Ellis' theorem, minimal left ideals, and minimal/maximal idempotents without \(\mathsf{AC}\)

Abstract

In [18], we showed that the Boolean prime ideal theorem (\(\mathsf{BPI}\)) suffices to prove the celebrated theorem of R. Ellis, which states: ``Every compact Hausdorff right topological semigroup has an idempotent element''. However, the natural and intriguing question of the status of the reverse implication remained open until now. We resolve this open problem in the setting of \(\mathsf{ZFA}\) (Zermelo–Fraenkel set theory with atoms), namely we establish that Ellis' theorem does not imply \(\mathsf{BPI}\) in \(\mathsf{ZFA}\), and thus is strictly weaker than \(\mathsf{BPI}\) in \(\mathsf{ZFA}\). From the above paper, we also answer two more open questions and strengthen some theorems.

Typical results are:

1. Ellis' theorem is true in the Basic Fraenkel Model, and thus Ellis' theorem does not imply \(\mathsf{BPI}\) in \(\mathsf{ZFA}\).

2. In \(\mathsf{ZF}\) (Zermelo–Fraenkel set theory without the Axiom of Choice (\(\mathsf{AC}\))), if \(S\) is a compact Hausdorff right topological semigroup with \(S\) well orderable, then every left ideal of \(S\) contains a minimal left ideal and a minimal idempotent element. In addition, every such semigroup \(S\) has a maximal idempotent element.

3. In \(\mathsf{ZF}\), if \(S\) is a compact Hausdorff right topological abelian semigroup, then every left ideal of \(S\) contains a minimal left ideal.

4. In \(\mathsf{ZF}\), \(\mathsf{BPI}\) implies ``Every compact Hausdorff right topological abelian semigroup \(S\) has a minimal idempotent element''.

5. In \(\mathsf{ZFA}\), the Axiom of Multiple Choice (\(\mathsf{MC}\)) implies ``Every compact Hausdorff right topological abelian semigroup \(S\) has a minimal idempotent element''.

6. In \(\mathsf{ZFA}\), \(\mathsf{MC}\) implies ``Every compact Hausdorff right topological semigroup \(S\) with \(S\) linearly orderable, has a minimal idempotent element''.