ccc and retral properties for Maltsev spaces under compact-cover hypotheses

Investigate whether the following hold: (i) every σ-pseudocompact Maltsev space is ccc and whether it is retral; (ii) every Maltsev space with a K(M)-ordered compact cover, where M is separable metrizable, is ccc and whether it is retral; and (iii) every E(ℵ0) Maltsev space is ccc and whether it is retral.

Background

Maltsev spaces are spaces admitting a continuous ternary operation M(x,y,z) with M(x,y,y)=x=M(y,y,x). Retral spaces are retracts of topological groups; every retral space is Maltsev, and pseudocompact Maltsev spaces are retral, but not all Maltsev spaces are retral.

The paper establishes ccc results for retral and E(ℵ0) retral spaces via calibre arguments. The authors ask whether analogous ccc and retral conclusions extend to broader classes of Maltsev spaces under σ-pseudocompactness, K(M)-ordered compact covers, or E(ℵ0) hypotheses.