Pseudocompactness from separately continuous extension of the Mal’cev operation on βX for semitopological Mal’cev algebras

Determine whether every semitopological Mal’cev algebra X such that the Mal’cev operation extends to a separately continuous operation on the Stone–Čech compactification βX must be pseudocompact. Equivalently, ascertain whether under this hypothesis βX is a Dugundji compactum and the extended operation on βX is a separately continuous Mal’cev operation.

Background

In the semitopological setting, Theorem 3.3 gives conditions ensuring that if the Mal’cev operation extends separately continuously to βX and X is pseudocompact (with certain additional extendability/quasicontinuity properties), then βX is Dugundji and powers of X are pseudocompact.

The open question asks whether pseudocompactness of X already follows from the sole assumption that the Mal’cev operation extends separately continuously to βX, mirroring the stronger conclusions available in the topological case.

References

Let (P_3) be the condition of Problem \ref{q:main:3}. Note that the following conjectures are equivalent: if (P_3), then $X$ is pseudocompact; if (P_3), then $ \beta X$ is Dugundji; if (P_3), then the operation on $ \beta X$ is a separately continuous Mal'cev operation, $X$ is pseudocompact, and $ \beta X$ is Dugundji.

— Extensions and factorizations of topological and semitopological universal algebras (2402.01418 - Reznichenko, 2 Feb 2024) in Following Question q:main:3, Section 3.1 (Main results: Extension of operations on X)