Ellis’ Theorem (i.e., “every compact Hausdorff right topological semigroup has an idempotent element”) is known to be proved only under the assumption of the full Axiom of Choice (AC); AC is used in the proof in the disguise of Zorn’s Lemma.
In this article, we prove that in ZF, Ellis’ Theorem follows from the Boolean Prime Ideal Theorem (BPI), and hence is strictly weaker than AC in ZF. In fact, we establish that BPI implies the formally stronger (than Ellis’ Theorem) statement “for every family
${\cal A} = \{ ({S_i},{ \cdot _i},{{\cal T}_i}):i \in I\}$
of nontrivial compact Hausdorff right topological semigroups, there exists a function f with domain I such that
$f\left( i \right)$
is an idempotent of
${S_i}$
, for all
$i \in I$
”, which in turn implies ACfin (i.e., AC for sets of nonempty finite sets).
Furthermore, we prove that in ZFA, the Axiom of Multiple Choice (MC) implies Ellis’ Theorem for abelian semigroups (i.e., “every compact Hausdorff right topological abelian semigroup has an idempotent element”) and that the strictly weaker than MC (in ZFA) principle LW (i.e., “every linearly ordered set can be well-ordered”) implies Ellis’ Theorem for linearly orderable semigroups (i.e., “every compact Hausdorff right topological linearly orderable semigroup has an idempotent element”); thus the latter formally weaker versions of Ellis’ Theorem are strictly weaker than BPI in ZFA. Yet, it is shown that no choice is required in order to prove Ellis’ Theorem for well-orderable semigroups.
We also show that each one of the (strictly weaker than AC) statements “the Tychonoff product
$2^{\Cal R} $
is compact and Loeb” and
$BPI_{\Cal R}$
(BPI for filters on
${\Cal R}$
) implies “there exists a free idempotent ultrafilter on ω” (which in turn is not provable in ZF). Moreover, we prove that the latter statement does not imply
$BP{I_\omega }$
(BPI for filters on ω) in ZF, hence it does not imply any of
$AC_{\Cal R} $
(AC for sets of nonempty sets of reals) and
$BPI_{\Cal R} $
in ZF, either.
In addition, we prove that the statements “there exists a free ultrafilter on ω”, “there exists a free ultrafilter on ω which is not idempotent”, and “for every IP set
$A \subseteq \omega$
, there exists a free ultrafilter
${\cal F}$
on ω such that
$A \in {\cal F}$
” are pairwise equivalent in ZF.