Identities on free products of bands (Q1118698)

The author considers the question whether for any normal bands \(U\) and \(V\), the free product \(U*_{{\mathcal B}}V\) of \(U\) and \(V\) within the variety \({\mathcal B}\) of bands belongs to a proper subvariety of \(\mathcal B\). It is shown that, if \(U\) and \(V\) are rectangular bands, then \(U*_{{\mathcal B}}V\) belongs to the variety which in the notation of \textit{C. F. Fennemore} [Math. Nachr. 48, 237-252, 253-262 (1971; Zbl 0194.02703 and Zbl 0194.02801)] is denoted by \([R_ 3d\bar R_ 3=Q_ 3d\bar Q_ 3]\). An example with \(U\) normal and \(V\) trivial shows that \(U*_{{\mathcal B}}V\) need not be contained in any proper subvariety of \(\mathcal B\). The main result states that if \(U\) is right normal and \(V\) a rectangular band, then \(U*_{{\mathcal B}}V\) is in the variety \([R_ 4=S_ 4]\).