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Cyclic order

In mathematics, a cyclic order (also known as a circular order) on a set X is a ternary relation \beta \subseteq X^3 that captures the relative positions of three distinct elements as if arranged on a circle, where \beta(u, v, w) holds if the directed arc from u to w passes through v in a fixed orientation (e.g., counterclockwise). This structure satisfies Huntington's axioms for cyclic orders: for distinct elements, exactly one of \beta(u, v, w) or \beta(w, v, u) holds (totality and asymmetry); \beta(u, v, w) implies \beta(v, w, u) (cyclic symmetry); and \beta(u, v, w) together with \beta(u, w, x) implies \beta(u, v, x) (transitivity). Unlike a linear order, which imposes a "first" and "last" element, a cyclic order has no endpoints and treats the arrangement as rotationally symmetric up to reversal. Cyclic orders can be defined on various mathematical objects, including abstract sets, groups, and topological spaces. For a finite set of n elements, the number of distinct cyclic orders is (n-1)!, corresponding to the ways to arrange them clockwise around a circle, modulo rotations. A classic example is the set of points on the unit circle in the Euclidean plane, where the relation \beta(a, b, c) holds if b lies between a and c in the counterclockwise direction. In group theory, a left-invariant cyclic order on a group G is one preserved under left multiplication, satisfying additional invariance axioms such as \triangleleft(ga, gb, gc) if and only if \triangleleft(a, b, c) for all g \in G.^[1] Such orders exist on free groups and are linked to the geometry of the group acting on the circle.^[2] The concept originates from early 20th-century work in geometry and order theory, with formal axiomatization by Edward V. Huntington in 1916 for abstract cyclic structures.^[3] Cyclic orders generalize betweenness relations in the plane and are foundational in projective geometry, where they describe collinear points or conic sections. In topology, they appear in the study of orientable manifolds and the fundamental group of the circle, which carries a natural cyclic structure.^[4] Applications of cyclic orders span multiple fields. In graph theory, they characterize circular-arc graphs, where vertices represent arcs on a circle and edges indicate intersections, aiding in scheduling and resource allocation problems.^[4] For groups, bi-invariant cyclic orders (invariant under both left and right multiplication) classify certain solvable groups and appear in the study of dynamics on the circle.^[1] In model theory, expansions of fields by cyclic orders enable the analysis of tame geometries and o-minimal structures, with implications for algebraic geometry over real closed fields.^[5] More recently, in data science, algorithms for recognizing and computing strict cyclic orders facilitate circular seriation, used in visualizing cyclic patterns in gene expression data, DNA sequencing, and tomographic reconstruction.^[3] These tools run in optimal time complexity, such as O(n \log n) for recognition on n elements.^[6]

Basic Concepts

Definition

A cyclic order on a set X is defined by a ternary relation [a, b, c] for distinct elements a, b, c \in X, indicating that b lies between a and c in the clockwise direction when the elements are arranged on a circle.^[7] This concept was first axiomatized by Edward V. Huntington in his 1916 paper, where he introduced independent postulates for cyclic order using a triadic relation on a class of elements to model circular arrangements, such as those arising in cyclic permutations.^[8] Huntington expanded on this in 1924, providing sets of completely independent postulates that further refined the ternary framework for cyclic structures.^[9] The standard modern axioms for a total cyclic order via the ternary relation C(a, b, c) (equivalent to [a, b, c]) are as follows:

Cyclicity: If C(a, b, c), then C(b, c, a) and C(c, a, b).
Asymmetry: If C(a, b, c), then \neg C(a, c, b).
Transitivity: If C(a, b, c) and C(a, c, d), then C(a, b, d).
Totality: For any distinct a, b, c \in X, either C(a, b, c) or C(c, b, a).^[7]

