Decidable theories of non-projectable \(l\)-groups of continuous functions (Q876381)

An (abelian) \(\ell \)-group \(G\) is called projectable if for every \(0<a\in G\) the interval \([0,a]\) is pseudocomplemented. If \(X\) is a topological space then \(C(X)\) denotes the \(\ell \)-group of all continuous real-valued functions on \(X\). The author studies the class of \(\ell \)-groups \(C(X)\) such that \(X\) is a so-called essential \(P\)-space. In the paper it is shown that many of such \(\ell \)-groups are non-projectable and that their elementary theories may often be reduced to those of associated Boolean algebras with distinguished ideals. This is used to establish the decidability of two classes of \(\ell \)-groups of continuous functions which, among others, are not projectable.