{"history":[],"article":{"id":"5ebf5f3c-1fed-4b75-9033-31b8701ac810","title":"Cyclic order","slug":"cyclic-order","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **cyclic order** is a way to arrange a set of objects in a [circle](https://bliptext.com/articles/circle).\nUnlike most structures in [order theory](https://bliptext.com/articles/order-theory), a cyclic order is not modeled as a [binary relation](https://bliptext.com/articles/binary-relation), such as \"\na < b\n\". One does not say that east is \"more clockwise\" than west. Instead, a cyclic order is defined as a [ternary relation](https://bliptext.com/articles/ternary-relation)\n[a, b, c]\n, meaning \"after a, one reaches b before c\".\nFor example, [June, October, February], but not [June, February, October], cf.\npicture.\nA ternary relation is called a cyclic order if it is [cyclic, asymmetric, transitive, and connected](https://bliptext.com/articles/).\nDropping the \"connected\" requirement results in a [partial cyclic order](https://bliptext.com/articles/partial-cyclic-order).\nA [set](https://bliptext.com/articles/set-mathematics) with a cyclic order is called a **cyclically ordered [set](https://bliptext.com/articles/set-mathematics)** or simply a **cycle**.\nSome familiar cycles are discrete, having only a [finite number](https://bliptext.com/articles/finite-set) of [elements](https://bliptext.com/articles/element-mathematics): there are seven [days of the week](https://bliptext.com/articles/days-of-the-week), four [cardinal directions](https://bliptext.com/articles/cardinal-direction), twelve notes in the [chromatic scale](https://bliptext.com/articles/chromatic-scale), and three plays in [rock-paper-scissors](https://bliptext.com/articles/rock-paper-scissors).\nIn a finite cycle, each element has a \"next element\" and a \"previous element\".\nThere are also cyclic orders with infinitely many elements, such as the oriented [unit circle](https://bliptext.com/articles/unit-circle) in the plane.\nCyclic orders are closely related to the more familiar [linear orders](https://bliptext.com/articles/[line](https://bliptext.com/articles/line-geometry)ar-order), which arrange objects in a [line](https://bliptext.com/articles/line-geometry).\nAny linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line.\nThese 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.\nCycles have more symmetries than linear orders, and they often naturally occur as residues of linear structures, as in the [finite cyclic groups](https://bliptext.com/articles/finite-cyclic-group) or the [real projective line](https://bliptext.com/articles/real-projective-line).\n# [The months are a cyclic order. | upload.wikimedia.org/wikipedia/commons/4/40/Months///circle///svg.svg]\n\n# Finite cycles\nA cyclic order on a set X with n elements is like an arrangement of X on a clock face, for an n-hour clock.\nEach 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\nx(1), x(2), ..., x(n)\n.\nThere are a few equivalent ways to state this definition.\nA cyclic order on X is the same as a [permutation](https://bliptext.com/articles/permutation) that makes all of X into a single [cycle](https://bliptext.com/articles/cycles-and-fixed-points), which is a special type of [permutation](https://bliptext.com/articles/permutation) - a circular [permutation](https://bliptext.com/articles/permutation).\nAlternatively, a cycle with n elements is also a\nZn\n-[torsor](https://bliptext.com/articles/torsor): a set with a free transitive [action](https://bliptext.com/articles/group-action-mathematics) by a [finite cyclic group](https://bliptext.com/articles/finite-cyclic-group).\nAnother formulation is to make X into the standard [directed cycle graph](https://bliptext.com/articles/directed-cycle-graph) on n vertices, by some matching of elements to vertices.\nIt can be instinctive to use cyclic orders for [symmetric functions](https://bliptext.com/articles/symmetric-function), for example as in\nxy + yz + zx\nwhere writing the final [monomial](https://bliptext.com/articles/monomial) as\nxz\nwould distract from the pattern.\nA substantial use of cyclic orders is in the determination of the [conjugacy classes](https://bliptext.com/articles/conjugacy-class) of [free groups](https://bliptext.com/articles/free-group).\nTwo 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\ny−1\nwith y in Y, and then those products are put in cyclic order, the cyclic orders are equivalent under the [rewriting](https://bliptext.com/articles/rewriting) rules that allow one to remove or add adjacent y and\ny−1\n.\nA cyclic order on a set X can be determined by a linear order on X, but not in a unique way.\nChoosing a linear order is equivalent to choosing a first element, so there are exactly n linear orders that induce a given cyclic order.\nSince there are\nn!\npossible linear orders (as in [permutations](https://bliptext.com/articles/permutation)), there are\n(n − 1)!\npossible cyclic orders (as in [circular permutations](https://bliptext.com/articles/permutation)).\n# [A 5-element cycle | upload.wikimedia.org/wikipedia/commons/f/f9/Orientovan%C3%A1///kru%C5%BEnice.svg]\n\n# Definitions\nAn [infinite set](https://bliptext.com/articles/infinite-set) can also be ordered cyclically.\nImportant examples of infinite cycles include the [unit circle](https://bliptext.com/articles/unit-circle),\nS1\n, and the [rational numbers](https://bliptext.com/articles/rational-number),\nQ\n. The basic idea is the same: we arrange elements of the set around a circle.\nHowever, in the infinite case we cannot rely upon an immediate successor relation, because points may not have successors.\nFor example, given a point on the unit circle, there is no \"next point\".\nNor can we rely upon a binary relation to determine which of two points comes \"first\".\nTraveling clockwise on a circle, neither east or west comes first, but each follows the other.\nInstead, we use a ternary relation denoting that elements a, b, c occur after each other (not necessarily immediately) as we go around the circle.\nFor example, in clockwise order, [east, south, west].\nBy [currying](https://bliptext.com/articles/currying) the arguments of the ternary relation\n[a, b, c]\n, 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.\n\n## The ternary relation\nThe general definition is as follows: a cyclic order on a set X is a relation\nC ⊂ X3\n, written\n[a, b, c]\n, that satisfies the following axioms:\n[a, b, c]\nthen\n[b, c, a]\n[a, b, c]\nthen not\n[c, b, a]\n[a, b, c]\nand\n[a, c, d]\nthen\n[a, b, d]\n[a, b, c]\nor\n[c, b, a]\nThe axioms are named by analogy with the [asymmetry](https://bliptext.com/articles/asymmetric-relation), [transitivity](https://bliptext.com/articles/transitive-relation), and [connectedness](https://bliptext.com/articles/connected-relation) axioms for a binary relation, which together define a [strict linear order](https://bliptext.com/articles/strict-linear-order).