English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

For a positive integer $n$, the full transformation semigroup $T_n$ consistsof all self maps of the set $\{1,\ldots,n\}$ under composition. Any finitesemigroup $S$ embeds in some $T_n$, and the least such $n$ is called the(minimum transformation) degree of $S$ and denoted $\mu(S)$. We find degreesfor various classes of finite semigroups, including rectangular bands,rectangular groups and null semigroups. The formulae we give involve naturalparameters associated to integer compositions. Our results on rectangular bandsanswer a question of Easdown from 1992, and our approach utilises some resultsof 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 fixedelement $a\in S$, has underlying set $S$ and operation $x\star y=xay$.) It hasbeen 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 $r\geqn-1$. Here we show that $\mu(T_n^a)=2n-r$ for $r\geq n-6$. In stark contrast tothis, when $r=1$, and the above inequality says $n\leq\mu(T_n^a)\leq 2n-1$, weshow 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.

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