On cardinal invariants and metrizability of topological inverse Clifford semigroups

Let S be a compact topological inverse Clifford semigroup S such that the maximal semilattice E and all maximal groups of S are metrizable. We prove that S is first countable and has countable cellularity; moreover, S is metrizable, provided one of the following conditions is satisfied: H) holds; Gδ-set in S; ero-dimensional; Lawson semilattice; ximal groups of S are Lie groups; yadic or scadic compact; fragmentable (or Rosenthal) monolithic compactum; Corson (or Rosenthal) compactum with countable spread. CH two (separable and unseparable) compact non-metrizable topological inverse commutative semigroups with metrizable subsemilattices and subgroups are constructed. One of these semigroups is a first-countable ccc Corson compact space satisfying the properties (M), (∗) and (Kn) for all n⩾2 but failing (∗∗).