Construction of Semigroups with Some Exotic Properties

What is it about?

The "growth function'' $g_U(n)$ of a finitely generated semigroup $S$ is defined as the number of distinct products in $S$ of length at most $n$, with respect to a specified finite generating set. The Gelʹfand-Kirillov dimension of $S$ is defined by ${\rm GK} \dim (S) = \limsup_{n \rightarrow \infty} (\log_n g_U (n))$. The authors first construct a finitely presented semigroup with the "exotic'' property that its Gelʹfand-Kirillov dimension is 6.5. This semigroup is generated by 25 elements, subject to a collection of 64 finite sets of relations. They then proceed by similar methods to prove that, in fact, for any integer $q$ there is a finitely presented semigroup whose Gelʹfand-Kirillov dimension has $1/q$ as its fractional part. Reviewed by Peter R. Jones

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