Hereditarily finitely based monoids of extensive transformations.
Let \(E_n\) denote the monoid of `extensive' transformations of the ordered set \(\{1,2,\dots,n\}\), that is, the transformations \(\alpha\) with the property that \(i\alpha\geq i\) for all \(i\). The author proves that \(E_n\) is hereditarily finitely based if and only if \(n\leq 3\). By far the most difficult part of the proof is to show that \(E_3\) is hereditarily finitely based. The author notes that \(E_n\) is itself not finitely based, for \(n>4\), by a recent result of \textit{I. A. Goldberg} [in Semigroups and formal languages. Proceedings of the international conference in honour of the 65th birthday of Donald B. McAlister, Lisboa, Portugal, 2005. Hackensack, NJ: World Scientific. 101-110 (2007; Zbl 1131.20043)]. Whether \(E_4\) is finitely based was undetermined at the time the paper was written.
scientific article; zbMATH DE number 5638420