Lattice of subalgebras of the ring of continuous functions and Hewitt spaces (Q1277577)

Suppose that \(X\), \(Y\) are realcompact spaces and \(C(X)\) is the ring of continuous mappings from \(X\) into the real numbers \(\mathbb{R}\). Let \(A(X)\) denote the lattice of all subalgebras of \(C(X)\) under the inclusion relation. The author's main result is that if \(A(X)\), \(A(Y)\) are lattice isomorphic, then \(X\), \(Y\) are homeomorphic. To this end, he shows that the atoms of \(A(X)\) have the form \(eR\), where \(e\in C(X)\) is an idempotent. He describes various properties of disconnected spaces \(X\) in terms of the atoms of \(A(X)\), and he determines when \(A(X)\) is modular or distributive. He poses the open question as to whether all maximal subalgebras of \(A(X)\) are either real maximal ideals or else have the form \(A_{x,y}= \{f\in C(X): f(x)= f(y)\}\) for some points \(x\neq y\) in \(X\).