These axioms ensure the relation captures a consistent orientation around a circle, with the transitivity applying locally to points sharing a common reference arc (under the condition that the intervals do not wrap around in a way that violates the order).^[7] Unlike a linear order, which is a binary relation with a total, antisymmetric, and transitive structure that imposes a sequence with endpoints, a cyclic order emphasizes the absence of such endpoints and the inherent rotational symmetry of a circle, allowing generalization to infinite sets without a beginning or end.^[7] For instance, the unit circle with its standard counterclockwise orientation exemplifies an infinite cyclic order on the real numbers modulo $2\pi.^[7]

Examples

Finite cyclic orders arise in everyday and combinatorial settings where elements repeat in a loop without endpoints. For instance, the seven days of the week—Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday—form a discrete cyclic order, where each day follows the next in clockwise sequence, returning to Sunday after Saturday.^[10] Similarly, the twelve notes of the chromatic scale (C, C♯/D♭, D, D♯/E♭, E, F, F♯/G♭, G, G♯/A♭, A, A♯/B♭, B) are arranged cyclically on a circle due to octave equivalence, enabling modular progression in musical composition.^[10] A classic application is the clock face, which embodies a finite cyclic order. On a 12-hour analog clock, the hours occupy 12 positions in a circular arrangement, with each hour succeeding the previous in a fixed rotation; after 12 comes 1 again.^[10] The 24-hour clock extends this by doubling the positions, forming a cyclic order that double-covers the 12-hour version, as each 12-hour mark aligns with two 24-hour points (e.g., 12:00 and 00:00 both map to the top).^[10] Infinite cyclic orders appear in geometric and algebraic structures. The unit circle S^1, parameterized by angles in [0, 2\pi), induces a cyclic order on its points via counterclockwise orientation: for distinct points a, b, c represented by angles \theta_a, \theta_b, \theta_c, the order [a, b, c] holds if traversing from a to b to c follows the positive direction without exceeding a full rotation.^[11] The rational numbers modulo 1, consisting of fractions p/q with $0 \leq p/q < 1, form a dense subset of this order on S^1, inheriting the angular cyclic structure while being countable and everywhere dense.^[12] The real projective line \mathbb{RP}^1, obtained by identifying antipodal points on S^1, homeomorphic to the circle, supports a cyclic order obtained from the quotient.^[13] These examples illustrate the ternary relation axioms of cyclic orders, where any three distinct elements admit exactly one of the two possible orientations.^[10]

Properties

Intervals and Cuts

In a cyclically ordered set (G, C), where C is a ternary relation satisfying asymmetry, cyclicity, and transitivity, the open interval (a, b) for distinct a, b \in G is defined as the set of all points x \in G \setminus \{a, b\} such that (a, x, b) \in C, meaning x lies strictly between a and b in the positive orientation of the cyclic order.^[14] This captures the points on one of the two arcs determined by a and b on the underlying circle, excluding the endpoints. The closed interval [a, b] extends this by including the endpoints, so [a, b] = (a, b) \cup \{a, b\}, while half-open intervals such as [a, b) or (a, b] include one endpoint and exclude the other, adapting the linear interval conventions to the cyclic structure with wrapping around the circle.^[14] These intervals are convex in the sense that for any u, v \in (a, b) with u preceding v relative to a, the subinterval (u, v) is contained in (a, b).^[14] Cuts in a cyclically ordered set (G, C) generalize Dedekind cuts from linear orders and are defined as partitions (L, R) of G into disjoint subsets L \cup R = G such that every element of L precedes every element of R in the cyclic order, meaning that for all l \in L and r \in R, the orientation places l before r without elements of R intervening in the arc from l to r.^[15] Equivalently, a cut corresponds to a compatible linear order < on G where x < y < z implies (x, y, z) \in C, effectively "cutting" the cycle at a boundary between L and R to yield a linearization.^[15] Cuts are classified by their boundaries: a jump cut has both least and greatest elements, a gap cut has neither, and a Dedekind cut has exactly one boundary element.^[15] Distinct cuts <_1 and <_2 partition G into complementary initial segments, with (G, <_1) = L \oplus R and (G, <_2) = R \oplus L.^[15] The open intervals (a, b) in a cyclic order generate the order topology on G, forming a basis for the topology where every open set is a union of such intervals, provided the space is cyclically orderable.^[16] This basis ensures that the topology respects the circular arrangement, with neighborhoods around points defined by small arcs. Cuts facilitate embedding cyclic orders into linear ones: selecting a cut (L, R) induces a linear order on G compatible with C, allowing the cyclic structure to be represented as a linear extension "unrolled" at the cut point.^[15] The set of all cuts on (G, C) itself forms a cyclically ordered set under the induced relation.^[15]

