Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups

TL;DR

This paper determines the minimum degrees of various finite semigroups, including rectangular bands, null semigroups, and variants of full transformation semigroups, using combinatorial and hypergraph coloring techniques.

Contribution

It provides explicit formulas for the degrees of these semigroups, answers a longstanding question about rectangular bands, and analyzes degrees of semigroup variants with different ranks.

Findings

Degrees for rectangular bands, rectangular groups, and null semigroups are explicitly calculated.

The degree of a variant $T_n^a$ is shown to be $2n-r$ for certain ranks, with asymptotic behavior analyzed.

Classified 3-nilpotent subsemigroups of $T_n$ and determined their maximum size.

Abstract

For a positive integer $n$, the full transformation semigroup $T_n$ consists of all self maps of the set $\{1,\ldots,n\}$ under composition. Any finite semigroup $S$ embeds in some $T_n$, 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 $T_n^a$. (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 $n\leq \mu(T_n^a)\leq 2n-r$ if the sandwich element $a$ has rank $r$, and the upper bound of $2n-r$ is known to be sharp if $r\geq n-1$. Here we show that $\mu(T_n^a)=2n-r$ for $r\geq n-6$. In stark contrast to this, when $r=1$, and the above inequality says $n\leq\mu(T_n^a)\leq 2n-1$, we show that $\mu(T_n^a)/n\to1$ and $\mu(T_n^a)-n\to\infty$ as $n\to\infty$. Among other results, we also classify the $3$-nilpotent subsemigroups of $T_n$, and calculate the maximum size of such a subsemigroup.