mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a cyclic order is a way to arrange a set of objects in a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

. Unlike most structures in

order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

, a cyclic order is not modeled as a

binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...

, such as "". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a

ternary relation In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relati ...

, meaning "after , one reaches before ". For example, une, October, February but not une, 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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 A day is the time period of a full rotation of the Earth with respect to the Sun. On average, this is 24 hours, 1440 minutes, or 86,400 seconds. In everyday life, the word "day" often refers to a solar day, which is the length between two ...

, four

cardinal direction The four cardinal directions, or cardinal points, are the four main compass directions: north, east, south, and west, commonly denoted by their initials N, E, S, and W respectively. Relative to north, the directions east, south, and west are ...

s, twelve notes in the

chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce th ...

, 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 mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

in the plane. Cyclic orders are closely related to the more familiar

linear order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...

s, which arrange objects in a

line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

. 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 In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not int ...

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 In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...

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 function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...

s, for example as in : where writing the final

monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...

as would distract from the pattern. A substantial use of cyclic orders is in the determination of the conjugacy classes of

free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...

s. 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 In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduc ...

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.

Definitions

An infinite set can also be ordered cyclically. Important examples of infinite cycles include the

, , and the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

s, . 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 , , occur after each other (not necessarily immediately) as we go around the circle. For example, in clockwise order, ast, south, west By

currying In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f tha ...

the arguments of the ternary relation , 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 , called ''intervals''.

The ternary relation

The general definition is as follows: a cyclic order on a set is a relation , written , that satisfies the following axioms: #Cyclicity: If then #Asymmetry: If then not #Transitivity: If and then #Connectedness: If , , and are distinct, then either or The axioms are named by analogy with the

asymmetry Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection). Symmetry is an important property of both physical and abstract systems and it may be displayed in pre ...

, transitivity, and connectedness axioms for a binary relation, which together define a strict linear order. 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 , the cyclic order on induced by is defined as follows: : if and only if or or 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 (, each element defines a natural linear order on the remainder of the set, , by the following rule: : if and only if . Moreover, can be extended by adjoining as a least element; the resulting linear order on is called the principal cut with least element . Likewise, adjoining as a greatest element results in a cut .

Intervals

Given two elements , the

open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...

from to , written , is the set of all such that . 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 has a natural linear order given by . One can define half-closed and closed intervals , , and by adjoining as a least element and/or as a greatest element. As a special case, the open interval is defined as the cut . More generally, a proper subset ''S'' of ''K'' is called

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

if it contains an interval between every pair of points: for , either or must also be in ''S''. A convex set is linearly ordered by the cut for any not in the set; this ordering is independent of the choice of .

Automorphisms

As a circle has a

clockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...

order and a counterclockwise order, any set with a cyclic order has two senses. A

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

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 In mathematics, a separation relation is a formal way to arrange a set of objects in an unoriented circle. It is defined as a quaternary relation ' satisfying certain axioms, which is interpreted as asserting that ''a'' and ''c'' separate ''b'' fro ...

to describe cyclic order, and this relation is strong enough to distinguish the two senses of cyclic order. The

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...

s 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 In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

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, , is called a ''

monotonic function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...

'' or a ''

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

'' if it pulls back the ordering on : whenever , one has . Equivalently, is monotone if whenever and , and are all distinct, then . A typical example of a monotone function is the following function on the cycle with 6 elements: : : : A function is called an ''

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is g ...

'' if it is both monotone and

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

. Equivalently, an embedding is a function that pushes forward the ordering on : whenever , one has . As an important example, if is a subset of a cyclically ordered set , and is given its natural ordering, then the

inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota ...

is an embedding. Generally, an injective function from an unordered set to a cycle induces a unique cyclic order on that makes an embedding.

Functions on finite sets

A cyclic order on a finite set can be determined by an injection into the unit circle, . 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 In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

known as a

cyclohedron In geometry, the cyclohedron is a d-dimensional polytope where d can be any non-negative integer. It was first introduced as a combinatorial object by Raoul Bott and Clifford Taubes and, for this reason, it is also sometimes called the Bott–T ...

