Metrizability of Clifford topological semigroups

We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E={e\in S:ee=e}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.

💡 Research Summary

The paper investigates the metrizability problem for topological Clifford semigroups—semigroups equipped with a continuous associative multiplication in which every element belongs to some subgroup, and where each element has a unique inverse. While the classical Birkhoff‑Kakutani theorem characterizes metrizability of topological groups by first‑countability, the situation for semigroups is more delicate because the inversion need not be continuous.

The authors first distinguish between “Clifford topological semigroups” (algebraically Clifford but without assuming continuity of inversion) and “topological Clifford semigroups” (where inversion is continuous). They recall known results: for compact Clifford semigroups the inversion is automatically continuous, and for countably compact Clifford semigroups it is sequentially continuous. Moreover, several sufficient conditions (e.g., Tychonoff plus pseudo‑compact square, countable compactness of the square, sequentiality, or topological periodicity with first‑countability at each idempotent) guarantee that a countably compact Clifford semigroup is actually a topological Clifford semigroup.

The core of the paper is a metrizability criterion expressed in terms of two topological notions: M‑spaces and the structure of the idempotent set (E={e\in S:e^2=e}). An M‑space, following Morita’s characterization, is a space admitting a closed continuous map onto a metrizable space with countably compact fibers; such spaces are known to be metrizable whenever they have a (G_\delta)-diagonal.

Section 2 develops cardinal invariants for topological Clifford semigroups. The authors prove the inequality (\Delta(S)\le\Delta(E)\cdot\psi(E,S)), where (\Delta) denotes the diagonal number (pseudo‑character of the diagonal) and (\psi(E,S)) the pseudo‑character of (E) in (S). Using Arkhangel’skii’s formula (w(X)=l(X)\cdot\Delta(X)) for locally compact spaces, they deduce that for a locally compact Clifford semigroup the weight satisfies (w(S)=l(S)\cdot w(E)\cdot\psi(E,S)). Consequently, if (E) is a metrizable (G_\delta)-set, then (\Delta(E)) and (\psi(E,S)) are countable, forcing (\Delta(S)) to be countable and making (S) an M‑space with a (G_\delta)-diagonal.

The main result of this section (Theorem 2.4) states: a topological Clifford semigroup (S) is metrizable iff (i) (S) is an M‑space, and (ii) the idempotent set (E) is a metrizable (G_\delta)-subset of (S). The “only‑if” direction is straightforward: a metrizable space is an M‑space and any closed (G_\delta) subset of a metrizable space is metrizable. The “if” direction combines the cardinal invariant inequality with Morita’s theorem to obtain a closed map onto a metrizable space with compact fibers, and then uses the known fact that an M‑space with a (G_\delta)-diagonal is metrizable.

Section 3 extends the criterion to countably compact Clifford semigroups, where the M‑space condition is automatically satisfied (countably compact spaces are M‑spaces). Theorem 3.1 asserts that a countably compact Clifford semigroup (S) is metrizable iff its idempotent set (E) is a metrizable (G_\delta)-set. The proof proceeds through several claims:

1. Since (E) is a (G_\delta)-set, each idempotent (e) has a countable neighborhood base, i.e., (S) is first‑countable at every idempotent.
2. By Theorem 1.2, inversion is sequentially continuous on countably compact semigroups; combined with first‑countability at idempotents, inversion becomes continuous at each idempotent.
3. For each idempotent (e), the maximal subgroup (H_e={x\in S:xx^{-1}=e}) is a paratopological group. Continuity of inversion at (e) upgrades (H_e) to a topological group, and first‑countability at (e) makes (H_e) first‑countable, so the Birkhoff‑Kakutani theorem yields that (H_e) is metrizable.
4. Using Zorn’s lemma and the natural partial order on (E) (defined by (x\le y\iff xy=x= yx)), the authors show that for any element (a\in S) the set of accumulation points of its powers contains a minimal idempotent (e). This minimal idempotent is shown to be topologically periodic.
5. By Theorem 1.3(4), a countably compact Clifford semigroup in which every idempotent is topologically periodic and first‑countable is a topological Clifford semigroup.

Thus (S) becomes a topological Clifford semigroup, which is countably compact and therefore an M‑space. Applying Theorem 2.4 yields metrizability.

The paper also provides a counterexample (Example 2.5) showing that the M‑space hypothesis cannot be omitted: a countable, commutative Clifford semigroup whose idempotent set is compact, metrizable, and open, yet the whole semigroup fails to be metrizable because the underlying space (X) is non‑metrizable.

In summary, the authors establish a clean and elegant metrizability criterion for Clifford semigroups: the global topology of the semigroup is completely governed by the topological nature of its idempotent set and the M‑space property. For countably compact semigroups, the M‑space condition is automatic, so the idempotent set alone determines metrizability. This work extends classical metrizability results from topological groups to a broader algebraic setting, offering new tools for the study of semitopological algebraic structures.