Abstract
Strictly zero-dimensional biframes were introduced by Banaschewski and Brümmer as a class of strongly zero-dimensional biframes including the congruence biframes. We consider the category of strictly zero-dimensional biframes and show it is both complete and cocomplete. We characterise the extremal epimorphisms in this category and explore the special position that congruence biframes hold in it. Finally, we provide an internal characterisation of congruence biframes, and hence, of congruence frames.
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References
Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories: The Joy of Cats. Dover Publications, Mineola (2009)
Banaschewski, B., Brümmer, G.C.L.: Stably continuous frames. Math. Proc. Camb. Philos. Soc. 104(1), 7–19 (1988)
Banaschewski, B., Brümmer, G.C.L.: Strong zero-dimensionality of biframes and bispaces. Quaest. Math. 13(3–4), 273–290 (1990)
Banaschewski, B., Brümmer, G.C.L., Hardie, K.A.: Biframes and bispaces. Quaest. Math. 6(1–3), 13–25 (1983)
Banaschewski, B., Frith, J.L., Gilmour, C.R.A.: On the congruence lattice of a frame. Pac. J. Math. 130(2), 209–213 (1987)
Dow, A.S., Watson, S.W.: Skula spaces. Comment. Math. Univ. Carol. 31(1), 27–31 (1990)
Frith, J.L.: Structured frames. Ph.D. Thesis, University of Cape Town (1987)
Frith, J.L., Schauerte, A.: An asymmetric characterization of the congruence frame. Topol. Appl. 158(7), 939–944 (2011)
Isbell, J.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)
Isbell, J.: On dissolute spaces. Topol. Appl. 40, 63–70 (1991)
Joyal, A., Tierney, M.: An extension of the Galois theory of Grothendieck. Mem. Am. Math. Soc. 51(309), 1–71 (1984)
Manuell, G.: Congruence frames of frames and \(\kappa \)-frames. Master’s Thesis, University of Cape Town (2015)
Manuell, G.: A special class of congruences on \(\kappa \)-frames. Algebra Universalis 78(2), 125–130 (2017)
Picado, J., Pultr, A.: Frames and Locales: Topology Without Points. Frontiers in Mathematics. Springer, Basel (2012)
Tholen, W.: Semi-topological functors I. J. Pure Appl. Algebra 15(1), 53–73 (1979)
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Communicated by B. Banaschewski.
I acknowledge financial assistance from the National Research Foundation of South Africa (Grant No. 89726).
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Manuell, G. Strictly Zero-Dimensional Biframes and a Characterisation of Congruence Frames. Appl Categor Struct 26, 645–655 (2018). https://doi.org/10.1007/s10485-017-9507-x
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DOI: https://doi.org/10.1007/s10485-017-9507-x