In mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "a < b". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation[a, b, c], meaning "after a, one reaches b before c". For example, [June, October, February], but not [June, February, October], cf. picture. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and connected. Dropping the "connected" requirement results in a partial cyclic order.
A set with a cyclic order is called a cyclically ordered set or simply a cycle.[nb] Some familiar cycles are discrete, having only a finite number of elements: there are seven days of the week, four cardinal directions, twelve notes in the chromatic scale, and three plays in rock-paper-scissors. In a finite cycle, each element has a "next element" and a "previous element". There are also cyclic orders with infinitely many elements, such as the oriented unit circle in the plane.
Cyclic orders are closely related to the more familiar linear orders, which arrange objects in a line. Any linear order can be bent into a circle, and any cyclic order can be cut at a point, resulting in a line. These operations, along with the related constructions of intervals and covering maps, mean that questions about cyclic orders can often be transformed into questions about linear orders. Cycles have more symmetries than linear orders, and they often naturally occur as residues of linear structures, as in the finite cyclic groups or the real projective line.
Finite cycles
A 5-element cycle
A cyclic order on a set X with n elements is like an arrangement of X on a clock face, for an n-hour clock. Each element x in X has a "next element" and a "previous element", and taking either successors or predecessors cycles exactly once through the elements as x(1), x(2), ..., x(n).
There are a few equivalent ways to state this definition. A cyclic order on X is the same as a permutation that makes all of X into a single cycle, which is a special type of permutation - a circular permutation. Alternatively, a cycle with n elements is also a Zn-torsor: a set with a free transitive action by a finite cyclic group.[1] Another formulation is to make X into the standard directed cycle graph on n vertices, by some matching of elements to vertices.
It can be instinctive to use cyclic orders for symmetric functions, for example as in
xy + yz + zx
where writing the final monomial as xz would distract from the pattern.
A substantial use of cyclic orders is in the determination of the conjugacy classes of free groups. Two elements g and h of the free group F on a set Y are conjugate if and only if, when they are written as products of elements y and y−1 with y in Y, and then those products are put in cyclic order, the cyclic orders are equivalent under the rewriting rules that allow one to remove or add adjacent y and y−1.
A cyclic order on a set X can be determined by a linear order on X, but not in a unique way. Choosing a linear order is equivalent to choosing a first element, so there are exactly n linear orders that induce a given cyclic order. Since there are n! possible linear orders (as in permutations), there are (n− 1)! possible cyclic orders (as in circular permutations).
Definitions
An infinite set can also be ordered cyclically. Important examples of infinite cycles include the unit circle, S1, and the rational numbers, Q. The basic idea is the same: we arrange elements of the set around a circle. However, in the infinite case we cannot rely upon an immediate successor relation, because points may not have successors. For example, given a point on the unit circle, there is no "next point". Nor can we rely upon a binary relation to determine which of two points comes "first". Traveling clockwise on a circle, neither east or west comes first, but each follows the other.
Instead, we use a ternary relation denoting that elements a, b, c occur after each other (not necessarily immediately) as we go around the circle. For example, in clockwise order, [east, south, west]. By currying the arguments of the ternary relation [a, b, c], one can think of a cyclic order as a one-parameter family of binary order relations, called cuts, or as a two-parameter family of subsets of K, called intervals.
The ternary relation
The general definition is as follows: a cyclic order on a set X is a relation C ⊂ X3, written [a, b, c], that satisfies the following axioms:[nb]
Cyclicity: If [a, b, c] then [b, c, a]
Asymmetry: If [a, b, c] then not [c, b, a]
Transitivity: If [a, b, c] and [a, c, d] then [a, b, d]
Connectedness: If a, b, and c are distinct, then either [a, b, c] or [c, b, a]
The axioms are named by analogy with the asymmetry, transitivity, and connectedness axioms for a binary relation, which together define a strict linear order. EdwardHuntington(1916, 1924) considered other possible lists of axioms, including one list that was meant to emphasize the similarity between a cyclic order and a betweenness relation. A ternary relation that satisfies the first three axioms, but not necessarily the axiom of totality, is a partial cyclic order.
