Refined semilattices of semigroups (Q2747695)

The authors introduce a generalization of the concept of strong semilattice of semigroups called refined semilattice of semigroups. Using this concept they prove that a band is regular if and only if it is a refined semilattice of rectangular bands. This result is a contribution to the study of the structure of bands. Previously \textit{A. H. Clifford} [Ann. Math., II. Ser. 42, 1037-1049 (1941)] and \textit{D. Mclean} [Am. Math. Mon. 61, 110-113 (1954; Zbl 0055.01404)] proved that every band is a semilattice of rectangular bands and \textit{M. Yamada} and \textit{N. Kimura} [Proc. Japan Acad. 34, 110-112 (1958; Zbl 0081.25501)] proved that a band is normal if and only if it is a strong semilattice of rectangular bands.