Lyndon's groupoid is not inherently nonfinitely based
The first example of a finite algebra which does not have a finite basis for its laws was given by \textit{R. C. Lyndon} [Proc. Am. Math. Soc. 5, 8-9 (1954; Zbl 0055.027)]. Since then, other examples have been given, many of which have been inherently nonfinitely based, that is, have not belonged to any locally finite finitely based variety. The question naturally arises as to whether Lyndon's algebra \({\mathbb{L}}\) is inherently nonfinitely based. In this paper it is shown that it is not, by exhibiting a finitely based finite algebra \({\mathbb{B}}\) of which \({\mathbb{L}}\) is a subalgebra. \{The same result was given by the second author and \textit{C. R. Shallon} [Lect. Notes Math. 1004, 206-231 (1983; Zbl 0513.08003)] but, unfortunately, there was an error in their proof.\}
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