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Volume 2
Summer 2007
Number 1

A Basis for the k-Normalization of Semigroups

C. Masse, H. Wang and S. L. Wismath

Abstract: The depth of a term is an inductively defined measure of its complexity. For any natural number $k \geq 1$, an identity $s \approx t$ is said to be $k$-normal, with respect to the depth measurement, if either $s = t$ or both $s$ and $t$ have depth at least $k$. A variety $V$ of algebras is said to be $k$-normal if all the identities satisfied by $V$ are $k$-normal. For any variety $V$ of algebras, the $k$-normalization of $V$ is the variety defined by all the $k$-normal identities satisfied in $V$. This is the smallest $k$-normal variety to contain $V$.
A semigroup is an algebra with one binary operation which satisfies the associative law. Let $Sem$ be the variety of all semigroups and let $N_k(Sem)$ be the $k$-normalization of $Sem$. The variety $N_k(Sem)$ is the equational class of algebras that satisfy all $k$-normal consequences of associativity. In this paper we produce a finite equational basis for $N_k(Sem)$, for $k \geq 3$.