a Small Almost Cross Variety
Document Type
Article
Publication Date
3-2009
Publication Title
Algebra Universalis
Keywords
Lyndon's Groupoid, Groupoids, Varieties, Finitely Based, Finitely Generated, Small, Cross, Almost Cross
ISSN
0002-5240
Volume
60
Issue/No.
2
First Page
239
Last Page
246
Abstract
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.
NSUWorks Citation
Lee, Edmond W. H., "Lyndon's Groupoid Generates a Small Almost Cross Variety" (2009). Mathematics Faculty Articles. 154.
https://nsuworks.nova.edu/math_facarticles/154
ORCID ID
0000-0002-1662-3734
ResearcherID
I-6970-2013
DOI
10.1007/s00012-009-2120-6