This frame arose in our studies of completeness theory, seeking the extent of the Artin-gluing based completion method which was found to work for the Fine general frame.
Details of the construction
An integer partition λ is a finite nonincreasing list of positive row lengths. Draw those rows as a Young diagram.
The lattice corresponding to this order is an object of study in algebraic combinatorics, and other fields. One key use is in irreducible representations of symmetric groups of order n.
For our purposes, its complex branching structure, paired with sufficient regularity, made it an attractive example to test various conjectures in superintuitionistic logic.
This Fine-style analogue attaches a Rieger-Nishimura poset to Young’s lattice We enumerate all finite partitions by increasing box count, breaking ties lexicographically. If λ has enumeration index k, attach a support gλ below b and both ladder points at level k.
xλ ≤ xμ ⇔ λ ⊆ μ
xλ < gμ ⇔ λ ⊆ μ
There is an additional point z which is incomparable with the Rieger-Nishimura, and above every antichain point.
Choose a support or mast point to inspect its partition as a Young diagram.
References
- On Quantitative Substitutional Analysis.Proceedings of the London Mathematical Society, Series 1, 33, 97–146. Communicated in 1900; revised in 1901. Young diagrams appear in §15, p. 133.DOI
- Enumerative Combinatorics, Volume 2.Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press; Chapter 7, Symmetric Functions.DOI
- Young Tableaux: With Applications to Representation Theory and Geometry.London Mathematical Society Student Texts, vol. 35. Cambridge University Press.DOI