\(A\)-compactness and minimal subalgebras of \(C(X)\) (Q812477)

Let \(X\) be a Tychonoff space and \(\Sigma(X)\) denote the set of subalgebras \(A\) of the ring \(C(X)\) of continuous mappings from \(X\) into the real field \(\mathbb R\). A maximal ideal \(M\) of \(A\) is said to be real in case \(A/M = \mathbb R\); otherwise \(M\) is hyperreal. We say that \(X\) is \(A\)-compact in case \(M\) real implies \(M\) fixed. Thus \(X\) realcompact implies that \(X\) is \(C\)-compact. The following question has been floating for some time. If \(X\) is realcompact, then is there a minimal member \(A\in \Sigma(X)\) such that \(X\) is \(A\)-compact? \textit{S. K. Acharyya, K. C. Chattopadhyay}, and \textit{D. P. Ghosh} [Proc. Am. Math. Soc. 125, 611--615 (1997; Zbl 0858.54014)] defined a pre-order on \(\Sigma(X)\) different from set inclusion which provides an affirmative answer. They also conjectured that the answer is generally negative with respect to set inclusion. In the present paper the authors show that if \(X\) is realcompact but not compact, then the answer is indeed negative with respect to set inclusion.