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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''. #< < = , 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
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