Rolling and cuts
Given a linear order < on a set X, the cyclic order on X induced by < is defined as follows:[2]
[a, b, c] if and only if a < b < c or b < c < a or c < a < b
Two linear orders induce the same cyclic order if they can be transformed into each other by a cyclic rearrangement, as in cutting a deck of cards.[3] One may define a cyclic order relation as a ternary relation that is induced by a strict linear order as above.[4]
Cutting a single point out of a cyclic order leaves a linear order behind. More precisely, given a cyclically ordered set (K, [ ]), each element a ∈ K defines a natural linear order <a on the remainder of the set, K ∖ a, by the following rule:[5]
x <ay if and only if [a, x, y].
Moreover, <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.[6]
Intervals
Given two elements a ≠ b ∈ K, the open interval from a to b, written (a, b), is the set of all x ∈ K such that [a, x, b]. The system of open intervals completely defines the cyclic order and can be used as an alternate definition of a cyclic order relation.[7]
An interval (a, b) has a natural linear order given by <a. One can define half-closed and closed intervals [a, b), (a, b], and [a, b] by adjoining a as a least element and/or b as a greatest element.[8] As a special case, the open interval (a, a) is defined as the cut K ∖ a.
More generally, a proper subset S of K is called convex if it contains an interval between every pair of points: for a ≠ b ∈ S, either (a, b) or (b, a) must also be in S.[9] A convex set is linearly ordered by the cut <x for any x not in the set; this ordering is independent of the choice of x.
Automorphisms
As a circle has a clockwise order and a counterclockwise order, any set with a cyclic order has two senses. A bijection of the set that preserves the order is called an ordered correspondence. If the sense is maintained as before, it is a direct correspondence, otherwise it is called an opposite correspondence.[10] Coxeter uses a separation relation to describe cyclic order, and this relation is strong enough to distinguish the two senses of cyclic order. The automorphisms of a cyclically ordered set may be identified with C2, the two-element group, of direct and opposite correspondences.
Monotone functions
The "cyclic order = arranging in a circle" idea works because any subset of a cycle is itself a cycle. In order to use this idea to impose cyclic orders on sets that are not actually subsets of the unit circle in the plane, it is necessary to consider functions between sets.
A function between two cyclically ordered sets, f: X → Y, is called a monotonic function or a homomorphism if it pulls back the ordering on Y: whenever [f(a), f(b), f(c)], one has [a, b, c]. Equivalently, f is monotone if whenever [a, b, c] and f(a), f(b), and f(c) are all distinct, then [f(a), f(b), f(c)]. A typical example of a monotone function is the following function on the cycle with 6 elements:
f(0) = f(1) = 4,
f(2) = f(3) = 0,
f(4) = f(5) = 1.
A function is called an embedding if it is both monotone and injective.[nb] Equivalently, an embedding is a function that pushes forward the ordering on X: whenever [a, b, c], one has [f(a), f(b), f(c)]. As an important example, if X is a subset of a cyclically ordered set Y, and X is given its natural ordering, then the inclusion mapi: X → Y is an embedding.
Generally, an injective function f from an unordered set X to a cycle Y induces a unique cyclic order on X that makes f an embedding.
Functions on finite sets
A cyclic order on a finite set X can be determined by an injection into the unit circle, X → S1. 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 (n − 1)-dimensionalpolytope known as a cyclohedron. Cyclohedra were first applied to the study of knot invariants;[11] they have more recently been applied to the experimental detection of periodically expressedgenes in the study of biological clocks.[12]
A cut with both a least element and a greatest element is called a jump. For example, every cut of a finite cycle Zn is a jump. A cycle with no jumps is called dense.[13][14]
A cut with neither a least element nor a greatest element is called a gap. For example, the rational numbers Q 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.[15][14]
A cut with exactly one endpoint is called a principal or Dedekind cut. For example, every cut of the circle S1 is a principal cut. A cycle where every cut is principal, being both dense and complete, is called continuous.[16][14]
[<1, <2, <3] and [x, y, z]
The set of all cuts is cyclically ordered by the following relation: [<1, <2, <3] if and only if there exist x, y, z such that:[17]
x <1y <1z,
x <1y <2z <2x, and
x <1y <1z <3x <3y.
A certain subset of this cycle of cuts is the Dedekind completion of the original cycle.
Further constructions
Unrolling and covers
Starting from a cyclically ordered set K, one may form a linear order by unrolling it along an infinite line. This captures the intuitive notion of keeping track of how many times one goes around the circle. Formally, one defines a linear order on the Cartesian productZ × K, where Z is the set of integers, by fixing an element a and requiring that for all i:[18]
If [a, x, y], then ai < xi < yi < ai + 1.
