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In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation , meaning "after , one reaches before ". For example, (October, February ). A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and total. Dropping the "total" requirement results in a partial cyclic order. A set with a cyclic order is called a cyclically ordered set or simply a cycle. Some familiar cycles are discrete, having only a finite number of elements: there are seven days of the week, four cardinal directions, twelve notes in the chromatic scale, and three plays in rock-paper-scissors. In a finite cycle, each element has a "next element" and a "previous element". There are also continuously variable cycles with infinitely many elements, such as the oriented unit circle in the plane. Cyclic orders are closely related to the more familiar linear orders, which arrange objects in a line. Any linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line. These operations, along with the related constructions of intervals and covering maps, mean that questions about cyclic orders can often be transformed into questions about linear orders. Cycles have more symmetries than linear orders, and they often naturally occur as residues of linear structures, as in the finite cyclic groups or the real projective line. ==Finite cycles== A cyclic order on a set with elements is like an arrangement of on a clock face, for an -hour clock. Each element in has a "next element" and a "previous element", and taking either successors or predecessors cycles exactly once through the elements as . There are a few equivalent ways to state this definition. A cyclic order on is the same as a permutation that makes all of into a single cycle. A cycle with elements is also a -torsor: a set with a free transitive action by a finite cyclic group. Another formulation is to make into the standard directed cycle graph on vertices, by some matching of elements to vertices. It can be instinctive to use cyclic orders for symmetric functions, for example as in : where writing the final monomial as would distract from the pattern. A substantial use of cyclic orders is in the determination of the conjugacy classes of free groups. Two elements and of the free group on a set are conjugate if and only if, when they are written as products of elements and with in , and then those products are put in cyclic order, the cyclic orders are equivalent under the rewriting rules that allow one to remove or add adjacent and . A cyclic order on a set can be determined by a linear order on , but not in a unique way. Choosing a linear order is equivalent to choosing a first element, so there are exactly linear orders that induce a given cyclic order. Since there are possible linear orders, there are possible cyclic orders. 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「Cyclic order」の詳細全文を読む スポンサード リンク
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