The variety generated by all monoids of order four is finitely based. (Q2786994)

The authors prove the theorem stated in the title by exhibiting an explicit basis. Let \(\mathbf{M_n}\) denote the variety of monoids generated by all monoids of order \(n\). The ubiquitous six-element Brandt monoid shows that \(\mathbf{M_n}\) is not finitely based for \(n\geq 6\). In conjunction with the work of \textit{Y. Luo} and \textit{W. Zhang} [J. Algebra 334, No. 1, 1-30 (2011; Zbl 1243.20071)], the authors' theorem leaves only the basis property for \(\mathbf{M_5}\) to be decided. They provide evidence to suggest that, again, an explicit basis will be needed, if the variety is indeed finitely based. (If \(\mathbf{S_n}\) denotes the analogous variety of semigroups, the case is closed, as a result of the work of the first author, \textit{Y. Luo} and \textit{W. Zhang} [Algebra Univers. 73, No. 3-4, 225-248 (2015; Zbl 1328.20076)]: \(\mathbf{S_n}\) is finitely based if and only if \(n\leq 4\).)