C-connected frame congruences

Document Type: Research Paper

Authors

1 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.

2 Department of Applied Mathematics and CE-ITI, MFF, Charles University, Malostransk'e n'am. 24, 11800 Praha 1, Czech Republic.

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


[1] Aull, C.E. and W.J. Thron, Separation axioms between T0 and T1, Indag. Math. 24 (1962), 26-37.
[2] Ball, R.N., J. Picado, and A. Pultr, On an aspect of scatteredness in the point-free setting, Port. Math. 73(2) (2016), 139-152.
[3] Banaschewski, B., J.L. Frith, and C.R.A. Gilmour, On the congruence lattice of a frame, Pacific J. Math. 130(2) (1987), 209-213.
[4] Banaschewski, B. and A. Pultr, Pointfree aspects of the TD axiom of classical topology, Quaest. Math. 33(3)  (2010), 369-385.
[5] Birkhof, G., Lattice Theory", Amer. Math. Soc. Colloq. Publ. Vol. 25, Third edition, American Mathematical Society, 1967.
[6] Chen, X., On the local connectedness of frames, J. Pure Appl. Algebra 79 (1992), 35-43.
[7] Dube, T., Submaximality in locales, Topology Proc. 29 (2005), 431-444.
[8] Gratzer, G., General Lattice Theory", Academic Press, 1978.
[9] Isbell, J.R., Atomless parts of spaces, Math. Scand. 31 (1972), 5-32.
[10] Johnstone, P.T., Stone Spaces", Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, Cambridge, 1982.
[11] Picado, J. and A. Pultr, Frames and Locales: Topology without Points", Frontiers in Mathematics 28, Springer, Basel, 2012.
[12] Picado, J. and A. Pultr, Still more about subfitness, to appear in Appl. Categ. Structures.
[13] Plewe, T., Sublocale lattices, J. Pure and Appl. Algebra 168 (2002), 309-326.