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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 362068, 12 pages
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., Canada T1K 3M4

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.


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 𝑉 and 𝑉 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.