The main topic of the paper is the investigation of the connection between subalgebras and Riesz subspaces of an Archimedean \(f\)-algebra with unit element and also of the relationship between algebra homomorphisms and Riesz homomorphisms between such \(f\)-algebras, respectively. Some of the results are:
(1) Let \(A\) be an Archimedean \(f\)-algebra with unit element \(e\).
(a) The uniformly closed Riesz subspace \(L\) of \(A\) is a subalgebra if and only if \(v\wedge e\in L^+\) for all \(v\in L^+\). (The latter condition is called the “Stone condition”.)
(b) The uniformly closed subalgebra \(L\) of \(A\) is a Riesz subspace if and only if \(L\) satisfies the Stone condition.
(2) Let \(A\) and \(B\) be Archimedean \(f\)-algebras with unit elements \(e_ A\) and \(e_ B\), respectively. Denote by \({\mathcal H}\) the convex set of all Markov operators from \(A\) into \(B\), so \({\mathcal H}=\{S:A\to B; S \text{ linear}, S\geq 0, Se_ A=e_ B\}\). If \(T\in {\mathcal H}\), then \(T\) is an algebra homomorphism if and only if \(T\) is a Riesz homomorphism. This result generalizes the result for the \(C(X)-C(Y)\) case, proved by A. J. Ellis. Furthermore, \(T\) is an extremal point of \({\mathcal H}\) if and only if \(T\) is a Riesz (algebra) homomorphism for which \(Te_ A=e_ B\). The latter equivalence is for the \(C(X)-C(Y)\) case due to A. and C. Ionescu-Tulcea, R. R. Phelps and J. Tate.