\nconsidered other possible lists of axioms, including one list that was meant to emphasize the similarity between a cyclic order and a [betweenness relation](https://bliptext.com/articles/betweenness-relation).\nA ternary relation that satisfies the first three axioms, but not necessarily the axiom of totality, is a [partial cyclic order](https://bliptext.com/articles/partial-cyclic-order).\n\n## Rolling and cuts\nGiven a linear order\n<\non a set X, the cyclic order on X induced by\n<\nis defined as follows:\n[a, b, c]\nif and only if\na < b < c\nor\nb < c < a\nor\nc < a < b\nTwo 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](https://bliptext.com/articles/cut-cards).\nOne may define a cyclic order relation as a ternary relation that is induced by a strict linear order as above.\nCutting a single point out of a cyclic order leaves a linear order behind.\nMore precisely, given a cyclically ordered set, each element a\\in K defines a natural linear order <_a on the remainder of the set, , by the following rule:\nMoreover, <_a can be extended by adjoining a as a least element; the resulting linear order on K is called the principal cut with least element a. Likewise, adjoining a as a greatest element results in a cut <^a.\n\n## Intervals\nGiven two elements a\\ne b\\in K, the [open interval](https://bliptext.com/articles/open-interval) from a to b, written (a,b), is the set of all x\\in K such that [a,x,b].\nThe system of open intervals completely defines the cyclic order and can be used as an alternate definition of a cyclic order relation.\nAn interval (a,b) has a natural linear order given by <_a.\nOne can define half-closed and closed intervals [a,b), (a,b], and [a,b] by adjoining a as a [least element](https://bliptext.com/articles/least-element) and/or b as a [greatest element](https://bliptext.com/articles/greatest-element).\nAs a special case, the open interval (a,a) is defined as the cut.\nMore generally, a proper subset S of K is called [convex](https://bliptext.com/articles/convex-set) if it contains an interval between every pair of points: for a\\ne b\\in S, either (a,b) or (b,a) must also be in S. A [convex](https://bliptext.com/articles/convex-set) set is linearly ordered by the cut <_x for any x not in the set; this ordering is independent of the choice of x.\n\n## Automorphisms\nAs a circle has a [clockwise](https://bliptext.com/articles/clockwise) order and a counter[clockwise](https://bliptext.com/articles/clockwise) order, any set with a cyclic order has two **senses**.\nA [bijection](https://bliptext.com/articles/bijection) of the set that preserves the order is called an **ordered correspondence**.\nIf the sense is maintained as before, it is a **direct correspondence**, otherwise it is called an **opposite correspondence**.\nCoxeter uses a [separation relation](https://bliptext.com/articles/separation-relation) to describe cyclic order, and this relation is strong enough to distinguish the two senses of cyclic order.\nThe [automorphisms](https://bliptext.com/articles/automorphism) of a cyclically ordered set may be identified with C2, the two-element group, of direct and opposite correspondences.\n\n# Monotone functions\nThe \"cyclic order = arranging in a circle\" idea works because any [subset](https://bliptext.com/articles/subset) of a cycle is itself a cycle.\nIn 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](https://bliptext.com/articles/function-mathematics) between sets.\nA function between two cyclically ordered sets,\nf : X → Y\n, is called a [monotonic function](https://bliptext.com/articles/monotonic-function) or a [homomorphism](https://bliptext.com/articles/morphism) if it pulls back the ordering on Y: whenever\n[f(a), f(b), f(c)]\n, one has\n[a, b, c]\n. Equivalently, f is monotone if whenever\n[a, b, c]\nand\nf(a), f(b)\n, and\nf(c)\nare all distinct, then\n[f(a), f(b), f(c)]\n. A typical example of a monotone function is the following function on the cycle with 6 elements:\nf(0) = f(1) = 4,\nf(2) = f(3) = 0,\nf(4) = f(5) = 1.\nA function is called an [embedding](https://bliptext.com/articles/embedding) if it is both monotone and [injective](https://bliptext.com/articles/injective-function).\nEquivalently, an embedding is a function that pushes forward the ordering on X: whenever\n[a, b, c]\n, one has\n[f(a), f(b), f(c)]\n. 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](https://bliptext.com/articles/inclusion-map)\ni : X → Y\nis an embedding.\nGenerally, 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.\n\n## Functions on finite sets\nA cyclic order on a finite set X can be determined by an injection into the unit circle,\nX → S1\n. There are many possible functions that induce the same cyclic order—in fact, infinitely many.\nIn order to quantify this redundancy, it takes a more complex combinatorial object than a simple number.\nExamining the [configuration space](https://bliptext.com/articles/configuration-space-mathematics) of all such maps leads to the definition of an (n − 1)\n-dimensional [polytope](https://bliptext.com/articles/polytope) known as a [cyclohedron](https://bliptext.com/articles/cyclohedron).\nCyclohedra were first applied to the study of [knot invariants](https://bliptext.com/articles/knot-invariant); they have more recently been applied to the experimental detection of [periodically expressed](https://bliptext.com/articles/periodically-expressed-gene) [genes](https://bliptext.com/articles/gene) in the study of [biological clocks](https://bliptext.com/articles/circadian-rhythm).\nThe category of homomorphisms of the standard finite cycles is called the [cyclic category](https://bliptext.com/articles/cyclic-category); it may be used to construct [Alain Connes](https://bliptext.com/articles/alain-connes)' [cyclic homology](https://bliptext.com/articles/cyclic-homology).\nOne may define a degree of a function between cycles, analogous to the [degree of a continuous mapping](https://bliptext.com/articles/degree-of-a-continuous-mapping).\nFor example, the natural map from the [circle of fifths](https://bliptext.com/articles/circle-of-fifths) to the [chromatic circle](https://bliptext.com/articles/chromatic-circle) is a map of degree 7.\nOne may also define a [rotation number](https://bliptext.com/articles/rotation-number).\n\n## Completion\nZn\nis a jump.\nA cycle with no jumps is called [dense](https://bliptext.com/articles/dense-order).\nQ\nhave a gap at every irrational number.\nThey also have a gap at infinity, i.e. the usual ordering.\nA cycle with no gaps is called [complete](https://bliptext.com/articles/completeness-order-theory).\nS1\nis a principal cut.\nA cycle where every cut is principal, being both dense and complete, is called continuous.