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

Determine whether every topological Mal’cev algebra X whose 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, or whether the extended operation on βX is actually a continuous Mal’cev operation and X is pseudocompact with βX Dugundji.

Background

For topological Mal’cev algebras, the authors prove that continuous extendability of the Mal’cev operation to βX is equivalent to X being pseudocompact and βX being a Dugundji compactum (Theorem 3.2).

They then consider the weaker assumption that the Mal’cev operation only extends separately continuously to βX, and formulate equivalent conjectures expressing that this should still force pseudocompactness (and Dugundji compactness of βX). Establishing this would generalize the continuous-extension result to the separately continuous setting.

References

Let (P_2) be the condition of Problem \ref{q:main:2}. Note that the following conjectures are equivalent: if (P_2), then $X$ is pseudocompact; if (P_2), then $ \beta X$ is Dugundji; if (P_2), then the operation on $ \beta X$ is a 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:2, Section 3.1 (Main results: Extension of operations on X)