. Cyclohedra were first applied to the study of

knot invariant In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...

s; they have more recently been applied to the experimental detection of periodically expressed

gene In biology, the word gene (from , ; "...Wilhelm Johannsen coined the word gene to describe the Mendelian units of heredity..." meaning ''generation'' or ''birth'' or ''gender'') can have several different meanings. The Mendelian gene is a b ...

s 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 In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independen ...

. 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 In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval of ...

to the chromatic circle is a map of degree 7. One may also define a rotation number.

Completion

*A cut with both a least element and a greatest element is called a ''jump''. For example, every cut of a finite cycle is a jump. A cycle with no jumps is called '' dense''. *A cut with neither a least element nor a greatest element is called a ''gap''. For example, the rational numbers have a gap at every irrational number. They also have a gap at infinity, i.e. the usual ordering. A cycle with no gaps is called ''

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

''. *A cut with exactly one endpoint is called a ''principal'' or ''Dedekind'' cut. For example, every cut of the circle is a principal cut. A cycle where every cut is principal, being both dense and complete, is called ''continuous''. The set of all cuts is cyclically ordered by the following relation: if and only if there exist such that: :, :, and :. 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 , 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 In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...

, where is the set of

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s, by fixing an element and requiring that for all : :If , then . For example, the months , , , and occur in that order. This ordering of is called the

universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...

of . Its

order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y suc ...

is independent of the choice of , but the notation is not, since the integer coordinate "rolls over" at . 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 and an order-preserving

with unbounded orbits, the

orbit space In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...

is cyclically ordered by the requirement: :If , then . In particular, one can recover by defining on . There are also -fold coverings for finite ; in this case, one cyclically ordered set covers another cyclically ordered set. For example, the is a double cover of the . In geometry, the

pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a tra ...

ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gr ...

s emanating from a point in the oriented plane is a double cover of the pencil of unoriented

s passing through the same point.; These covering maps can be characterized by lifting them to the universal cover.

Products and retracts

Given a cyclically ordered set and a linearly ordered set , the (total) lexicographic product is a cyclic order on the product set , defined by if one of the following holds: * * and * and * and * and The lexicographic product globally looks like and locally looks like ; it can be thought of as copies of . This construction is sometimes used to characterize cyclically ordered groups. One can also glue together different linearly ordered sets to form a circularly ordered set. For example, given two linearly ordered sets and , one may form a circle by joining them together at positive and negative infinity. A circular order on the disjoint union is defined by , where the induced ordering on is the opposite of its original ordering. For example, the set of all

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...

s is circularly ordered by joining together all points west and all points east, along with the

prime meridian A prime meridian is an arbitrary meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in a 360°-system) form a great ...

and the

180th meridian The 180th meridian or antimeridian is the meridian 180° both east and west of the prime meridian in a geographical coordinate system. The longitude at this line can be given as either east or west. On Earth, these two meridians form a ...

. use this construction while characterizing the spaces of orderings and real places of double

formal Laurent series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...

over a

real closed field In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. D ...

Topology

The open intervals form a base for a natural

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, the cyclic

order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...

. The

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

s in this topology are exactly those sets which are open in ''every'' compatible linear order. To 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. Interesting examples of cyclically ordered spaces include the conformal boundary of a simply connected Lorentz surface and the leaf space of a lifted essential lamination of certain 3-manifolds. Discrete dynamical systems on cyclically ordered spaces have also been studied. The interval topology forgets the original orientation of the cyclic order. This 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. In other words, a cyclically ordered set can be thought of as a locally linearly ordered space: an object like a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

, but with order relations instead of coordinate charts. This viewpoint makes it easier to be precise about such concepts as covering maps. The generalization to a locally partially ordered space is studied in ; see also '' Directed topology''.

Related structures

Groups

cyclically ordered group In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger ...

is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order. Cyclically ordered groups were first studied in depth by Ladislav Rieger in 1947. They are a generalization of

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

s: the infinite cyclic group and the finite cyclic groups . Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of

linearly ordered group In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G'', ≤) is a: * le ...

s: the

s , the real numbers , and so on. Some of the most important cyclically ordered groups fall into neither previous category: the

circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...

and its subgroups, such as the subgroup of rational points. Every cyclically ordered group can be expressed as a quotient , where is a linearly ordered group and is a cyclic cofinal subgroup of . Every cyclically ordered group can also be expressed as a subgroup of a product , where is a linearly ordered group. If a cyclically ordered group is Archimedean or compact, it can be embedded in itself.