Automorphisms

An automorphism of a cyclic order on a set X is a bijection f: X \to X that preserves the ternary relation defining the order, meaning that for all distinct x, y, z \in X, the triple (x, y, z) satisfies the positive orientation if and only if (f(x), f(y), f(z)) does.^[17] These automorphisms form a group under function composition, known as the automorphism group \operatorname{Aut}(X), which acts on X by permuting elements while respecting the cyclic structure.^[17] In finite cyclic orders with n elements, such as points arranged in a cyclic sequence, the automorphism group is the cyclic group of order n, generated by rotations (cyclic shifts) that preserve the orientation. Reflections reverse the orientation and are not automorphisms of the fixed cyclic order, though they belong to the dihedral group D_n of order $2n, the full symmetry group including the opposite orientation. For the simplest non-trivial case with three points, the group is isomorphic to C_3, generated by a 120-degree rotation. For dense cyclic orders, such as the countable dense circular order on \mathbb{Q}/\mathbb{Z}, the automorphism group is highly transitive: it acts transitively on finite subsets of the same cardinality and preserves the order type, reflecting the ultrahomogeneity of the structure. This group is a Polish group with additional topological properties, including Roelcke precompactness and property (T).^[17] Automorphisms of a cyclic order necessarily preserve the derived betweenness relation, defined such that y lies between x and z if either (x, y, z) or (z, y, x) holds in the ternary relation, ensuring that the cyclic arrangement and interval structures are maintained under the mapping.^[17]

Finite Cyclic Orders

Circular Permutations

In the context of finite cyclic orders, circular permutations provide a combinatorial framework for understanding arrangements of labeled elements on a circle, where the structure is invariant under rotation but preserves orientation. For a set of n distinct labeled elements, a cyclic order corresponds precisely to an equivalence class of linear arrangements under cyclic shifts, yielding (n-1)! distinct such orders. This count arises because fixing one element's position eliminates rotational redundancy from the n! linear permutations, effectively modeling necklaces with fixed beads and a distinguished direction.^[18]^[19] Such cyclic orders can be represented as the rotations of any underlying linear order on the set, without designating a canonical starting point. Specifically, given a total linear order, its cyclic variants form a single cyclic order by considering all possible "cuts" along the sequence, which wrap around to maintain the relative succession. This representation underscores the transition from linear to circular structures, where the absence of endpoints distinguishes cyclic orders from their linear counterparts. For instance, the linear order $1 < 2 < 3 < \cdots < n generates the cyclic order where each element follows the previous in sequence, looping back to the first.^[20] A deeper algebraic connection links these cyclic orders to group theory: each cyclic order on the labeled set corresponds to a conjugacy class of reduced words in the free group freely generated by those elements. Here, the order is encoded by a cyclically reduced word that traverses all generators exactly once (a primitive full-cycle word), with the conjugacy class capturing all rotations of that word while preserving the sequential relations. This bijection highlights how cyclic orders encode rotational invariance akin to conjugacy, facilitating applications in combinatorial group theory and word problems.^[21]

