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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.〔K. Devlin. ''Constructibility''. Springer, Berlin, 1984.〕 Velleman 〔D. Velleman. Simplified Morasses, ''Journal of Symbolic Logic'' 49, No. 1 (1984), pp 257–271.〕 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,〔D. Velleman. Simplified Morasses, ''Journal of Symbolic Logic'' 49, No. 1 (1984), pp 257–271.〕 where he also constructed (ω0,1) simplified morasses in ZFC. In 〔D. Velleman. Simplified Gap-2 Morasses, ''Annals of Pure and Applied Logic'' 34, (1987), pp 171–208.〕 he gave similar simple definitions for gap-2 ''simplified morasses'', and in 〔D. Velleman. Gap-2 Morasses of Height ω0, ''Journal of Symbolic Logic'' 52, (1987), pp 928–938.〕 he constructed (ω0,2) simplified morasses in ZFC. Higher gap simplified morasses for any ''n'' ≥ 1 were defined by Morgan 〔Ch. Morgan. ''The Equivalence of Morasses and Simplified Morasses in the Finite Gap Case'', PhD.Thesis, Merton College, UK, 1989.〕 and Szalkai,.〔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〕 Roughly speaking: a (κ,''n'' + 1)-simplified morass (of Szalkai) M = < ''M''→, ''F''⇒ > contains a sequence ''M''→ = < ''M''''β'' : β ≤ κ > of (< ''κ'',''n'')-simplified morass-like structures for ''β'' < ''κ'' , ''M''''κ'' is a (''κ''+,''n'') -simplified morass, and a double sequence F⇒ = < ''F''α,β : ''α'' < ''β'' ≤ κ > where ''F''''α'',''β'' are collections of mappings from ''M''''α'' to ''M''''β'' for α < β ≤ κ with specific conditions. Quagmires are similar, morass-like structures in set theory. == References == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Simplified morass」の詳細全文を読む スポンサード リンク
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