Alternative mathematical ordering
In mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as ""a" ; "b"". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation ["a", "b", "c"], meaning "after a, one reaches b before c". For example, [June, October, February], but not [June, February, October], cf. picture. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and connected. Dropping the "connected" requirement results in a partial cyclic order.
A set with a cyclic order is called a cyclically ordered set or simply a cycle.[nb] 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 cyclic orders 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 X with n elements is like an arrangement of X on a clock face, for an n-hour clock. Each element x in X has a "next element" and a "previous element", and taking either successors or predecessors cycles exactly once through the elements as "x"(1), "x"(2), ..., "x"("n").
There are a few equivalent ways to state this definition. A cyclic order on X is the same as a permutation that makes all of X into a single cycle, which is a special type of permutation - a circular permutation. Alternatively, a cycle with n elements is also a Z"n"-torsor: a set with a free transitive action by a finite cyclic group. Another formulation is to make X into the standard directed cycle graph on n vertices, by some matching of elements to vertices.
It can be instinctive to use cyclic orders for symmetric functions, for example as in
"xy" + "yz" + "zx"
where writing the final monomial as "xz" would distract from the pattern.
A substantial use of cyclic orders is in the determination of the conjugacy classes of free groups. Two elements g and h of the free group F on a set Y are conjugate if and only if, when they are written as products of elements y and "y−1" with y in Y, 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 y and "y"−1.
A cyclic order on a set X can be determined by a linear order on X, but not in a unique way. Choosing a linear order is equivalent to choosing a first element, so there are exactly n linear orders that induce a given cyclic order. Since there are "n"! possible linear orders (as in permutations), there are ("n" − 1)! possible cyclic orders (as in circular permutations).
Definitions.
An infinite set can also be ordered cyclically. Important examples of infinite cycles include the unit circle, "S"1, and the rational numbers, Q. The basic idea is the same: we arrange elements of the set around a circle. However, in the infinite case we cannot rely upon an immediate successor relation, because points may not have successors. For example, given a point on the unit circle, there is no "next point". Nor can we rely upon a binary relation to determine which of two points comes "first". Traveling clockwise on a circle, neither east or west comes first, but each follows the other.
Instead, we use a ternary relation denoting that elements a, b, c occur after each other (not necessarily immediately) as we go around the circle. For example, in clockwise order, [east, south, west]. By currying the arguments of the ternary relation ["a", "b", "c"], one can think of a cyclic order as a one-parameter family of binary order relations, called "cuts", or as a two-parameter family of subsets of K, called "intervals".
The ternary relation.
The general definition is as follows: a cyclic order on a set X is a relation "C" ⊂ "X"3, written ["a", "b", "c"], that satisfies the following axioms:[nb]
The axioms are named by analogy with the asymmetry, transitivity, and connectedness axioms for a binary relation, which together define a strict linear order. Edward Huntington (1916, 1924) considered other possible lists of axioms, including one list that was meant to emphasize the similarity between a cyclic order and a betweenness relation. A ternary relation that satisfies the first three axioms, but not necessarily the axiom of totality, is a partial cyclic order.
Rolling and cuts.
Given a linear order ; on a set X, the cyclic order on X induced by ; is defined as follows:
["a", "b", "c"] if and only if "a" ; "b" ; "c" or "b" ; "c" ; "a" or "c" ; "a" ; "b"
Two linear orders induce the same cyclic order if they can be transformed into each other by a cyclic rearrangement, as in
cutting a deck of cards. One may define a cyclic order relation as a ternary relation that is induced by a strict linear order as above.
Cutting a single point out of a cyclic order leaves a linear order behind. More precisely, given a cyclically ordered set formula_1, each element formula_2 defines a natural linear order formula_3 on the remainder of the set, formula_4, by the following rule:
 ;templatestyles src="Block indent/styles.css"/ ;formula_5 if and only if formula_6.
Moreover, formula_3 can be extended by adjoining formula_8 as a least element; the resulting linear order on formula_9 is called the principal cut with least element formula_8. Likewise, adjoining formula_8 as a greatest element results in a cut formula_12.
Intervals.
Given two elements formula_13, the open interval from formula_8 to formula_15, written formula_16, is the set of all formula_17 such that formula_18. The system of open intervals completely defines the cyclic order and can be used as an alternate definition of a cyclic order relation.
An interval formula_16 has a natural linear order given by formula_3. One can define half-closed and closed intervals formula_21, formula_22, and formula_23 by adjoining formula_8 as a least element and/or formula_15 as a greatest element. As a special case, the open interval formula_26 is defined as the cut formula_27.
