Simplified morass
 | It has been suggested that this article be merged into Morass (set theory). (Discuss) Proposed since September 2014. |
 | This article needs attention from an expert in mathematics. Please add a reason or a talk parameter to this template to explain the issue with the article. (August 2014) |
 | This article may be too technical for most readers to understand. (August 2014) |
In mathematics, a (κ,n)-morass is a specific structure M of "height" κ and "gap" n for any uncountable regular cardinal κ and natural number n ≥ 1.
The original definition and applications of gap-1 and higher gap (ordinary) morasses, invented by Ronald Jensen, are complicated ones, see eg.[1]
Velleman [2] defined much simpler structures for n = 1 and showed that the existence of gap-1 morasses is equivalent to the existence of gap-1 simplified morasses.
Roughly speaking: a (κ,1)-simplified morass M = < φ→, F⇒ > contains a sequence φ→ = < φβ : β ≤ κ > of ordinals such that φβ < κ for β < κ and φκ = κ+, and a double sequence F⇒ = < Fα,β : α < β ≤ κ > where Fα,β are collections of monotone mappings from φα to φβ for α<β ≤ κ with specific (easy but important) conditions.
Velleman's clear definition can be found in,[3] where he also constructed (ω0,1) simplified morasses in ZFC. In [4] he gave similar simple definitions for gap-2 simplified morasses, and in [5] he constructed (ω0,2) simplified morasses in ZFC.
Higher gap simplified morasses for any n ≥ 1 were defined by Morgan [6] and Szalkai,.[7][8]
Roughly speaking: a (κ,n + 1)-simplified morass (of Szalkai) M = < M→, F⇒ > contains a sequence M→ = < Mβ : β ≤ κ > of (< κ,n)-simplified morass-like structures for β < κ , Mκ is a (κ+,n) -simplified morass, and a double sequence F⇒ = < Fα,β : α < β ≤ κ > where Fα,β are collections of mappings from Mα to Mβ for α < β ≤ κ with specific conditions.
Quagmires are similar, morass-like structures in set theory.[9]
References
- ^ K. Devlin. Constructibility. Springer, Berlin, 1984.
- ^ D. Velleman. Simplified Morasses, Journal of Symbolic Logic 49, No. 1 (1984), pp 257–271.
- ^ D. Velleman. Simplified Morasses, Journal of Symbolic Logic 49, No. 1 (1984), pp 257–271.
- ^ D. Velleman. Simplified Gap-2 Morasses, Annals of Pure and Applied Logic 34, (1987), pp 171–208.
- ^ D. Velleman. Gap-2 Morasses of Height ω0, Journal of Symbolic Logic 52, (1987), pp 928–938.
- ^ Ch. Morgan. The Equivalence of Morasses and Simplified Morasses in the Finite Gap Case, PhD.Thesis, Merton College, UK, 1989.
- ^ I. Szalkai. Higher Gap Simplified Morasses and Combinatorial Applications, PhD-Thesis (in Hungarian), ELTE, Budapest, 1991. English abstract: http://math.uni-pannon.hu/~szalkai/Szalkai-1991d-MorassAbst-.pdf
- ^ I. Szalkai. An Inductive Definition of Higher Gap Simplified Morasses, Publicationes Mathematicae Debrecen 58 (2001), pp 605–634. http://math.uni-pannon.hu/~szalkai/Szalkai-2001a-IndMorass.pdf
- ^ Kanamori, Akihiro (1983). "Morasses in combinatorial set theory". In Mathias, A.R.D. (ed.). Surveys in set theory. London Mathematical Society Lecture Note Series. Vol. 87. Cambridge: Cambridge University Press. pp. 167–196. ISBN 0-521-27733-7. Zbl 0525.03036.