A study of a new dimension for frames (Q1985618)

The authors introduce a new type of dimension for frames, \(\mathsf{dim}_{\mathsf q}\), called quasi covering dimension, and study some of its properties in different classes of frames. The new definition is obtained from the familiar definition of covering dimension by replacing covers with quasi-covers, that is, subsets of the frame whose supremum is a dense element of the frame. The two notions are related in a frame \(L\) by the identity \[\dim_{\mathsf q}(L)=\bigvee\{\dim(\downarrow x) :\ x\mbox{ is a dense element of }L \}.\]