Combinatorial Theory

For a positive integer $n$, the full transformation semigroup ${\mathcal{T}}_n$ consists of all self maps of the set $\{1,\dots ,n\}$ under composition. Any finite semigroup $S$ embeds in some ${\mathcal{T}}_n$, and the least such $n$ is called the (minimum transformation) degree of $S$ and denoted $\mu (S)$. We find degrees for various classes of finite semigroups, including rectangular bands, rectangular groups and null semigroups. The formulae we give involve natural parameters associated to integer compositions. Our results on rectangular bands answer a question of Easdown from 1992, and our approach utilises some results of independent interest concerning partitions/colourings of hypergraphs.

As an application, we prove some results on the degree of a variant ${\mathcal{T}}_n^a$. (The variant $S^a=(S,\star )$ of a semigroup $S$, with respect to a fixed element $a\in S$, has underlying set $S$ and operation $x\star y=xay$.) It has been previously shown that $n\le \mu ({\mathcal{T}}_n^a)\le 2n-r$ if the sandwich element $a$ has rank $r$, and the upper bound of $2n-r$ is known to be sharp if $r\ge n-1$. Here we show that $\mu ({\mathcal{T}}_n^a)=2n-r$ for $r\ge n-6$. In stark contrast to this, when $r=1$, and the above inequality says $n\le \mu ({\mathcal{T}}_n^a)\le 2n-1$, we show that $\mu ({\mathcal{T}}_n^a)/n\to 1$ and $\mu ({\mathcal{T}}_n^a)-n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$.

Among other results, we also classify the $3$-nilpotent subsemigroups of ${\mathcal{T}}_n$, and calculate the maximum size of such a subsemigroup.

Mathematics Subject Classifications: 20M20, 20M15, 20M30, 05E16, 05C65

Keywords: Transformation semigroup, transformation representation, semigroup variant, rectangular band, nilpotent semigroup, hypergraph