\n[[Image:CyclicOrderingOfCuts.svg|thumb|\n[<1, <2, <3]\nand\n[x, y, z]\n]]\nThe set of all cuts is cyclically ordered by the following relation:\n[<1, <2, <3]\nif and only if there exist\nx, y, z\nsuch that:\nx <1 y <1 z\n,\nx <1\ny <2 z <2 x\n, and\nx <1 y <1\nz <3 x <3 y\n.\nA certain subset of this cycle of cuts is the [Dedekind completion](https://bliptext.com/articles/dedekind-completion) of the original cycle.\n\n# Further constructions\n\n## Unrolling and covers\nStarting from a cyclically ordered set K, one may form a linear order by unrolling it along an infinite line.\nThis captures the intuitive notion of keeping track of how many times one goes around the circle.\nFormally, one defines a linear order on the [Cartesian product](https://bliptext.com/articles/cartesian-product)\nZ × K\n, where\nZ\nis the set of [integers](https://bliptext.com/articles/integer), by fixing an element a and requiring that for all i:\n[a, x, y]\n, then\nai < xi < yi < ai+1\n. For example, the months 1 2025, 5 2025, 9 2025, and 1 occur in that order.\nThis ordering of\nZ × K\nis called the [universal cover](https://bliptext.com/articles/universal-cover) of K. Its [order type](https://bliptext.com/articles/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](https://bliptext.com/articles/pitch-class) is compatible with the A-to-G alphabetical order, C is chosen to be the first note in each octave, so in [note-octave](https://bliptext.com/articles/note-octave) notation, B3 is followed by C4.\nThe inverse construction starts with a linearly ordered set and coils it up into a cyclically ordered set.\nGiven a linearly ordered set L and an order-preserving [bijection](https://bliptext.com/articles/bijection)\nT : L → L\nwith unbounded orbits, the [orbit space](https://bliptext.com/articles/orbit-space)\nL / T\nis cyclically ordered by the requirement:\na < b < c < T(a)\n, then\na], [b], [c\n. In particular, one can recover K by defining\nT(xi) = xi+1\non\nZ × K\n.\nThere are also n-fold coverings for finite n; in this case, one cyclically ordered set covers another cyclically ordered set.\nFor example, the 24-hour clock is a double cover of the 12-hour clock.\nIn geometry, the [pencil](https://bliptext.com/articles/pencil-mathematics) of [rays](https://bliptext.com/articles/ray-mathematics) emanating from a point in the oriented plane is a double cover of the [pencil](https://bliptext.com/articles/pencil-mathematics) of unoriented [lines](https://bliptext.com/articles/line-geometry) passing through the same point.\nThese covering maps can be characterized by lifting them to the universal cover.\n\n## Products and retracts\nGiven a cyclically ordered set\n(K, [ ])\nand a linearly ordered set\n(L, <)\n, the (total) lexicographic product is a cyclic order on the [product set](https://bliptext.com/articles/cartesian-product)\nK × L\n, defined by\n[(a, x), (b, y), (c, z)]\nif one of the following holds:\n[a, b, c]\na = b ≠ c\nand\nx < y\nb = c ≠ a\nand\ny < z\nc = a ≠ b\nand\nz < x\na = b = c\nand\n[x, y, z]\nThe lexicographic product\nK × L\nglobally looks like K and locally looks like L; it can be thought of as K copies of L. This construction is sometimes used to characterize cyclically ordered groups.\nOne can also glue together different linearly ordered sets to form a circularly ordered set.\nFor example, given two linearly ordered sets\nL1\nand\nL2\n, one may form a circle by joining them together at positive and negative infinity.\nA circular order on the disjoint union\nL1 ∪ L2 ∪ {–∞, ∞}\nis defined by\n∞ < L1 < –∞ < L2 < ∞\n, where the induced ordering on\nL1\nis the opposite of its original ordering.\nFor example, the set of all [longitudes](https://bliptext.com/articles/longitude) is circularly ordered by joining all points west and all points east, along with the [prime meridian](https://bliptext.com/articles/prime-meridian) and the [180th meridian](https://bliptext.com/articles/180th-meridian).\nuse this construction while characterizing the spaces of orderings and [real places](https://bliptext.com/articles/real-place) of double [formal Laurent series](https://bliptext.com/articles/formal-laurent-series) over a [real closed field](https://bliptext.com/articles/real-closed-field).\n# [400px | upload.wikimedia.org/wikipedia/commons/d/d5/CyclicLinearProductLabels.svg]\n\n# Topology\nThe open intervals form a [base](https://bliptext.com/articles/base-[topology](https://bliptext.com/articles/topological-space)) for a natural [topology](https://bliptext.com/articles/topological-space), the cyclic order [topology](https://bliptext.com/articles/topological-space).\nThe [open sets](https://bliptext.com/articles/open-set) in this topology are exactly those sets which are open in every compatible linear order.\nTo illustrate the difference, in the set [0, 1), the subset [0, 1/2) is a neighborhood of 0 in the linear order but not in the cyclic order.\nInteresting examples of cyclically ordered spaces include the conformal boundary of a [simply connected](https://bliptext.com/articles/simply-connected) [Lorentz surface](https://bliptext.com/articles/lorentz-surface) and the [leaf space](https://bliptext.com/articles/leaf-space) of a lifted [essential lamination](https://bliptext.com/articles/essential-lamination) of certain 3-manifolds.\n[Discrete dynamical systems](https://bliptext.com/articles/discrete-dynamical-system) on cyclically ordered spaces have also been studied.\nThe interval topology forgets the original orientation of the cyclic order.\nThis orientation can be restored by enriching the intervals with their induced linear orders; then one has a set covered with an atlas of linear orders that are compatible where they overlap.\nIn other words, a cyclically ordered set can be thought of as a locally linearly ordered space: an object like a [manifold](https://bliptext.com/articles/manifold), but with order relations instead of coordinate charts.\nThis viewpoint makes it easier to be precise about such concepts as covering maps.\nThe generalization to a locally partially ordered space is studied in ; see also [Directed topology](https://bliptext.com/articles/directed-topology).\n\n# Related structures\n\n## Groups\nA [cyclically ordered group](https://bliptext.com/articles/cyclically-ordered-group) is a set with both a [group structure](https://bliptext.com/articles/group-mathematics) and a cyclic order, such that left and right multiplication both preserve the cyclic order.\nCyclically ordered groups were first studied in depth by [Ladislav Rieger](https://bliptext.com/articles/ladislav-rieger) in 1947.\nThey are a generalization of [cyclic groups](https://bliptext.com/articles/cyclic-group): the [infinite cyclic group](https://bliptext.com/articles/infinite-cyclic-group)\nZ\nand the [finite cyclic groups](https://bliptext.com/articles/finite-cyclic-group)\nZ/n\n. Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of [linearly ordered groups](https://bliptext.com/articles/linearly-ordered-group): the [rational numbers](https://bliptext.com/articles/rational-number)\nQ\n, the real numbers\nR\n, and so on.\nSome of the most important cyclically ordered groups fall into neither previous category: the [circle group](https://bliptext.com/articles/circle-group)\nT\nand its subgroups, such as the [subgroup of rational points](https://bliptext.com/articles/group-of-rational-points-on-the-unit-circle).\nEvery cyclically ordered group can be expressed as a quotient\nL / Z\n, where L is a linearly ordered group and Z is a cyclic cofinal subgroup of L. Every cyclically ordered group can also be expressed as a subgroup of a product\nT × L\n, where L is a linearly ordered group.\nIf a cyclically ordered group is Archimedean or compact, it can be embedded in\nT\nitself.\n\n## Modified axioms\nA [partial cyclic order](https://bliptext.com/articles/partial-cyclic-order) is a ternary relation that generalizes a (total) cyclic order in the same way that a [partial order](https://bliptext.com/articles/partial-order) generalizes a [total order](https://bliptext.com/articles/total-order).\nIt is cyclic, asymmetric, and transitive, but it need not be total.\nAn [order variety](https://bliptext.com/articles/order-variety) is a partial cyclic order that satisfies an additional spreading axiom.\nReplacing the asymmetry axiom with a complementary version results in the definition of a co-cyclic order.\nAppropriately total co-cyclic orders are related to cyclic orders in the same way that\n≤\nis related to\n<\n.\nA cyclic order obeys a relatively strong 4-point transitivity axiom.\nOne structure that weakens this axiom is a [CC system](https://bliptext.com/articles/cc-system): a ternary relation that is cyclic, asymmetric, and total, but generally not transitive.\nInstead, a CC system must obey a 5-point transitivity axiom and a new interiority axiom, which constrains the 4-point configurations that violate cyclic transitivity.\nA cyclic order is required to be symmetric under cyclic permutation,\n[a, b, c] ⇒ [b, c, a]\n, and asymmetric under reversal:\n[a, b, c] ⇒ ¬[c, b, a]\n. A ternary relation that is asymmetric under cyclic permutation and symmetric under reversal, together with appropriate versions of the transitivity and totality axioms, is called a [betweenness relation](https://bliptext.com/articles/betweenness-relation).\nA [separation relation](https://bliptext.com/articles/separation-relation) is a [quaternary relation](https://bliptext.com/articles/quaternary-relation) that can be thought of as a cyclic order without an orientation.\nThe relationship between a circular order and a [separation relation](https://bliptext.com/articles/separation-relation) is analogous to the relationship between a linear order and a betweenness relation.\n\n# Symmetries and model theory\nprovide a model-theoretic description of the covering maps of cycles.\nstudies groups of automorphisms of cycles with various [transitivity](https://bliptext.com/articles/transitive-action) properties.\ncharacterize cycles whose full automorphism groups act [freely and transitively](https://bliptext.com/articles/free-transitive-action).\ncharacterize [countable](https://bliptext.com/articles/countable-set) [colored](https://bliptext.com/articles/colored-set) cycles whose automorphism groups act transitively.\nstudies the automorphism group of the unique (up to isomorphism) countable dense cycle.\nstudy [minimality](https://bliptext.com/articles/minimality) conditions on circularly ordered [structures](https://bliptext.com/articles/structure-mathematical-logic), i.e. models of first-order languages that include a cyclic order relation.\nThese conditions are analogues of [o-minimality](https://bliptext.com/articles/o-minimal-theory) and weak [o-minimality](https://bliptext.com/articles/o-minimal-theory) for the case of linearly ordered structures.\ncontinues with some characterizations of [ω-categorical](https://bliptext.com/articles/omega-categorical-theory) structures.\n\n# Cognition\n[Hans Freudenthal](https://bliptext.com/articles/hans-freudenthal) has emphasized the role of cyclic orders in cognitive development, as a contrast to [Jean Piaget](https://bliptext.com/articles/jean-piaget) who addresses only linear orders.\nSome experiments have been performed to investigate the mental representations of cyclically ordered sets, such as the months of the year.\n\n# Notes on usage\nThe relation may be called a cyclic order, a circular order , a cyclic ordering , or a circular ordering.\nSome authors call such an ordering a total cyclic order, a complete cyclic order , a linear cyclic order , or an l-cyclic order or ℓ-cyclic order , to distinguish from the broader class of [partial cyclic orders](https://bliptext.com/articles/partial-cyclic-order), which they call simply cyclic orders.\nFinally, some authors may take cyclic order to mean an unoriented quaternary [separation relation](https://bliptext.com/articles/separation-relation).\nA set with a cyclic order may be called a cycle or a circle.\nThe above variations also appear in adjective form: cyclically ordered set (cyklicky uspořádané množiny, ), circularly ordered set, total cyclically ordered set, complete cyclically ordered set, linearly cyclically ordered set, l-cyclically ordered set, ℓ-cyclically ordered set.\nAll authors agree that a cycle is totally ordered.\nThere are a few different symbols in use for a cyclic relation.\nuses concatenation:\nABC\n. and use ordered triples and the set membership symbol:\n(a, b, c) ∈ C\n. uses concatenation and set membership:\nabc ∈ C\n, understanding\nabc\nas a cyclically ordered triple.\nThe literature on groups, such as and, tend to use square brackets:\n[a, b, c]\n. use round parentheses:\n(a, b, c)\n, reserving square brackets for a betweenness relation.\nuse a function-style notation:\nR(a, b, c)\n. [Rieger (1947)](https://bliptext.com/articles/), cited after ) uses a \"less-than\" symbol as a delimiter:\n< x, y, z <\n. Some authors use infix notation:\na < b < c\n, with the understanding that this does not carry the usual meaning of\na < b\nand\nb < c\nfor some binary relation <.\nemphasizes the cyclic nature by repeating an element:\np ↪ r ↪ q ↪ p\n.\ncalls an embedding an \"isomorphic embedding\".\nIn this case, write that K is L \"rolled up\".\nThe map T is called archimedean by, coterminal by , and a translation by.\ncalls\nZ × K\nthe \"universal cover\" of K. write that K is\nZ × K\n\"coiled\".\ncall\nZ × K\nthe \"∞-times covering\" of K. Often this construction is written as the anti-lexicographic order on\nK × Z\n.