The word problem in regular band free products (Q1820247)

At the time of review, the problem of finding a normal form for, or otherwise describing, the elements in the free product of two bands U and V, in the variety \({\mathcal B}\) of all bands, is unsolved. The author shows here, however, that at least for the free product within the variety \({\mathcal R}{\mathcal B}\) of ''regular'' bands - those satisfying the identity \(xyxzx=xyzx\)- such a normal form exists. Any member of this free product has as normal form either \[ (i)\quad...v_{p-1}u_{t-1}v_ pu_ tv_{p+1}u_{p+1}v_{p+2}..., \] where \(u_{t-1},u_ t\) and \(u_ t,u_{t+1}..\). belong to strictly descending and strictly ascending chains of \({\mathcal D}\)-classes of U, respectively, and similarly for \(...v_{p-1}\), \(v_ p\) and \(v_ p,v_{p+1}..\). in V (except that possibly \(v_ p=v_{p+1})\), or (ii) a form obtained by interchanging the roles of U and V in (i). If U and V are semilattices this normal form is unique and in fact describes the free product of U and V in \({\mathcal B}\), by a result of the reviewer [Semigroup Forum 20, 335-341 (1980; Zbl 0448.20056)]. Otherwise uniqueness is ensured by the imposition of further restrictions.