Lattice of subalgebras of the ring of continuous functions and Hewitt spaces

The lattice $A(X)$ of all possible subalgebras of the ring of all continuous $\mathbb R$-valued functions defined on an $\mathbb R$-separated space $X$ is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line $\mathbb R$. The main achievement of the paper is the proof of the fact that any Hewitt space $X$ is determined by the lattice $A(X)$. An original technique of minimal and maximal subalgebras is applied. It is shown that the lattice $A(X)$ is regular if and only if $X$ contains at most two points.