翻訳と辞書
Words near each other
・ Fencing at the 1952 Summer Olympics – Men's team sabre
・ Fencing at the 1952 Summer Olympics – Men's team épée
・ Fencing at the 1952 Summer Olympics – Men's épée
・ Fencing at the 1952 Summer Olympics – Women's foil
・ Fencing at the 1956 Summer Olympics
・ Fencing at the 1956 Summer Olympics – Men's foil
・ Fencing at the 1956 Summer Olympics – Men's sabre
・ Fencing at the 1956 Summer Olympics – Men's team foil
・ Fencamine
・ Fence
・ Fence (community), Wisconsin
・ Fence (criminal)
・ Fence (disambiguation)
・ Fence (finance)
・ Fence (magazine)
Fence (mathematics)
・ Fence (woodworking)
・ Fence at Alamo Cement Company
・ Fence Colliery
・ Fence Cutting Wars
・ Fence for Life
・ Fence insert
・ Fence Lake, New Mexico
・ Fence lizard
・ Fence Records
・ Fence River
・ Fence Viewer
・ Fence, Lancashire
・ Fence, Wisconsin
・ Fencehouses


Dictionary Lists
翻訳と辞書 辞書検索 [ 開発暫定版 ]
スポンサード リンク

Fence (mathematics) : ウィキペディア英語版
Fence (mathematics)

In mathematics, a fence, also called a zigzag poset, is a partially ordered set in which the order relations form a path with alternating orientations:
:''a'' < ''b'' > ''c'' < ''d'' > ''e'' < ''f'' > ''h'' < ''i'' ...
or
:''a'' > ''b'' < ''c'' > ''d'' < ''e'' > ''f'' < ''h'' > ''i'' ...
A fence may be finite, or it may be formed by an infinite alternating sequence extending in both directions. The incidence posets of path graphs form examples of fences.
A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been studied since the 19th century.〔.〕 The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are
:1, 1, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042 .
The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation theorem, has as its graph the Fibonacci cube.〔 calls the fact that this lattice has a Fibonacci number of elements a “well known fact,” while asks for a description of it in an exercise. See also , , and .〕
A partially ordered set is series-parallel if and only if it does not have four elements forming a fence.〔.〕
Several authors have also investigated the number of order-preserving maps from fences to themselves, or to fences of other sizes.〔; ; ; ; .〕
An up-down poset ''Q''(''a'',''b'') is a generalization of a zigzag poset in which there are ''a'' downward orientations for every upward one and ''b'' total elements.〔.〕 For instance, ''Q''(2,9) has the elements and relations
:''a'' > ''b'' > ''c'' < ''d'' > ''e'' > ''f'' < ''g'' > ''h'' > ''i''.
In this notation, a fence is a partially ordered set of the form ''Q''(1,''n'').
==Equivalent conditions==
The following conditions are equivalent for a poset ''P'':
#''P'' is a disjoint union of zigzag posets.
#If ''a'' ≤ ''b'' ≤ ''c'' in ''P'', either ''a'' = ''b'' or ''b'' = ''c''.
#< \circ < = \emptyset, i.e. it is never the case that ''a'' < ''b'' and ''b'' < ''c'', so that < is vacuously transitive.
#''P'' has dimension at most one (defined analogously to the Krull dimension of a commutative ring).
#Every element of ''P'' is either maximal or minimal.
#The slice category Pos/''P'' is cartesian closed.
The prime ideals of a commutative ring ''R'', ordered by inclusion, satisfy the equivalent conditions above if and only if ''R'' has Krull dimension at most one.

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「Fence (mathematics)」の詳細全文を読む



スポンサード リンク
翻訳と辞書 : 翻訳のためのインターネットリソース

Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.