Monotone Functions on Finite Sets

In finite cyclic orders, a monotone function f: S \to T between cyclically ordered finite sets S and T preserves the ternary cyclic relation, meaning that if [a, b, c]_S holds in S (indicating b follows a before c in the cyclic arrangement), then [f(a), f(b), f(c)]_T holds in T.^[22] This preservation ensures that the circular arrangement in the domain is respected in the codomain, distinguishing monotone functions from arbitrary maps. For finite sets with at least three elements, the irreflexivity of the cyclic relation implies that monotone functions are often injective under suitable conditions, such as when the relation is total.^[23] Embeddings in this context are injective monotone functions, which faithfully embed the cyclic structure of S into T without distortion. These embeddings maintain the discrete circular permutation structure of finite cyclic orders, where the domain can be viewed as a circular arrangement of n distinct points.^[22] A key property is the preservation of intervals and cuts: an interval in a cyclic order, defined as the arc between two points excluding the complementary arc, maps to an interval in the codomain, while a cut (a partition into two complementary intervals) maps to a corresponding cut, ensuring structural integrity.^[22] This preservation is crucial for inductive constructions and comparability between different finite cyclic orders. Embeddings play a central role in building the cyclohedron, an (n-1)-dimensional polytope whose vertices and faces correspond to compatible embeddings of labeled subsets preserving the cyclic order, providing a geometric realization of cyclic bracketings and configurations on the circle. In applications, cyclohedra derived from such embeddings yield knot invariants; specifically, the self-linking number of a knot can be computed combinatorially via integrals over cyclohedral compactifications of configuration spaces. In biology, finite cyclic orders modeled via embeddings and monotone mappings help analyze periodic gene expression, as in the Oscope algorithm, which reconstructs cyclic orders of cell states from unsynchronized single-cell RNA data to identify oscillatory gene modules driving cycles like the cell cycle.

Infinite and General Cyclic Orders

Completion

The Dedekind completion of a cyclic order on a set X is obtained by adjoining points corresponding to all Dedekind cuts in X, yielding a dense and complete cyclic order that embeds X as a suborder.^[24] This construction parallels the Dedekind completion of linear orders, where cuts define missing limits, but adapted to the circular nature via compatible linearizations of the ternary relation.^[24] A Dedekind cut in a cyclically ordered set (X, \prec) is a linear order < on X such that for all distinct x, y, z \in X with x < y < z, the triple (x, y, z) satisfies the cyclic order \prec.^[24] The completion (X^*, \prec^*) is the set of all such cuts, equipped with an induced cyclic order defined by compatibility: for cuts \alpha, \beta, \gamma, \alpha \prec^* \beta \prec^* \gamma if there exist representatives a \in \alpha, b \in \beta, c \in \gamma with a \prec b \prec c.^[24] The original set embeds densely into X^* via the map sending each x \in X to the principal cut consisting of elements "before" x in a compatible linearization, provided (X, \prec) is dense (i.e., between any two distinct points there is a third).^[24] This completion satisfies a universal property: it is the unique (up to isomorphism over the embedding of X) complete cyclic order containing X as an initial suborder, meaning every cut in X^* is principal (has a least upper bound in the cyclic sense).^[24] Moreover, strictly monotone functions (order-preserving maps between cyclically ordered sets) extend uniquely to monotone functions on the completions, as such maps preserve compatible linear orders and thus induce maps on the sets of cuts.^[24]^[25] For instance, the Dedekind completion of a finite cyclic order on n points (the standard cycle) yields a dense complete cyclic order isomorphic to the circle S^1 = \mathbb{R}/\mathbb{Z}, filling the discrete gaps with continuum many points.^[24] Similarly, the cyclic order on the dense countable set \mathbb{Q}/\mathbb{Z} (rationals modulo 1, with the induced order from the circle) completes to \mathbb{R}/\mathbb{Z}, adding irrational limits via cuts to achieve completeness while preserving density.^[24]

