Two generalizations of homogeneity in groups with applications to regular semigroups (Q2796517)

Arguably the most exciting developments in the theory of semigroups in recent times has been to study how its structure is influenced by its group of units. The setup of such an investigation is typically that of a semigroup \(\langle G,a\rangle\) where \(G\) is a permutation group on \(\Omega\) and \(a\) an arbitrary map on \(\Omega\). These are in some sense minimal cases, as for a semigroup \(S\) with group of units \(G\) we have that \(S=\bigcup_{a\in S\setminus G} \langle G,a\rangle\).NEWLINENEWLINEThis approach allows the use of modern permutation group methods -- first and formost those arising from the classification of finite simple groups and the O'Nan-Scott theorem [\textit{L. L. Scott}, Proc. Symp. Pure Math. 37, 319--331 (1980; Zbl 0458.20039)], see [\textit{J. D. Dixon} and \textit{B. Mortimer}, Permutation groups. New York, NY: Springer-Verlag (1996; Zbl 0951.20001)] for a complete version, and including computational methods -- to study semigroups.NEWLINENEWLINEVice versa (if a property for \(\langle G,a\rangle\) only depends on \(G\) and not on \(a\)) it allows the translation of semigroup concepts to permutation groups, often resulting in questions that are of independent interest for group theory without ever touching semigroups, synchronization [\textit{F. Arnold} and \textit{B. Steinberg}, Theor. Comput. Sci. 359, No. 1--3, 101--110 (2006; Zbl 1097.68054)] arguably being the most prominent example thereof.NEWLINENEWLINEThe paper continues the authors' prior work following this paradigm. Concretely they prove numerous classification theorems concerning the question for which groups \(G\) the semigroup \(\langle G,a\rangle\) is regular (meaning that for every \(x\in \langle G,a\rangle\) there exists an \(y\) such that \(x=xyx\)); as well as classifications of groups satisfying the \(k\)-\textit{universal transversal property} (meaning that for every \(k\)-set \(I\subset\{1,\ldots n\}\) and any partion \(P\) of \(\{1,\ldots,n\}\) into \(k\) cells, there exists \(g\in G\) such that \(I^g\) is a section of \(P\)). This culminates in a theorem, significantly improving on [\textit{I. Levi} et al., Semigroup Forum 61, No. 3, 453--467 (2000; Zbl 0966.20031)], that shows that these two properties are idential.NEWLINENEWLINEGeneralizing work of [\textit{D. Livingstone} and \textit{A. Wagner}, Math. Z. 90, 393--403 (1965; Zbl 0136.28101)], the authors call a permutation group \(G\leq S_n\) as \((k,l)\)-\textit{homogeneous} (for \(k\leq l\)), if any \(k\)-element subset can be mapped by \(G\) into any \(l\)-element subset. As a result related to the work already described, they show that (with a small number of explicitly listed exceptions) any \((k,k+1)\)-homogeneous group is in fact \(k\)-homogeneous.NEWLINENEWLINEThe paper ends with a set of challenges, often involving related areas such as number theory or linear algebra.