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.

ORCID ID

0000-0002-1662-3734

ResearcherID

I-6970-2013

DOI

10.1007/s00012-009-2120-6

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