The P -frame Reflection of a Completely Regular Frame

Ompactification of Completely Regular Frames based on their Cozero Part

 Let L  be a frame. We denoted the set of all regular ideals of cozL by rId(cozL) . The aim of this paper is to study these ideals. For a  frame L , we show that  rId(cozL) is a compact completely regular frame and the map jc : rId(cozL)→L  given by jc (I)=⋁I   is a compactification of L which is isomorphism to its  Stone–Čech compactification and is proved that jc have a right adjoint rc : L →...

Compactification of κ-Frames

Concerning the frame of minimal prime ideals of pointfree function rings

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of continuous real-valued functions on $L$. We study the frame $mathfrak{O}(Min(mathcal{R}L))$ of minimal prime ideals of $mathcal{R}L$ in relation to $beta L$. For $Iinbeta L$, denote by $textit{textbf{O}}^I$ the ideal ${alphainmathcal{R}Lmidcozalphain I}$ of $mathcal{R}L$. We show that sending $I$ to the set of minimal prime ...

Uniformities and covering properties for partial frames (II)

This paper is a continuation of [Uniformities and covering properties for partial frames (I)], in which we make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting there our axiomatization of partial frames, which we call $sels$-frames, we added structure, in th...

INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...