Abstract
All the binomial lattices embed into \(Q_{\vee }(\mathbb {I})\), the complete lattice of sup-preserving endomaps of the unit interval—whose elements can be seen as continuous monotone paths from (0, 0) to (1, 1). This lattice is completely distributive. We give a general description of the complete congruences of completely distributive lattice s by means of an interior operator on the collection of closed subsets of an associated topological space. In particular, we show that these form a frame. We give a description of this frame for the unit interval lattice, showing that it is not a Boolean algebra nor a (co)spatial frame. For \(Q_{\vee }(\mathbb {I})\), we give a geometrical interpretation of these congruences by means of directed homotopies.
Research supported by the project LAMBDACOMB (ANR-21-CE48-0017).
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Acknowledgment
The authors are thankful to Mai Gehrke for the numerous hints, for the pointers to the literature, and for a careful reading of a first draft of this manuscript. The authors are also thankful to the anonymous referees for the valuable suggestions by which a first version of this manuscript could be improved.
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Calk, C., Santocanale, L. (2024). Complete Congruences of Completely Distributive Lattices. In: Fahrenberg, U., Fussner, W., Glück, R. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2024. Lecture Notes in Computer Science, vol 14787. Springer, Cham. https://doi.org/10.1007/978-3-031-68279-7_7
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