More generally, a proper subset formula_28 of formula_9 is called convex if it contains an interval between every pair of points: for formula_30, either formula_16 or formula_32 must also be in formula_28. A convex set is linearly ordered by the cut formula_34 for any formula_35 not in the set; this ordering is independent of the choice of formula_35.
Automorphisms.
As a circle has a clockwise order and a counterclockwise order, any set with a cyclic order has two senses. A bijection of the set that preserves the order is called an ordered correspondence. If the sense is maintained as before, it is a direct correspondence, otherwise it is called an opposite correspondence. Coxeter uses a separation relation to describe cyclic order, and this relation is strong enough to distinguish the two senses of cyclic order. The automorphisms of a cyclically ordered set may be identified with C2, the two-element group, of direct and opposite correspondences.
Monotone functions.
The "cyclic order = arranging in a circle" idea works because any subset of a cycle is itself a cycle. In order to use this idea to impose cyclic orders on sets that are not actually subsets of the unit circle in the plane, it is necessary to consider functions between sets.
A function between two cyclically ordered sets, "f" : "X" → "Y", is called a "monotonic function" or a "homomorphism" if it pulls back the ordering on Y: whenever ["f"("a"), "f"("b"), "f"("c")], one has ["a", "b", "c"]. Equivalently, f is monotone if whenever ["a", "b", "c"] and "f"("a"), "f"("b"), and "f"("c") are all distinct, then ["f"("a"), "f"("b"), "f"("c")]. A typical example of a monotone function is the following function on the cycle with 6 elements:
"f"(0) = "f"(1) = 4,
"f"(2) = "f"(3) = 0,
"f"(4) = "f"(5) = 1.
A function is called an "embedding" if it is both monotone and injective.[nb] Equivalently, an embedding is a function that pushes forward the ordering on X: whenever ["a", "b", "c"], one has ["f"("a"), "f"("b"), "f"("c")]. As an important example, if X is a subset of a cyclically ordered set Y, and X is given its natural ordering, then the inclusion map "i" : "X" → "Y" is an embedding.
Generally, an injective function f from an unordered set X to a cycle Y induces a unique cyclic order on X that makes f an embedding.
Functions on finite sets.
A cyclic order on a finite set X can be determined by an injection into the unit circle, "X" → "S"1. There are many possible functions that induce the same cyclic order—in fact, infinitely many. In order to quantify this redundancy, it takes a more complex combinatorial object than a simple number. Examining the configuration space of all such maps leads to the definition of an polytope known as a cyclohedron. Cyclohedra were first applied to the study of knot invariants; they have more recently been applied to the experimental detection of periodically expressed genes in the study of biological clocks.
The category of homomorphisms of the standard finite cycles is called the cyclic category; it may be used to construct Alain Connes' cyclic homology.
One may define a degree of a function between cycles, analogous to the degree of a continuous mapping. For example, the natural map from the circle of fifths to the chromatic circle is a map of degree 7. One may also define a rotation number.
Completion.
The set of all cuts is cyclically ordered by the following relation: [ ;1, ;2, ;3] if and only if there exist "x", "y", "z" such that:
 "x" ;1 "y" ;1 "z",
 "x" ;1"y" ;2 "z" ;2 "x", and
 "x" ;1 "y" ;1"z" ;3 "x" ;3 "y".
A certain subset of this cycle of cuts is the Dedekind completion of the original cycle.
Further constructions.
Unrolling and covers.
Starting from a cyclically ordered set K, one may form a linear order by unrolling it along an infinite line. This captures the intuitive notion of keeping track of how many times one goes around the circle. Formally, one defines a linear order on the Cartesian product Z × "K", where Z is the set of integers, by fixing an element a and requiring that for all i:
 If ["a", "x", "y"], then "a""i" ; "x""i" ; "y""i" ; "a""i"+1.
For example, the months 2025, 2025, 2025, and 2026 occur in that order.
This ordering of Z × "K" is called the universal cover of K.[nb] Its order type is independent of the choice of a, but the notation is not, since the integer coordinate "rolls over" at a. For example, although the cyclic order of pitch classes is compatible with the A-to-G alphabetical order, C is chosen to be the first note in each octave, so in note-octave notation, B3 is followed by C4.
The inverse construction starts with a linearly ordered set and coils it up into a cyclically ordered set. Given a linearly ordered set L and an order-preserving bijection "T" : "L" → "L" with unbounded orbits, the orbit space "L" / "T" is cyclically ordered by the requirement:[nb]
 If "a" ; "b" ; "c" ; "T"("a"), then [["a"], ["b"], ["c"]].
In particular, one can recover K by defining "T"("x""i") = "x""i"+1 on Z × "K".
There are also n-fold coverings for finite n; in this case, one cyclically ordered set covers another cyclically ordered set. For example, the 24-hour clock is a double cover