\n\n# Citations","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:36:42.628Z","updated_at":"2025-01-25T19:36:42.628Z","search_vector":"'cyclic':1 'order':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **cyclic order** is a way to arrange a set of objects in a [circle](https://bliptext.com/articles/circle).\nUnlike most structures in [order theory](https://bliptext.com/articles/order-theory), a cyclic order is not modeled as a [binary relation](https://bliptext.com/articles/binary-relation), such as \"\na < b\n\". One does not say that east is \"more clockwise\" than west. Instead, a cyclic order is defined as a [ternary relation](https://bliptext.com/articles/ternary-relation)\n[a, b, c]\n, meaning \"after a, one reaches b before c\".\nFor example, [June, October, February], but not [June, February, October], cf.\npicture.\nA ternary relation is called a cyclic order if it is [cyclic, asymmetric, transitive, and connected](https://bliptext.com/articles/).\nDropping the \"connected\" requirement results in a [partial cyclic order](https://bliptext.com/articles/partial-cyclic-order).\nA [set](https://bliptext.com/articles/set-mathematics) with a cyclic order is called a **cyclically ordered [set](https://bliptext.com/articles/set-mathematics)** or simply a **cycle**.\nSome familiar cycles are discrete, having only a [finite number](https://bliptext.com/articles/finite-set) of [elements](https://bliptext.com/articles/element-mathematics): there are seven [days of the week](https://bliptext.com/articles/days-of-the-week), four [cardinal directions](https://bliptext.com/articles/cardinal-direction), twelve notes in the [chromatic scale](https://bliptext.com/articles/chromatic-scale), and three plays in [rock-paper-scissors](https://bliptext.com/articles/rock-paper-scissors).\nIn a finite cycle, each element has a \"next element\" and a \"previous element\".\nThere are also cyclic orders with infinitely many elements, such as the oriented [unit circle](https://bliptext.com/articles/unit-circle) in the plane.\nCyclic orders are closely related to the more familiar [[line](https://bliptext.com/articles/line-geometry)ar orders](https://bliptext.com/articles/[line](https://bliptext.com/articles/line-geometry)ar-order), which arrange objects in a [line](https://bliptext.com/articles/line-geometry).\nAny linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line.\nThese 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.\nCycles have more symmetries than linear orders, and they often naturally occur as residues of linear structures, as in the [finite cyclic groups](https://bliptext.com/articles/finite-cyclic-group) or the [real projective line](https://bliptext.com/articles/real-projective-line).\n# [The months are a cyclic order. | upload.wikimedia.org/wikipedia/commons/4/40/Months///circle///svg.svg]\n\n# Finite cycles\nA cyclic order on a set X with n elements is like an arrangement of X on a clock face, for an n-hour clock.\nEach 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\nx(1), x(2), ..., x(n)\n.\nThere are a few equivalent ways to state this definition.\nA cyclic order on X is the same as a [permutation](https://bliptext.com/articles/permutation) that makes all of X into a single [cycle](https://bliptext.com/articles/cycles-and-fixed-points), which is a special type of [permutation](https://bliptext.com/articles/permutation) - a circular [permutation](https://bliptext.com/articles/permutation).\nAlternatively, a cycle with n elements is also a\nZn\n-[torsor](https://bliptext.com/articles/torsor): a set with a free transitive [action](https://bliptext.com/articles/group-action-mathematics) by a [finite cyclic group](https://bliptext.com/articles/finite-cyclic-group).\nAnother formulation is to make X into the standard [directed cycle graph](https://bliptext.com/articles/directed-cycle-graph) on n vertices, by some matching of elements to vertices.\nIt can be instinctive to use cyclic orders for [symmetric functions](https://bliptext.com/articles/symmetric-function), for example as in\nxy + yz + zx\nwhere writing the final [monomial](https://bliptext.com/articles/monomial) as\nxz\nwould distract from the pattern.\nA substantial use of cyclic orders is in the determination of the [conjugacy classes](https://bliptext.com/articles/conjugacy-class) of [free groups](https://bliptext.com/articles/free-group).\nTwo 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\ny−1\nwith y in Y, and then those products are put in cyclic order, the cyclic orders are equivalent under the [rewriting](https://bliptext.com/articles/rewriting) rules that allow one to remove or add adjacent y and\ny−1\n.\nA cyclic order on a set X can be determined by a linear order on X, but not in a unique way.\nChoosing a linear order is equivalent to choosing a first element, so there are exactly n linear orders that induce a given cyclic order.\nSince there are\nn!\npossible linear orders (as in [permutations](https://bliptext.com/articles/permutation)), there are\n(n − 1)!\npossible cyclic orders (as in [circular permutations](https://bliptext.com/articles/permutation)).\n# [A 5-element cycle | upload.wikimedia.org/wikipedia/commons/f/f9/Orientovan%C3%A1///kru%C5%BEnice.svg]\n\n# Definitions\nAn [infinite set](https://bliptext.com/articles/infinite-set) can also be ordered cyclically.\nImportant examples of infinite cycles include the [unit circle](https://bliptext.com/articles/unit-circle),\nS1\n, and the [rational numbers](https://bliptext.com/articles/rational-number),\nQ\n. The basic idea is the same: we arrange elements of the set around a circle.\nHowever, in the infinite case we cannot rely upon an immediate successor relation, because points may not have successors.\nFor example, given a point on the unit circle, there is no \"next point\".\nNor can we rely upon a binary relation to determine which of two points comes \"first\".\nTraveling clockwise on a circle, neither east or west comes first, but each follows the other.\nInstead, we use a ternary relation denoting that elements a, b, c occur after each other (not necessarily immediately) as we go around the circle.\nFor example, in clockwise order, [east, south, west].\nBy [currying](https://bliptext.com/articles/currying) the arguments of the ternary relation\n[a, b, c]\n, 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.\n\n## The ternary relation\nThe general definition is as follows: a cyclic order on a set X is a relation\nC ⊂ X3\n, written\n[a, b, c]\n, that satisfies the following axioms:\n[a, b, c]\nthen\n[b, c, a]\n[a, b, c]\nthen not\n[c, b, a]\n[a, b, c]\nand\n[a, c, d]\nthen\n[a, b, d]\n[a, b, c]\nor\n[c, b, a]\nThe axioms are named by analogy with the [asymmetry](https://bliptext.com/articles/asymmetric-relation), [transitivity](https://bliptext.com/articles/transitive-relation), and [connectedness](https://bliptext.com/articles/connected-relation) axioms for a binary relation, which together define a [strict linear order](https://bliptext.com/articles/strict-linear-order).