On maximal \(E\)-compactification (Q2748172)

The concept of \(E\)-compactness where \(E\) is a Hausdorff space, generalizing the concept of compactness was introduced by \textit{R. Engelking} and \textit{S. Mrówka} [Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 6, 429-436 (1958; Zbl 0083.17402)]. Further, the theory of \(E\)-compact spaces was developed and, in particular the maximal \(E\)-compact extension \(\beta_E(X)\) of an \(E\)-regular space -- a certain analogue of Stone-Čech compactification of a completely regular space, was constructed and studied in a series of papers [\textit{S. Mrówka}, Acta Math. 120, 161-185 (1968; Zbl 0179.51202)]. The aim of this paper is to study \(\beta_E(X)\) in the special case when \(E\) is a topological field. In particular the author is interested in characterizing \(\beta_E(X)\) by means of \(E\)-\(Z\)-ultrafilters in this case. The paper contains some results about \(\beta_E(X)\), \(E\) being a topological field. However, most of these results are already known.