Order and interval topologies on complete Boolean algebras
Dominic van der Zypen
TL;DR
This paper investigates the relationship between order and interval topologies on complete Boolean algebras, providing a negative answer to a longstanding open problem about their equivalence.
Contribution
It introduces and examines order convergence and the interval topology on posets, and resolves Birkhoff's problem by showing these topologies do not coincide on complete Boolean algebras.
Findings
Order and interval topologies differ on complete Boolean algebras
Answer to Birkhoff's Problem 76 is negative
Provides new insights into topological structures on Boolean algebras
Abstract
We introduce and examine order convergence and the interval topology on partially ordered sets in general. Problem 76 of Birkhoff's "Lattice Theory" asks whether for complete Boolean algebras the order topology and the interval topology coincide. We answer this question in the negative.
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Taxonomy
TopicsAdvanced Algebra and Logic · Advanced Topology and Set Theory · Constraint Satisfaction and Optimization