\nconsidered other possible lists of axioms, including one list that was meant to emphasize the similarity between a cyclic order and a [betweenness relation](https://bliptext.com/articles/betweenness-relation).\nA ternary relation that satisfies the first three axioms, but not necessarily the axiom of totality, is a [partial cyclic order](https://bliptext.com/articles/partial-cyclic-order).\n\n## Rolling and cuts\nGiven a linear order\n<\non a set X, the cyclic order on X induced by\n<\nis defined as follows:\n[a, b, c]\nif and only if\na < b < c\nor\nb < c < a\nor\nc < a < b\nTwo 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](https://bliptext.com/articles/cut-cards).\nOne may define a cyclic order relation as a ternary relation that is induced by a strict linear order as above.\nCutting a single point out of a cyclic order leaves a linear order behind.\nMore precisely, given a cyclically ordered set, each element a\\in K defines a natural linear order <_a on the remainder of the set, , by the following rule:\nMoreover, <_a can be extended by adjoining a as a least element; the resulting linear order on K is called the principal cut with least element a. Likewise, adjoining a as a greatest element results in a cut <^a.\n\n## Intervals\nGiven two elements a\\ne b\\in K, the [open interval](https://bliptext.com/articles/open-interval) from a to b, written (a,b), is the set of all x\\in K such that [a,x,b].\nThe system of open intervals completely defines the cyclic order and can be used as an alternate definition of a cyclic order relation.\nAn interval (a,b) has a natural linear order given by <_a.\nOne can define half-closed and closed intervals [a,b), (a,b], and [a,b] by adjoining a as a [least element](https://bliptext.com/articles/least-element) and/or b as a [greatest element](https://bliptext.com/articles/greatest-element).\nAs a special case, the open interval (a,a) is defined as the cut.\nMore generally, a proper subset S of K is called [convex](https://bliptext.com/articles/convex-set) if it contains an interval between every pair of points: for a\\ne b\\in S, either (a,b) or (b,a) must also be in S. A [convex](https://bliptext.com/articles/convex-set) set is linearly ordered by the cut <_x for any x not in the set; this ordering is independent of the choice of x.\n\n## Automorphisms\nAs a circle has a [clockwise](https://bliptext.com/articles/clockwise) order and a counter[clockwise](https://bliptext.com/articles/clockwise) order, any set with a cyclic order has two **senses**.\nA [bijection](https://bliptext.com/articles/bijection) of the set that preserves the order is called an **ordered correspondence**.\nIf the sense is maintained as before, it is a **direct correspondence**, otherwise it is called an **opposite correspondence**.\nCoxeter uses a [separation relation](https://bliptext.com/articles/separation-relation) to describe cyclic order, and this relation is strong enough to distinguish the two senses of cyclic order.\nThe [automorphisms](https://bliptext.com/articles/automorphism) of a cyclically ordered set may be identified with C2, the two-element group, of direct and opposite correspondences.\n\n# Monotone functions\nThe \"cyclic order = arranging in a circle\" idea works because any [subset](https://bliptext.com/articles/subset) of a cycle is itself a cycle.\nIn 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](https://bliptext.com/articles/function-mathematics) between sets.\nA function between two cyclically ordered sets,\nf : X → Y\n, is called a [monotonic function](https://bliptext.com/articles/monotonic-function) or a [homomorphism](https://bliptext.com/articles/morphism) if it pulls back the ordering on Y: whenever\n[f(a), f(b), f(c)]\n, one has\n[a, b, c]\n. Equivalently, f is monotone if whenever\n[a, b, c]\nand\nf(a), f(b)\n, and\nf(c)\nare all distinct, then\n[f(a), f(b), f(c)]\n. A typical example of a monotone function is the following function on the cycle with 6 elements:\nf(0) = f(1) = 4,\nf(2) = f(3) = 0,\nf(4) = f(5) = 1.\nA function is called an [embedding](https://bliptext.com/articles/embedding) if it is both monotone and [injective](https://bliptext.com/articles/injective-function).\nEquivalently, an embedding is a function that pushes forward the ordering on X: whenever\n[a, b, c]\n, one has\n[f(a), f(b), f(c)]\n. 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](https://bliptext.com/articles/inclusion-map)\ni : X → Y\nis an embedding.\nGenerally, 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.\n\n## Functions on finite sets\nA cyclic order on a finite set X can be determined by an injection into the unit circle,\nX → S1\n. There are many possible functions that induce the same cyclic order—in fact, infinitely many.\nIn order to quantify this redundancy, it takes a more complex combinatorial object than a simple number.\nExamining the [configuration space](https://bliptext.com/articles/configuration-space-mathematics) of all such maps leads to the definition of an (n − 1)\n-dimensional [polytope](https://bliptext.com/articles/polytope) known as a [cyclohedron](https://bliptext.com/articles/cyclohedron).\nCyclohedra were first applied to the study of [knot invariants](https://bliptext.com/articles/knot-invariant); they have more recently been applied to the experimental detection of [periodically expressed](https://bliptext.com/articles/periodically-expressed-gene) [genes](https://bliptext.com/articles/gene) in the study of [biological clocks](https://bliptext.com/articles/circadian-rhythm).\nThe category of homomorphisms of the standard finite cycles is called the [cyclic category](https://bliptext.com/articles/cyclic-category); it may be used to construct [Alain Connes](https://bliptext.com/articles/alain-connes)' [cyclic homology](https://bliptext.com/articles/cyclic-homology).\nOne may define a degree of a function between cycles, analogous to the [degree of a continuous mapping](https://bliptext.com/articles/degree-of-a-continuous-mapping).\nFor example, the natural map from the [circle of fifths](https://bliptext.com/articles/circle-of-fifths) to the [chromatic circle](https://bliptext.com/articles/chromatic-circle) is a map of degree 7.\nOne may also define a [rotation number](https://bliptext.com/articles/rotation-number).\n\n## Completion\nZn\nis a jump.\nA cycle with no jumps is called [dense](https://bliptext.com/articles/dense-order).\nQ\nhave a gap at every irrational number.\nThey also have a gap at infinity, i.e. the usual ordering.\nA cycle with no gaps is called [complete](https://bliptext.com/articles/completeness-order-theory).\nS1\nis a principal cut.\nA cycle where every cut is principal, being both dense and complete, is called continuous.