Modified axioms

A partial cyclic order is a ternary relation that generalizes a (total) cyclic order in the same way that a

partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

generalizes a

total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...

. It is cyclic, asymmetric, and transitive, but it need not be total. An order variety is a partial cyclic order that satisfies an additional ''spreading'' axiom . Replacing the asymmetry axiom with a complementary version results in the definition of a ''co-cyclic order''. Appropriately total co-cyclic orders are related to cyclic orders in the same way that is related to . A cyclic order obeys a relatively strong 4-point transitivity axiom. One structure that weakens this axiom is a CC system: a ternary relation that is cyclic, asymmetric, and total, but generally not transitive. Instead, 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. A cyclic order is required to be symmetric under cyclic permutation, , and asymmetric under reversal: . 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. A

is a quaternary relation that can be thought of as a cyclic order without an orientation. The relationship between a circular order and a

is analogous to the relationship between a linear order and a betweenness relation.

Symmetries and model theory

provide a model-theoretic description of the covering maps of cycles. studies groups of automorphisms of cycles with various transitivity properties. characterize cycles whose full automorphism groups act freely and transitively. characterize

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...

colored ''Colored'' (or ''coloured'') is a racial descriptor historically used in the United States during the Jim Crow Era to refer to an African American. In many places, it may be considered a slur, though it has taken on a special meaning in Sout ...

cycles whose automorphism groups act transitively. studies the automorphism group of the unique (up to isomorphism) countable dense cycle. study minimality conditions on circularly ordered

structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...

s, i.e. models of first-order languages that include a cyclic order relation. These conditions are analogues of

o-minimality In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) which is totally ordered by < is called an o-minimal structure if and only if every weak o-minimality for the case of linearly ordered structures. continues with some characterizations of ω-categorical structures.

Cognition

Hans Freudenthal has emphasized the role of cyclic orders in cognitive development, as a contrast to

Jean Piaget Jean William Fritz Piaget (, , ; 9 August 1896 – 16 September 1980) was a Swiss psychologist known for his work on child development. Piaget's theory of cognitive development and epistemological view are together called "genetic epistemolo ...

who addresses only linear orders. Some experiments have been performed to investigate the mental representations of cyclically ordered sets, such as the months of the year.

Notes on usage

The relation may be called a ''cyclic order'' , a ''circular order'' , a ''cyclic ordering'' , or a ''circular ordering'' . Some 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, which they call simply ''cyclic orders''. Finally, some authors may take ''cyclic order'' to mean an unoriented quaternary

. A set with a cyclic order may be called a ''cycle'' or a ''circle'' . The 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''. All authors agree that a cycle is totally ordered. There are a few different symbols in use for a cyclic relation. uses concatenation: . and use ordered triples and the set membership symbol: . uses concatenation and set membership: , understanding as a cyclically ordered triple. The literature on groups, such as and , tend to use square brackets: . use round parentheses: , reserving square brackets for a betweenness relation. use a function-style notation: . Rieger (1947), cited after ) uses a "less-than" symbol as a delimiter: . Some authors use infix notation: , with the understanding that this does not carry the usual meaning of and for some binary relation < . emphasizes the cyclic nature by repeating an element: . calls an embedding an "isomorphic embedding". In this case, write that is "rolled up". The map ''T'' is called ''archimedean'' by , ''coterminal'' by , and a ''translation'' by . calls the "universal cover" of . write that is "coiled". call the "∞-times covering" of . Often this construction is written as the anti-lexicographic order on .

References

;Citations ;Bibliography * * * * * * * * * * * * * * * * * * * * * * * * * * * **Translation of * * * * * * * * * * * * * * * * **Translation of * **Translation of * * *

External links

* * {{DEFAULTSORT:Cyclic Order Order theory Circles Combinatorics