Absolutely closed semigroups

Abstract

Let C be a class of topological semigroups. A semigroupLet T1S, T2S, and TzS be the classes of T1, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that any ideally (resp. injectively) TzS-closed semigroup has group-bounded (resp. group-finite) center Z(X). If a viable semigroup X is ideally TzS-closed, then (1) each maximal subgroup He of X is projectively TzS-closed, (2) X contains no strictly decreasing chains of idempotents, (3) the center Z(X) of X is chain-finite, group-bounded and Clifford+finite, (4) Z(X) is projectively T1S-closed. A commutative semigroup X is absolutely TzS-closed if and only if X is absolutely T2S-closed if and only if X is chain-finite, bounded, group-finite and Clifford+finite. On the other hand, a commutative semigroup X is absolutely T1S-closed if and only if X is finite.