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Green's-Like Relations on Algebras and Varieties

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  • K. Denecke
  • S. L. Wismath

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 ð œ . Some sort of (super-)associativity is needed for such definitions to work, and we consider algebras which are clones of terms of type ð œ , where the clone axioms including superassociativity hold. This allows us to define for any variety 𠑉 of type ð œ two Green's-like relations â„’ 𠑉 and â„› 𠑉 on the term clone of type ð œ . We prove a number of properties of these two relations, and describe their behaviour when 𠑉 is a variety of semigroups.

Suggested Citation

  • K. Denecke & S. L. Wismath, 2008. "Green's-Like Relations on Algebras and Varieties," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2008, pages 1-12, February.
    • Handle: RePEc:hin:jijmms:362068
    DOI: 10.1155/2008/362068
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