Topology

The order topology on a set equipped with a cyclic order is generated by taking as a basis the collection of all open intervals (a, b), where a and b are distinct elements such that the interval does not contain all elements of the set. These open intervals consist of all points strictly between a and b in the cyclic sense, excluding a and b themselves. This basis induces a topology that is Hausdorff provided the underlying set has at least three elements, as distinct points can be separated by disjoint open neighborhoods formed from such intervals.^[26]^[27] Geometrically, this topology models a circle-like space, where the cyclic order captures the circular arrangement without a distinguished starting point, and the resulting space is compact and connected when the cyclic order is complete. In Lorentzian surfaces, the cyclic order arises naturally from the light cones at each point, which divide the tangent space into future and past null directions, inducing a cyclic ordering on the spatial directions orthogonal to the time-like direction. This structure equips the surface with an order topology that reflects the causal geometry, ensuring the space is Hausdorff and locally resembling a circle in its null boundary components. Similarly, locally linearly ordered spaces, where the order is linear in small neighborhoods but globalizes to cyclic, inherit this topology, providing a framework for analyzing oriented structures in semi-Riemannian geometries.^[28] Discrete dynamical systems preserving a cyclic order, such as iterations on a circularly ordered set, generate periodic orbits that respect the interval topology, leading to tame dynamics where minimal subsystems are representable as skew products over irrational rotations. These systems exhibit periodic points whose orbits maintain the cyclic betweenness, and the Hausdorff topology ensures convergence properties for such orbits, facilitating the study of ergodic measures and symbolic representations.^[26] Cyclic orders on manifolds induce topologies that align with standard structures on spaces like the real projective line \mathbb{RP}^1, which is homeomorphic to the circle S^1 and thus carries the cyclic order topology as its standard Hausdorff topology. On the Möbius strip, the cyclic order along the central curve or boundary circle generates a topology compatible with the strip's non-orientable structure, where open intervals correspond to twisted arcs, preserving the overall manifold topology while highlighting the identification of antipodal points in the projective sense.^[29]^[27]

Constructions

Unrolling and Covers

One key construction in the theory of cyclic orders is the unrolling, which lifts a cyclic order on a set K to a linear order on its universal cover \mathbb{Z} \times K. This structure endows \mathbb{Z} \times K with a total order \leq_R defined lexicographically: for distinct elements (m, x), (n, y) \in \mathbb{Z} \times K, (m, x) \leq_R (n, y) if m < n, or if m = n and either x = y or the cyclic order on K satisfies R(x, y, e) where e is a fixed reference point, ensuring the order respects the cyclic arrangement within each copy of K. The projection map \pi: \mathbb{Z} \times K \to K given by \pi(m, x) = x is surjective and monotone, meaning it preserves the order relations in the sense that intervals in the linear order map to arcs in the cyclic order, effectively "winding" the infinite line around the circle infinitely many times. This unrolling is universal in the sense that any linear extension of the cyclic order on K factors through it. When K carries a compatible group structure, the equivalence relation for the projection corresponds to the discrete central subgroup generated by (1, e), which is cofinal and makes the quotient recover the original cyclic order. For instance, the standard cyclic order on the circle corresponds to the universal cover by the real line, where the projection is the exponential map preserving local order. In the discrete case, the cyclic order on \mathbb{Z}/n\mathbb{Z} unrolls to the standard order on \mathbb{Z}, with the projection modulo n. More generally, an n-fold cover of a cyclic order on K is a cyclic order on a set L equipped with a surjective projection p: L \to K such that each fiber p^{-1}(x) has exactly n elements, p is monotone (preserving cyclic intervals), and it is locally an isomorphism, meaning that for any small arc in K, the preimage is n disjoint arcs in L each order-isomorphic to the base arc. A concrete example is the cyclic order on the 24 positions of a 24-hour clock covering the 12-hour clock, where the projection identifies antipodal points (e.g., 1 AM with 1 PM), resulting in 2-to-1 fibers while locally preserving the clockwise order around each hour mark. Covers of cyclic orders inherit key properties from their linear counterparts: they are monotone and locally isomorphic, ensuring that the covering space captures the same local structure as the base but globally unwraps the cyclicity. These constructions play a role in homotopy theory, where cyclic orders model the fundamental group of the circle, and covers correspond to subgroups of \mathbb{Z}. In the topological realization, the universal cover induces a simply connected space whose deck transformations generate the cyclic structure. A retract in the context of cyclic orders is a suborder S \subseteq K such that there exists an idempotent projection \rho: K \to S (i.e., \rho \circ \rho = \rho) that is monotone with respect to the cyclic order and fixes every element of S. Such retracts arise naturally in covering constructions, where a finite subcover may retract onto a base via the projection map restricted to invariant subsets.

