On cardinal invariants and metrizability of topological inverse semigroups

Let S be a topological inverse semigroup, E={x∈S: xx=x} be the maximal semilattice in S, and C={x∈S: xe=ex for every idempotent e∈E} be the maximal Clifford semigroup of S. It is proven that a Lindelöf locally compact semigroup S is metrizable if and only if the maximal Clifford semigroup C is metrizable. We derive from this that a compact topological inverse semigroup S is metrizable, provided the maximal semilattice E and all maximal groups of S are metrizable and one of the following conditions is satisfied: (1) H) holds; Gδ-set in the maximal Clifford semigroup C of S; Lawson semilattice; ximal groups of C are Lie groups; yadic or scadic compact; fragmentable (or Rosenthal) monolithic compactum; Corson (or Rosenthal) compactum with countable spread.