Abstract
We discuss the congruences theta that are connected as elements of the (totally disconnected) congruence frame CL, 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 CL.
They are, among others, shown to be TD-spatial; characteristics for some special cases (Boolean, linear, scattered and Noetherian) are presented.