Compacta with continuous Mal'tsev's operation and retracts of topological groups (Q1175341)

A continuous mapping \(f\colon X^3\to X\) satisfying \(f(x,y,y)=f(y,y,x)=x\) for all \(x,y\in X\) is called Mal'tsev operation on a space \(X\). This notion was introduced by \textit{A. I. Mal'tsev} [Mat. Sb. 35(77), 3--20 (1954; Zbl 0057.02403)] who mentioned that every retract of a topological group has the Mal'tsev operation. It was proved by \textit{V. V. Uspenskiĭ} [Math. USSR, Sb. 67, No. 2, 555--580 (1990); translation from Mat. Sb. 180, No. 8, 1092--1118 (1989; Zbl 0684.22001)] that a compact space with the Mal'tsev operation is a Dugundji space, and hence dyadic. The main theorem of the paper under review states that every compact (even countably compact) space with the Mal'tsev operation is a retract of a topological group. However, it is an open problem whether the same conclusion holds for a pseudocompact space with the Mal'tsev operation.