A characterization of the cozero elements of a frame, without reference to the reals, is given and is used to obtain a characterization of pseudocompactness also independent of the reals. Applications are made to the congruence frame of a σ-frame and to Alexandroff spaces.

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...

A bijective correspondence between strong inclusions and compactifications in the setting of σ-frames is presented. The category of uniform σ-frames is defined and a description of the Samuel compactification is given. It is shown that the Samuel compactification of a uniform frame is completely determined by the σ-frame consisting of its uniform cozero part, and consequently, any compactificat...

We say that a Tychonoff space X has computable z-radicals if for all ideals a of C(X), the smallest z-ideal containing a is generated as an ideal by all the s◦f , where f is in a and s is a continuous function IR −→ IR with s−1(0) = {0}. We show that every cozero set of a compact space has computable z-radicals and that a subset X of IR has computable z-radicals if and only if X is locally closed.

We say that a Tychonoff space X has computable z-radicals if for all ideals a of C(X), the smallest z-ideal containing a is generated as an ideal by all the s ◦ f , where f is in a and s is a continuous function IR −→ IR with s−1(0) = {0}. We show that every cozero set of a compact space has computable z-radicals and that a subset X of IR has computable z-radicals if and only if X is locally cl...