Products and Retracts

The lexicographic product of a cyclically ordered set (C, R) and a linearly ordered set (L, \leq) is defined on the Cartesian product C \times L with a ternary relation R' that prioritizes the cyclic order on C: R'((c_1, l_1), (c_2, l_2), (c_3, l_3)) holds if (c_1, c_2, c_3) \in R when the c_i are distinct, or if the c_i are not all distinct, the relation is determined by treating equal c's as tied and broken by the linear order on the corresponding l's (e.g., if two c's equal and the third different, compare the differing c with the ordered pair from L). If all c_i equal, then l_1 < l_2 < l_3.^[30] This construction yields a cyclic order on the product set, generalizing the structure to incorporate both circular and linear aspects.^[31] A representative example is the cylinder S^1 \times \mathbb{R}, where S^1 carries the standard cyclic order induced by its circular topology and \mathbb{R} is equipped with the usual linear order; the resulting ternary relation mixes the cyclic comparison on the angular coordinate with linear ties broken by the height coordinate.^[30] Such products model scenarios like a clock with height, where time follows a cyclic order and position varies linearly, preserving the intuitive betweenness in combined dimensions. Products inherit key properties from their components: if the cyclic factor is connected (meaning the order has no gaps, allowing dense intervals between any pair), the product remains connected, and transitivity (local betweenness implying global) follows from the transitivity of both the cyclic and linear orders.^[31] Retracts in cyclic orders are defined analogously to those in partially ordered sets: a suborder S \subseteq T of a cyclically ordered set (T, R) is a retract if there exists an idempotent projection p: T \to S (satisfying p \circ p = p) that is monotone with respect to the ternary relation, meaning p preserves R in the sense that (p(a), p(b), p(c)) \in R|_S whenever (a, b, c) \in R, and p|_S = \mathrm{id}_S. This projection ensures the suborder inherits the cyclic structure while allowing deformation of the ambient order onto it without altering the core arrangement. In polyhedral decompositions, retracts arise as projections onto subcomplexes that maintain the cyclic ordering of vertices around faces, facilitating combinatorial analysis of geometric configurations. Monotone functions on such products extend naturally from the component orders, respecting the lexicographic priority.^[31]

Cyclically Ordered Groups

A cyclically ordered group is a group G equipped with a cyclic order, defined via a ternary relation \beta \subseteq G^3, that is compatible with the group operation, meaning that for all x, y, z \in G and g \in G, \beta(x, y, z) if and only if \beta(gx, gy, gz).^[30] This compatibility ensures that left multiplication by group elements preserves the cyclic order, generalizing the notion of linearly ordered groups to a circular setting. The cyclic order satisfies axioms of strictness, totality, cyclicity, and transitivity, with the additional group compatibility axiom. The concept was introduced and classified by Ladislav Rieger in a series of papers during the 1940s. Rieger proved that every cyclically ordered group arises as the "wound-round" quotient of a totally ordered group by a central, cofinal subgroup generated by an element z, specifically H / \langle z \rangle where H is the totally ordered group and the order on H descends to a cyclic order on the quotient.^[30] This classification highlights the intimate connection between cyclic and linear orders, showing that cyclically ordered groups can be constructed by "wrapping" linearly ordered structures around a circle.^[32] Common constructions of cyclically ordered groups include semidirect products L \rtimes \mathbb{Z}, where L is a linearly ordered group and \mathbb{Z} acts by automorphisms preserving the order, such as shifts. Another standard construction is the direct product \mathbb{T} \times L, where \mathbb{T} = \mathbb{R}/\mathbb{Z} is the circle group with its natural cyclic order induced from the reals modulo integers, and L is a linearly ordered group, equipped with the lexicographic cyclic order.^[30] These constructions allow for the generation of more complex examples from simpler linearly ordered components. Prominent examples include the circle group \mathbb{T} = \mathbb{R}/\mathbb{Z}, where the cyclic order is given by the projection of the standard order on \mathbb{R}, making it the prototypical abelian cyclically ordered group.^[30] Groups such as the projective special linear group \mathrm{PSL}(2, \mathbb{R}) and the group \mathrm{Homeo}^+(S^1) of orientation-preserving homeomorphisms of the circle act on the circle by preserving its natural cyclic order, and in certain contexts, these actions relate to cyclic orders on the groups themselves.^[33]^[1]