\n[[Image:CyclicOrderingOfCuts.svg|thumb|\n[<1, <2, <3]\nand\n[x, y, z]\n]]\nThe set of all cuts is cyclically ordered by the following relation:\n[<1, <2, <3]\nif and only if there exist\nx, y, z\nsuch that:\nx <1 y <1 z\n,\nx <1\ny <2 z <2 x\n, and\nx <1 y <1\nz <3 x <3 y\n.\nA certain subset of this cycle of cuts is the [Dedekind completion](https://bliptext.com/articles/dedekind-completion) of the original cycle.\n\n# Further constructions\n\n## Unrolling and covers\nStarting from a cyclically ordered set K, one may form a linear order by unrolling it along an infinite line.\nThis captures the intuitive notion of keeping track of how many times one goes around the circle.\nFormally, one defines a linear order on the [Cartesian product](https://bliptext.com/articles/cartesian-product)\nZ × K\n, where\nZ\nis the set of [integers](https://bliptext.com/articles/integer), by fixing an element a and requiring that for all i:\n[a, x, y]\n, then\nai < xi < yi < ai+1\n. For example, the months 1 2025, 5 2025, 9 2025, and 1 occur in that order.\nThis ordering of\nZ × K\nis called the [universal cover](https://bliptext.com/articles/universal-cover) of K. Its [order type](https://bliptext.com/articles/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](https://bliptext.com/articles/pitch-class) is compatible with the A-to-G alphabetical order, C is chosen to be the first note in each octave, so in [note-octave](https://bliptext.com/articles/note-octave) notation, B3 is followed by C4.\nThe inverse construction starts with a linearly ordered set and coils it up into a cyclically ordered set.\nGiven a linearly ordered set L and an order-preserving [bijection](https://bliptext.com/articles/bijection)\nT : L → L\nwith unbounded orbits, the [orbit space](https://bliptext.com/articles/orbit-space)\nL / T\nis cyclically ordered by the requirement:\na < b < c < T(a)\n, then\na], [b], [c\n. In particular, one can recover K by defining\nT(xi) = xi+1\non\nZ × K\n.\nThere are also n-fold coverings for finite n; in this case, one cyclically ordered set covers another cyclically ordered set.\nFor example, the 24-hour clock is a double cover of the 12-hour clock.\nIn geometry, the [pencil](https://bliptext.com/articles/pencil-mathematics) of [rays](https://bliptext.com/articles/ray-mathematics) emanating from a point in the oriented plane is a double cover of the [pencil](https://bliptext.com/articles/pencil-mathematics) of unoriented [lines](https://bliptext.com/articles/line-geometry) passing through the same point.\nThese covering maps can be characterized by lifting them to the universal cover.\n\n## Products and retracts\nGiven a cyclically ordered set\n(K, [ ])\nand a linearly ordered set\n(L, <)\n, the (total) lexicographic product is a cyclic order on the [product set](https://bliptext.com/articles/cartesian-product)\nK × L\n, defined by\n[(a, x), (b, y), (c, z)]\nif one of the following holds:\n[a, b, c]\na = b ≠ c\nand\nx < y\nb = c ≠ a\nand\ny < z\nc = a ≠ b\nand\nz < x\na = b = c\nand\n[x, y, z]\nThe lexicographic product\nK × L\nglobally looks like K and locally looks like L; it can be thought of as K copies of L. This construction is sometimes used to characterize cyclically ordered groups.\nOne can also glue together different linearly ordered sets to form a circularly ordered set.\nFor example, given two linearly ordered sets\nL1\nand\nL2\n, one may form a circle by joining them together at positive and negative infinity.\nA circular order on the disjoint union\nL1 ∪ L2 ∪ {–∞, ∞}\nis defined by\n∞ < L1 < –∞ < L2 < ∞\n, where the induced ordering on\nL1\nis the opposite of its original ordering.\nFor example, the set of all [longitudes](https://bliptext.com/articles/longitude) is circularly ordered by joining all points west and all points east, along with the [prime meridian](https://bliptext.com/articles/prime-meridian) and the [180th meridian](https://bliptext.com/articles/180th-meridian).\nuse this construction while characterizing the spaces of orderings and [real places](https://bliptext.com/articles/real-place) of double [formal Laurent series](https://bliptext.com/articles/formal-laurent-series) over a [real closed field](https://bliptext.com/articles/real-closed-field).\n# [400px | upload.wikimedia.org/wikipedia/commons/d/d5/CyclicLinearProductLabels.svg]\n\n# Topology\nThe open intervals form a [base](https://bliptext.com/articles/base-[topology](https://bliptext.com/articles/topological-space)) for a natural [topology](https://bliptext.com/articles/topological-space), the cyclic order [topology](https://bliptext.com/articles/topological-space).\nThe [open sets](https://bliptext.com/articles/open-set) in this topology are exactly those sets which are open in every compatible linear order.\nTo illustrate the difference, in the set [0, 1), the subset [0, 1/2) is a neighborhood of 0 in the linear order but not in the cyclic order.\nInteresting examples of cyclically ordered spaces include the conformal boundary of a [simply connected](https://bliptext.com/articles/simply-connected) [Lorentz surface](https://bliptext.com/articles/lorentz-surface) and the [leaf space](https://bliptext.com/articles/leaf-space) of a lifted [essential lamination](https://bliptext.com/articles/essential-lamination) of certain 3-manifolds.\n[Discrete dynamical systems](https://bliptext.com/articles/discrete-dynamical-system) on cyclically ordered spaces have also been studied.\nThe interval topology forgets the original orientation of the cyclic order.\nThis orientation can be restored by enriching the intervals with their induced linear orders; then one has a set covered with an atlas of linear orders that are compatible where they overlap.\nIn other words, a cyclically ordered set can be thought of as a locally linearly ordered space: an object like a [manifold](https://bliptext.com/articles/manifold), but with order relations instead of coordinate charts.\nThis viewpoint makes it easier to be precise about such concepts as covering maps.\nThe generalization to a locally partially ordered space is studied in ; see also [Directed topology](https://bliptext.com/articles/directed-topology).\n\n# Related structures\n\n## Groups\nA [cyclically ordered group](https://bliptext.com/articles/cyclically-ordered-group) is a set with both a [group structure](https://bliptext.com/articles/group-mathematics) and a cyclic order, such that left and right multiplication both preserve the cyclic order.\nCyclically ordered groups were first studied in depth by [Ladislav Rieger](https://bliptext.com/articles/ladislav-rieger) in 1947.\nThey are a generalization of [cyclic groups](https://bliptext.com/articles/cyclic-group): the [infinite cyclic group](https://bliptext.com/articles/infinite-cyclic-group)\nZ\nand the [finite cyclic groups](https://bliptext.com/articles/finite-cyclic-group)\nZ/n\n. Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of [linearly ordered groups](https://bliptext.com/articles/linearly-ordered-group): the [rational numbers](https://bliptext.com/articles/rational-number)\nQ\n, the real numbers\nR\n, and so on.\nSome of the most important cyclically ordered groups fall into neither previous category: the [circle group](https://bliptext.com/articles/circle-group)\nT\nand its subgroups, such as the [subgroup of rational points](https://bliptext.com/articles/group-of-rational-points-on-the-unit-circle).\nEvery cyclically ordered group can be expressed as a quotient\nL / Z\n, where L is a linearly ordered group and Z is a cyclic cofinal subgroup of L. Every cyclically ordered group can also be expressed as a subgroup of a product\nT × L\n, where L is a linearly ordered group.\nIf a cyclically ordered group is Archimedean or compact, it can be embedded in\nT\nitself.\n\n## Modified axioms\nA [partial cyclic order](https://bliptext.com/articles/partial-cyclic-order) is a ternary relation that generalizes a (total) cyclic order in the same way that a [partial order](https://bliptext.com/articles/partial-order) generalizes a [total order](https://bliptext.com/articles/total-order).\nIt is cyclic, asymmetric, and transitive, but it need not be total.\nAn [order variety](https://bliptext.com/articles/order-variety) is a partial cyclic order that satisfies an additional spreading axiom.\nReplacing the asymmetry axiom with a complementary version results in the definition of a co-cyclic order.\nAppropriately total co-cyclic orders are related to cyclic orders in the same way that\n≤\nis related to\n<\n.\nA cyclic order obeys a relatively strong 4-point transitivity axiom.\nOne structure that weakens this axiom is a [CC system](https://bliptext.com/articles/cc-system): a ternary relation that is cyclic, asymmetric, and total, but generally not transitive.\nInstead, a CC system must obey a 5-point transitivity axiom and a new interiority axiom, which constrains the 4-point configurations that violate cyclic transitivity.\nA cyclic order is required to be symmetric under cyclic permutation,\n[a, b, c] ⇒ [b, c, a]\n, and asymmetric under reversal:\n[a, b, c] ⇒ ¬[c, b, a]\n. A ternary relation that is asymmetric under cyclic permutation and symmetric under reversal, together with appropriate versions of the transitivity and totality axioms, is called a [betweenness relation](https://bliptext.com/articles/betweenness-relation).\nA [separation relation](https://bliptext.com/articles/separation-relation) is a [quaternary relation](https://bliptext.com/articles/quaternary-relation) that can be thought of as a cyclic order without an orientation.\nThe relationship between a circular order and a [separation relation](https://bliptext.com/articles/separation-relation) is analogous to the relationship between a linear order and a betweenness relation.\n\n# Symmetries and model theory\nprovide a model-theoretic description of the covering maps of cycles.\nstudies groups of automorphisms of cycles with various [transitivity](https://bliptext.com/articles/transitive-action) properties.\ncharacterize cycles whose full automorphism groups act [freely and transitively](https://bliptext.com/articles/free-transitive-action).\ncharacterize [countable](https://bliptext.com/articles/countable-set) [colored](https://bliptext.com/articles/colored-set) cycles whose automorphism groups act transitively.\nstudies the automorphism group of the unique (up to isomorphism) countable dense cycle.\nstudy [minimality](https://bliptext.com/articles/minimality) conditions on circularly ordered [structures](https://bliptext.com/articles/structure-mathematical-logic), i.e. models of first-order languages that include a cyclic order relation.\nThese conditions are analogues of [o-minimality](https://bliptext.com/articles/o-minimal-theory) and weak [o-minimality](https://bliptext.com/articles/o-minimal-theory) for the case of linearly ordered structures.\ncontinues with some characterizations of [ω-categorical](https://bliptext.com/articles/omega-categorical-theory) structures.\n\n# Cognition\n[Hans Freudenthal](https://bliptext.com/articles/hans-freudenthal) has emphasized the role of cyclic orders in cognitive development, as a contrast to [Jean Piaget](https://bliptext.com/articles/jean-piaget) who addresses only linear orders.\nSome experiments have been performed to investigate the mental representations of cyclically ordered sets, such as the months of the year.\n\n# Notes on usage\nThe relation may be called a cyclic order, a circular order , a cyclic ordering , or a circular ordering.\nSome authors call such an ordering a total cyclic order, a complete cyclic order , a linear cyclic order , or an l-cyclic order or ℓ-cyclic order , to distinguish from the broader class of [partial cyclic orders](https://bliptext.com/articles/partial-cyclic-order), which they call simply cyclic orders.\nFinally, some authors may take cyclic order to mean an unoriented quaternary [separation relation](https://bliptext.com/articles/separation-relation).\nA set with a cyclic order may be called a cycle or a circle.\nThe above variations also appear in adjective form: cyclically ordered set (cyklicky uspořádané množiny, ), circularly ordered set, total cyclically ordered set, complete cyclically ordered set, linearly cyclically ordered set, l-cyclically ordered set, ℓ-cyclically ordered set.\nAll authors agree that a cycle is totally ordered.\nThere are a few different symbols in use for a cyclic relation.\nuses concatenation:\nABC\n. and use ordered triples and the set membership symbol:\n(a, b, c) ∈ C\n. uses concatenation and set membership:\nabc ∈ C\n, understanding\nabc\nas a cyclically ordered triple.\nThe literature on groups, such as and, tend to use square brackets:\n[a, b, c]\n. use round parentheses:\n(a, b, c)\n, reserving square brackets for a betweenness relation.\nuse a function-style notation:\nR(a, b, c)\n. [Rieger (1947)](https://bliptext.com/articles/), cited after ) uses a \"less-than\" symbol as a delimiter:\n< x, y, z <\n. Some authors use infix notation:\na < b < c\n, with the understanding that this does not carry the usual meaning of\na < b\nand\nb < c\nfor some binary relation <.\nemphasizes the cyclic nature by repeating an element:\np ↪ r ↪ q ↪ p\n.\ncalls an embedding an \"isomorphic embedding\".\nIn this case, write that K is L \"rolled up\".\nThe map T is called archimedean by, coterminal by , and a translation by.\ncalls\nZ × K\nthe \"universal cover\" of K. write that K is\nZ × K\n\"coiled\".\ncall\nZ × K\nthe \"∞-times covering\" of K. Often this construction is written as the anti-lexicographic order on\nK × Z\n.\n\n# Citations","revisionCount":0},"hasMore":false,"nextCursor":null}

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