Free split bands.
As the authors write in the introduction: ``Recall that a band is a semigroup in which every element is an idempotent. The Green relation \(\mathcal D\) is the least semilattice congruence on a band and every band is a semilattice of its \(\mathcal D\)-classes; the \(\mathcal D\)-classes themselves form rectangular bands''. They are interested in the bands for which the congruence \(\mathcal D\) \textit{splits}, that is there exists a subsemilattice \(S^\circ\) in \(S\) which intersects every \(\mathcal D\)-class in exactly one element. Then we call this subsemilattice \(S^\circ\) a \textit{semilattice transversal} of \(S\). For a band \(S\) with semilattice transversal we have a unary operation \(a\mapsto a^\circ\), where \(a^\circ\) is the unique element of \(\mathcal D\)-class \(D_a\). The authors note in the introduction that ``\textit{the unary band} \((S,\cdot,^\circ)\) thus obtained satisfies, apart from the associative law, the identities \[ x\approx x^2,\quad xx^\circ x\approx x,\quad x^\circ xx^\circ\approx x^\circ ,\quad (xy)^\circ\approx x^\circ y^\circ\approx y^\circ x^\circ ,\quad (x^\circ)^\circ \approx x^\circ.\text{''}\tag{1} \] The authors call the unary bands which satisfy the identities (1) \textit{split bands}. The variety of all bands will be denoted by \(\mathbf B\) and the variety of all split bands will be denoted by \(\mathbf B^\circ\). As the authors note ``the aim of this paper is to solve the word problem for \(F_{\mathbf B^\circ}(X)\) and to show that \(F_{\mathbf B}(X)\) can be isomorphically embedded into the multiplicative reduction of \(F_{\mathbf B^\circ}(X)\)''.
scientific article; zbMATH DE number 6493122