For example, the months January2024, May2024, September2024, and January2025 occur in that order.
This ordering of Z × K is called the universal cover of K.[nb] Its order type is independent of the choice of a, but the notation is not, since the integer coordinate "rolls over" at a. For example, although the cyclic order of pitch classes is compatible with the A-to-G alphabetical order, C is chosen to be the first note in each octave, so in note-octave notation, B3 is followed by C4.
The inverse construction starts with a linearly ordered set and coils it up into a cyclically ordered set. Given a linearly ordered set L and an order-preserving bijectionT: L → L with unbounded orbits, the orbit spaceL / T is cyclically ordered by the requirement:[7][nb]
If a < b < c < T(a), then [[a], [b], [c]].
In particular, one can recover K by defining T(xi) = xi + 1 on Z × K.
There are also n-fold coverings for finite n; in this case, one cyclically ordered set covers another cyclically ordered set. For example, the 24-hour clock is a double cover of the 12-hour clock. In geometry, the pencil of rays emanating from a point in the oriented plane is a double cover of the pencil of unoriented lines passing through the same point.[19] These covering maps can be characterized by lifting them to the universal cover.[7]
Products and retracts
Given a cyclically ordered set (K, [ ]) and a linearly ordered set (L, <), the (total) lexicographic product is a cyclic order on the product setK × L, defined by [(a, x), (b, y), (c, z)] if one of the following holds:[20]
[a, b, c]
a = b ≠ c and x < y
b = c ≠ a and y < z
c = a ≠ b and z < x
a = b = c and [x, y, z]
The lexicographic product K × L globally 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.[21]
One can also glue together different linearly ordered sets to form a circularly ordered set. For example, given two linearly ordered sets L1 and L2, one may form a circle by joining them together at positive and negative infinity. A circular order on the disjoint union L1 ∪ L2 ∪ {–∞, ∞} is defined by ∞ < L1 < –∞ < L2 < ∞, where the induced ordering on L1 is the opposite of its original ordering. For example, the set of all longitudes is circularly ordered by joining all points west and all points east, along with the prime meridian and the 180th meridian. Kuhlmann, Marshall & Osiak (2011) use this construction while characterizing the spaces of orderings and real places of double formal Laurent series over a real closed field.[22]
Topology
The open intervals form a base for a natural topology, the cyclic order topology. The open sets in this topology are exactly those sets which are open in every compatible linear order.[23] 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.
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, 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 Roll (1993); see also Directed topology.
Every cyclically ordered group can be expressed as a quotient L / Z, 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 T × L, where L is a linearly ordered group. If a cyclically ordered group is Archimedean or compact, it can be embedded in T itself.[28]
Modified axioms
A partial cyclic order is a ternary relation that generalizes a (total) cyclic order in the same way that a partial order generalizes a total order. 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 [citation needed]. 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.[29]
A cyclic order is required to be symmetric under cyclic permutation, [a, b, c] ⇒ [b, c, a], and asymmetric under reversal: [a, b, c] ⇒ ¬[c, b, a]. 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 separation relation is a quaternary relation that can be thought of as a cyclic order without an orientation. The relationship between a circular order and a separation relation is analogous to the relationship between a linear order and a betweenness relation.[30]
Hans Freudenthal has emphasized the role of cyclic orders in cognitive development, as a contrast to Jean Piaget 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
^cyclic orderThe relation may be called a cyclic order(Huntington 1916, p.630), a circular order(Huntington 1916, p.630), a cyclic ordering(Kok 1973, p.6), or a circular ordering(Mosher 1996, p.109). Some authors call such an ordering a total cyclic order(Isli & Cohn 1998, p.643), a complete cyclic order(Novák 1982, p.462), a linear cyclic order(Novák 1984, p.323), or an l-cyclic order or ℓ-cyclic order(Černák 2001, p.32), 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 separation relation(Bowditch 1998, p.155).
^cycleA set with a cyclic order may be called a cycle(Novák 1982, p.462) or a circle(Giraudet & Holland 2002, p.1). The above variations also appear in adjective form: cyclically ordered set (cyklicky uspořádané množiny, Čech 1936, p.23), 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.
