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Meet-Semilattice Congruences on a Frame

  • John Frith
  • Anneliese Schauerte
Article
  • 29 Downloads

Abstract

The congruence lattice of a frame has long been an object of considerable interest, not least because it turns out to be a frame itself. Perhaps more surprisingly congruence lattices of, for instance, \(\sigma \)-frames, \(\kappa \)-frames and some partial frames also turn out to be frames. The situation for congruences of a meet-semilattice is notably different. In this paper we analyze the meet-semilattice congruence lattices of arbitrary frames and compare them with the corresponding lattices of frame congruences. In the course of this, we provide a structure theorem as well as many examples and counter-examples.

Keywords

Complete lattice Frame Partial frame \(\mathcal {S}\)-frame Frame congruence Meet-semilattice congruence Complements 

Mathematics Subject Classification

06B10 06D22 

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa

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