Fundamental semilattices of semigroups (Q519910)

In the paper, the notion of a fundamental semilattice of semigroups is introduced, which is a special case of a semilattice of semigroups and a generalization of a strong semilattice of semigroups. The relations among fundamental semilattice of semigroups, refined semilattice of semigroups (see [\textit L. Zhang et al., Algebra Colloq. 8, No. 1, 93--108 (2001; Zbl 0993.20038)]) and regular semilattice of semigroups (see [the author and \textit{Y.-L. Zhou}, Semigroup Forum 87, No. 2, 393--406 (2013; Zbl 1290.20049)]) are discussed. The author proves the theorem that a semigroup is completely regular if and only if it is a fundamental semilattice of completely simple semigroups. The relation between this structure theorem and the theorem on completely regular semigroups by \textit{G. Lallement} is also discussed [Ann. Mat. Pura Appl., IV. Ser. 77, 47--129 (1967; Zbl 0186.03302)]. In conclusion, the structure of some special classes of completely regular semigroups is described as fundamental semilattice of completely simple semigroups.