^ternary relationThere are a few different symbols in use for a cyclic relation. Huntington (1916, p.630) uses concatenation: ABC. Čech (1936, p.23) and (Novák 1982, p.462) use ordered triples and the set membership symbol: (a, b, c) ∈ C. Megiddo (1976, p.274) uses concatenation and set membership: abc ∈ C, understanding abc as a cyclically ordered triple. The literature on groups, such as Świerczkowski (1959a, p.162) and Černák & Jakubík (1987, p.157), tend to use square brackets: [a, b, c]. Giraudet & Holland (2002, p.1) use round parentheses: (a, b, c), reserving square brackets for a betweenness relation. Campero-Arena & Truss (2009, p.1) use a function-style notation: R(a, b, c). Rieger (1947), cited after Pecinová 2008, p.82) uses a "less-than" symbol as a delimiter: < x, y, z <. Some authors use infix notation: a < b < c, with the understanding that this does not carry the usual meaning of a < b and b < c for some binary relation < (Černy 1978, p.262). Weinstein (1996, p.81) emphasizes the cyclic nature by repeating an element: p ↪ r ↪ q ↪ p.
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσοςisos "equal", and μορφήmorphe "form" or "shape".
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation on some set , which satisfies the following for all and in :
(reflexive).
If and then (transitive).
If and then (antisymmetric).
or .
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain necessarily contains at least one maximal element.
In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.
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 introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group
In finite geometry, the Fano plane is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces, is PG(2, 2). Here, PG stands for "projective geometry", the first parameter is the geometric dimension and the second parameter is the order.
In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the smallest nonabelian simple group after the alternating group A5 with 60 elements, isomorphic to PSL(2, 5).
This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:
In order theory, a branch of mathematics, a linear extension of a partial order is a total order that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order.
In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements.
In order theory, the Szpilrajn extension theorem, proved by Edward Szpilrajn in 1930, states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable. The theorem is one of many examples of the use of the axiom of choice in the form of Zorn's lemma to find a maximal set with certain properties.
In order theory, a branch of mathematics, a semiorder is a type of ordering for items with numerical scores, where items with widely differing scores are compared by their scores and where scores within a given margin of error are deemed incomparable. Semiorders were introduced and applied in mathematical psychology by Duncan Luce as a model of human preference. They generalize strict weak orderings, in which items with equal scores may be tied but there is no margin of error. They are a special case of partial orders and of interval orders, and can be characterized among the partial orders by additional axioms, or by two forbidden four-item suborders.
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic. For instance, Minkowski's question-mark function produces an isomorphism between the numerical ordering of the rational numbers and the numerical ordering of the dyadic rationals.
Bass, Hyman; Otero-Espinar, Maria Victoria; Rockmore, Daniel; Tresser, Charles (1996), Cyclic Renormalization and Automorphism Groups of Rooted Trees, Lecture Notes in Mathematics, vol.1621, Springer, doi:10.1007/BFb0096321, ISBN978-3-540-60595-9
Courcelle, Bruno (21 August 2003), "2.3 Circular order"(PDF), in Berwanger, Dietmar; Grädel, Erich (eds.), Problems in Finite Model Theory, p.12, archived from the original(PDF) on 27 May 2011, retrieved 15 May 2011
Coxeter, H. S. M. (1949), "Chapter 3: Order and continuity", The Real Projective Plane
Evans, David M.; Macpherson, Dugald; Ivanov, Alexandre A. (1997), "Finite Covers", in Evans, David M. (ed.), Model theory of groups and automorphism groups: Blaubeuren, August 1995, London Mathematical Society Lecture Note Series, vol.244, Cambridge University Press, pp.1–72, ISBN978-0-521-58955-0, retrieved 5 May 2011
