Concerning \(P\)-frames, essential \(P\)-frames, and strongly zero-dimensional frames (Q1040656)

A topological space is said to be a \(P\)-space if every zero-set is open. This notion extends to the pointfree setting of frames in the following way: a frame \(L\) is a \(P\)-frame in case \(a\vee a^*=1\) for every cozero element \(a\) of \(L\). Indeed, a topological space \(X\) is a \(P\)-space if and only if the frame of open sets of \(X\) is a \(P\)-frame. In the present paper, the author introduces a generalization of \(P\)-frames, called essential \(P\)-frames, proves that they are always strongly zero-dimensional and presents several nice and useful characterizations of \(P\)-frames, essential \(P\)-frames and strongly zero-dimensional frames in terms of algebraic properties of the ring of continuous functions on the frame.