Modified Axioms

Partial cyclic orders generalize the standard ternary relation of a full cyclic order by defining it only on a subset of triples, without requiring the relation to be total or connected across the entire set. This weakened structure satisfies local axioms such as non-degeneracy (no point is between the other two in a triple), cyclicity (rotating the triple preserves the relation), and transitivity where defined, allowing for incomplete circular arrangements. Such orders are useful for modeling partial configurations where not all relative positions are specified, and they can be extended to total cyclic orders under certain conditions, like when the partial order admits a circular completion without contradictions.^[34] CC systems, introduced by Donald Knuth, provide another modification through betweenness relations that incorporate cyclicity and orientation properties, particularly for planar point configurations. These systems satisfy axioms including totality for distinct points, non-degeneracy, the Pasch axiom for convex quadrilaterals, and a cyclicity condition ensuring consistent counterclockwise orientations around points, without full transitivity of betweenness. CC systems model abstract order types in geometry, equivalent to uniform oriented matroids of rank 3, and are applied in enumerating non-isomorphic planar embeddings.^[35] Point-pair separation offers a quaternary axiom system for cyclic orders, focusing on distinguishing pairs of points via a separation relation rather than relying on full ternary transitivity. Defined by axioms such as reflexivity, antisymmetry, and compatibility with cyclic permutations, this relation holds that two distinct pairs (a,b) and (c,d) are separated if the points interleave in the cyclic arrangement, ensuring points are distinguishable without assuming a complete order. Originally axiomatized by Giovanni Vailati, it was later employed by H.S.M. Coxeter to describe oriented cyclic orders on lines or circles, serving as a foundational tool in projective geometry. (Coxeter, 1969) These modified axioms find applications in hyperbolic geometry, where partial cyclic orders describe orientations on the ideal boundary (a circle at infinity), facilitating the study of geodesic configurations and Fuchsian groups without Euclidean assumptions. In graph theory, they underpin partial orders for embedding graphs in the plane or on surfaces, enabling analysis of crossing numbers and planar representations through local cyclicity constraints. Full cyclic orders emerge as special cases when these partial structures satisfy completeness and global connectedness.^[14]^[2]