Isli, Amar; Cohn, Anthony G. (1998), "An algebra for cyclic ordering of 2D orientations"(PDF), AAAI '98/IAAI '98 Proceedings of the fifteenth national/tenth conference on Artificial intelligence/Innovative applications of artificial intelligence, ISBN978-0-262-51098-1, retrieved 23 May 2011
Kuhlmann, Salma; Marshall, Murray; Osiak, Katarzyna (1 June 2011), "Cyclic 2-structures and spaces of orderings of power series fields in two variables", Journal of Algebra, 335 (1): 36–48, doi:10.1016/j.jalgebra.2011.02.026
Mosher, Lee (1996), "A user's guide to the mapping class group: once-punctured surfaces", in Baumslag, Gilbert (ed.), Geometric and computational perspectives on infinite groups, DIMACS, vol.25, AMS Bookstore, pp.101–174, arXiv:math/9409209, Bibcode:1994math......9209M, ISBN978-0-8218-0449-0
Novák, Vítězslav; Novotný, Miroslav (1987), "On completion of cyclically ordered sets", Czechoslovak Mathematical Journal, 37 (3): 407–414, doi:10.21136/CMJ.1987.102168, hdl:10338.dmlcz/102168
Pecinová-Kozáková, Eliška (2005), "Ladislav Svante Rieger and His Algebraic Work", in Safrankova, Jana (ed.), WDS 2005 - Proceedings of Contributed Papers, Part I, Prague: Matfyzpress, pp.190–197, CiteSeerX10.1.1.90.2398, ISBN978-80-86732-59-6
Rieger, L. S. (1947), "О uspořádaných a cyklicky uspořádaných grupách II (On ordered and cyclically ordered groups II)", Věstník Královské české Spolecnosti Nauk, Třída Mathematicko-přírodovědná (Journal of the Royal Czech Society of Sciences, Mathematics and Natural History) (in Czech) (1): 1–33
Weinstein, Tilla (July 1996), An introduction to Lorentz surfaces, De Gruyter Expositions in Mathematics, vol.22, Walter de Gruyter, ISBN978-3-11-014333-1
Further reading
Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998), Notes on Infinite Permutation Groups, Lecture Notes in Mathematics, vol.1698, Springer, pp.108–109, doi:10.1007/BFb0092550, ISBN978-3-540-64965-6
Cameron, Peter J. (June 1976), "Transitivity of permutation groups on unordered sets", Mathematische Zeitschrift, 148 (2): 127–139, doi:10.1007/BF01214702, S2CID120757129
Cameron, Peter J. (June 1977), "Cohomological aspects of two-graphs", Mathematische Zeitschrift, 157 (2): 101–119, doi:10.1007/BF01215145, S2CID120726731
Cameron, Peter J. (1997), "The algebra of an age", in Evans, David M. (ed.), Model theory of groups and automorphism groups: Blaubeuren, August 1995, London Mathematical Society Lecture Note Series, vol.244, Cambridge University Press, pp.126–133, CiteSeerX10.1.1.39.2321, ISBN978-0-521-58955-0
Droste, M.; Giraudet, M.; Macpherson, D. (March 1995), "Periodic Ordered Permutation Groups and Cyclic Orderings", Journal of Combinatorial Theory, Series B, 63 (2): 310–321, doi:10.1006/jctb.1995.1022
Ivanov, A. A. (January 1999), "Finite Covers, Cohomology and Homogeneous Structures", Proceedings of the London Mathematical Society, 78 (1): 1–28, doi:10.1112/S002461159900163X, S2CID120545318
Kennedy, Christine Cowan (August 1955), On a cyclic ternary relation ... (M.A. Thesis), Tulane University, OCLC16508645
Kónya, Eszter Herendine (2006), "A mathematical and didactical analysis of the concept of orientation", Teaching Mathematics and Computer Science, 4 (1): 111–130, doi:10.5485/TMCS.2006.0108
Marongiu, Gabriele (1985), "Some remarks on the ℵ0-categoricity of circular orderings", Unione Matematica Italiana. Bollettino. B. Serie VI (in Italian), 4 (3): 883–900, MR0831297
McCleary, Stephen; Rubin, Matatyahu (6 October 2005), Locally Moving Groups and the Reconstruction Problem for Chains and Circles, arXiv:math/0510122, Bibcode:2005math.....10122M
Müller, G. (1974), "Lineare und zyklische Ordnung", Praxis der Mathematik, 16: 261–269, MR0429660
Rubin, M. (1996), "Locally moving groups and reconstruction problems", in Holland, W. Charles (ed.), Ordered groups and infinite permutation groups, Mathematics and Its Applications, vol.354, Kluwer, pp.121–157, ISBN978-0-7923-3853-6
Świerczkowski, S. (1956), "On cyclic ordering relations", Bulletin de l'Académie Polonaise des Sciences, Classe III, 4: 585–586
Truss, J.K. (July 1992), "Generic Automorphisms of Homogeneous Structures", Proceedings of the London Mathematical Society, 3, 65 (1): 121–141, doi:10.1112/plms/s3-65.1.121
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