International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 362068, 12 pages
doi:10.1155/2008/362068
Research Article
Green's-Like Relations on Algebras and Varieties
1Institut für Mathematik, Universität Potsdam, D-14415 Potsdam, PF 601553, Germany
2Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB. T1K-3M4, Canada
Received 13 September 2007; Accepted 30 October 2007
Academic Editor: Robert H. Redfield
Copyright © 2008 K. Denecke and S. L. Wismath. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
There are five equivalence relations known as Green's relations definable on any
semigroup or monoid, that is, on any algebra with a binary operation which is
associative. In this paper, we examine whether Green's relations can be defined
on algebras of any type τ.
Some sort of (super-)associativity is needed for such definitions to work, and we consider algebras which are clones of terms of type τ, where the clone axioms including superassociativity hold. This allows us to define for any variety V of type τ two Green's-like
relations ℒV and ℛV on the term clone of type τ. We prove a number of properties of these two relations, and describe their behaviour when V is a variety of semigroups.