The join of the pseudovarieties of idempotent semigroups and locally trivial semigroups (Q1891654)

This paper concerns the calculation of the join of the pseudovarieties \(\mathbf {LI}\) of locally trivial semigroups and \({\mathbf B}\) of bands, a problem which was proposed by the reviewer. A related pseudovariety is the Mal'cev product \({\mathbf B}\mathop {m} {\mathbf N}\) of \({\mathbf B}\) with the pseudovariety \({\mathbf N}\) of nilpotent semigroups. The latter is defined by the pseudoidentities \((x^\omega y)^2= x^\omega y\) and \((yx^\omega)^2= yx^\omega\). The main result in the paper states that \(\mathbf {LI}\vee {\mathbf B}= {\mathbf B}\mathop {m} {\mathbf N}\). As has been pointed out by M. V. Volkov in private correspondence, this result is actually false. The error can be located at the beginning of the proof of Lemma 3.7, where Lemma 3.6 is invoked incorrectly. Meanwhile, the calculation of \(\mathbf {LI} \vee {\mathbf B}\) can be easily completed as in the paper by adding the missing pseudoidentity \(x^\omega y^2 z^\omega= x^\omega y z^\omega\). A correction note by the author appears in the same journal [52, No. 2, 249 (1996)]. Some related joins are also calculated in the paper.