Construction of Semigroups with Some Exotic Properties

What is it about?

Let $R$ be an algebra, generated by some set $X$, over a field $K$. Let $V^n$ denote the subspace of $R$ spanned by all products of at most $n$ elements of $X$. Then the Gelʹfand-Kirillov (GK) dimension of $R$ is defined by $$ {\rm GK}\dim(R)=\lim\sup_{n\rightarrow\infty}(\log_n g_V(n)), $$ where $g_V(n)=\dim(V^n)$ is referred to as the growth function of $R$ with respect to $V$. It is clear that the GK dimension is independent of the choice of generating set. Moreover, the GK dimension of any finite-dimensional algebra is always equal to zero. This concept of a GK dimension first appeared in 1966 in the work of Gelʹfand and Kirillov which studied fields associated with enveloping algebras. It gradually came to be seen as a useful invariant and has been steadily explored (together with the broader notion of growth of algebras) since the mid seventies. Analogously, one may define the GK dimension of a semigroup. Let $S$ be a finitely generated semigroup. Then the number of non-equivalent words in $S$ of length at most $n$ is denoted by $g(n)$ and is known as the growth function of $S$. The Gelʹfand-Kirillov dimension of $S$ is defined by $$ {\rm GK}\dim(S)=\lim\sup_{n\rightarrow\infty}(\log_n g(n)). $$ In the paper under review, the authors introduce a finitely presented semigroup and prove that its GK dimension is 5.5. The techniques the authors apply, as well as the semigroup they construct, are similar to those employed in their previous paper [Comm. Algebra 31 (2003), no. 2, 673–696; MR1968920], where they created a semigroup with GK $\dim$ equal to 6.5. The authors then strengthen their result by showing that for any rational number $\alpha$ greater than 5 there exists a finitely presented semigroup $S$ such that GK $\dim(S)= \alpha$. Reviewed by Noelle C. Antony

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