C-connected frame congruences

Abstract

We discuss the congruences $theta$ that are connected as elements of the (totally disconnected) congruence frame $CF L$, and show that they are in a one-to-one correspondence with the completely prime elements of $L$, giving an explicit formula. Then we investigate those frames $L$ with enough connected congruences to cover the whole of $CF L$. They are, among others, shown to be $T_D$-spatial; characteristics for some special cases (Boolean, linear, scattered and Noetherian) are presented.

Keywords

• Frame
• frame congruence
• congruence and sublocale lattice
• connectedness
• $T_D$-spatiality