Rieger and Nishimura introduced this construction to study free Heyting algebras on one generator. In particular, the ladder displayed here is the Esakia dual to this algebra.
Details of the construction
The Rieger–Nishimura ladder can be described in several ways (see Bellissima’s construction). The one given here is taken from Esakia. We define the space RN = ℕ ∪ {∞}, with the following order.
RN = ℕ ∪ {∞}
i ≤ j ⇔ i = j or i ≥ j + 2
∞ ≤ i for every finite i
p = {1}
i ≤ j ⇔ i = j or i ≥ j + 2
∞ ≤ i for every finite i
p = {1}
The topology has a basis consisting of the finite subsets of ℕ and the cofinite subsets containing ∞. This makes the "finite points" converge to ∞. The clopen-upsets are then each finite upset of RN and the whole space.
References
- On the Lattice Theory of Brouwerian Propositional Logic.Acta Facultatis Rerum Naturalium Universitatis Carolinae, no. 189, pp. 1–40. F. Řivnáč, Prague.Bibliographic record, reference [3]
- On formulas of one variable in intuitionistic propositional calculus.The Journal of Symbolic Logic 25(4), 327–331.DOI
- Lattices of Intermediate and Cylindric Modal Logics.PhD thesis, Universiteit van Amsterdam. ILLC Dissertation Series DS-2006-02. ISBN 978-90-5776-147-8.Thesis
- Finitely generated free Heyting algebras: the well-founded initial segment.The Journal of Symbolic Logic 77(4), 1291–1307.DOI