Applications

Symmetries and Model Theory

In model theory, cyclic orders are studied through their first-order axiomatizations and structural properties, particularly for countable dense examples. A cyclic order can be expressed as a ternary relation C(x, y, z) satisfying first-order axioms: totality (for distinct x, y, z, exactly one of C(x, y, z), C(y, z, x), or C(z, x, y) holds), cyclicity (if C(x, y, z) then not C(x, z, y)), and transitivity (if C(x, y, z) and C(y, z, w) then C(x, y, w)). These axioms define the class of cyclic orders in first-order logic, making it an elementary class. For dense cyclic orders, additional axioms ensure no "gaps," analogous to dense linear orders, leading to a complete theory for the countable case.^[36] The rational circle \mathbb{Q}/\mathbb{Z}, denoted \mathbb{Q}^\circ, serves as the canonical countable dense cyclic order. It is the Fraïssé limit of the class of all finite cyclic orders equipped with the amalgamation property, yielding a unique (up to isomorphism) countable homogeneous structure where every isomorphism between finite substructures extends to an automorphism of the entire structure. This ultrahomogeneity implies that the automorphism group \mathrm{Aut}(\mathbb{Q}^\circ) acts transitively on finite configurations preserving the cyclic order, making \mathbb{Q}^\circ a homogeneous model in the model-theoretic sense. The theory of countable dense cyclic orders is \omega-categorical, meaning it has a unique countable model up to isomorphism, namely \mathbb{Q}^\circ. This follows from the Ryll-Nardzewski theorem applied to the oligomorphic permutation group \mathrm{Aut}(\mathbb{Q}^\circ), which has finitely many orbits on n-tuples for each n, reflecting the symmetry of the structure. Such \omega-categoricity underscores the rigid symmetries inherent in dense cyclic orders, distinguishing them from less symmetric orderings. Countable dense cyclic orders like \mathbb{Q}^\circ are minimal structures in model theory, possessing no proper elementary submodels. This minimality arises because any elementary substructure must be dense and thus coincide with the whole, preventing nontrivial definable subsets that could generate proper submodels. In the context of cyclically ordered groups, extensions to divisible abelian cases preserve this minimality when the structure is torsion-free and the quotients by convex subgroups are dense, ensuring strong structural homogeneity.

Cognition

In cognitive science, cyclic orders play a key role in how children develop an understanding of temporal and spatial structures, as highlighted in Hans Freudenthal's educational philosophy. Freudenthal emphasized that young children naturally encounter cyclic orders through everyday experiences, such as arranging themselves in a circle during counting activities, where the sequence loops back to the starting point, prompting questions like "didn't you forget anybody?" to reinforce the structure.^[37] He argued that these interactions allow children to mathematize their environment intuitively, progressing from imitation to reflective understanding of loops and repetitions. For instance, scanning the bars of a playpen to "close the cycle" illustrates how infants impose cyclic interpretations on linear arrangements, fostering early cognitive growth in pattern recognition.^[37] Children aged 5 to 11 demonstrate progressive mastery of cyclic temporal aspects, such as the recurrence of days or seasons, with significant improvements in ordering events within cycles by age 9, indicating a developmental shift toward abstract cyclic reasoning.^[38] Biological clocks, particularly circadian rhythms, impose cyclic temporal structures on cognitive processes, modulating attention and memory consolidation over 24-hour cycles to optimize performance during peak periods, as seen in enhanced problem-solving during diurnal phases.^[39] These rhythms function as inherent cognitive timers, anticipating environmental changes and integrating sensory inputs in looped patterns.^[40] Finite cyclic examples, like days of the week, further illustrate this perceptual tendency in everyday cognition.^[38]

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[8]
[PDF] chapter iv. degree one circle map
Note that R/Z can be identified with the unit circle in the complex plane under the ... This cyclic order type coincides with the cyclic order type for an orbit ...
[9]
First order theory of cyclically ordered groups - ResearchGate
... cyclic order (see [6, 13]). Indeed, one can view (G ∩ I, + 1 , − 1 , 0, <) as a subgroup of the unit circle (S 1 , ·) with the induced circular order ...
[10]
[PDF] arXiv:math/0511527v1 [math.GT] 21 Nov 2005
Nov 21, 2005 · Recall that a real projective line is homeomorphic to circle. A set of four points on a circle inherits two cyclic orders, each of which ...
[11]
None
### Summary of Definitions Related to Intervals in Cyclic Orders from arXiv:1706.09155
[12]
[PDF] Vнtězslav Novбk Cuts in cyclically ordered sets - DML-CZ
An ordered set is a pair (G, <) where G is a set and < is an order on G, i.e. an irreflexive and transitive binary relation on G. We write briefly G instead ...
[13]
[PDF] arXiv:2110.06979v3 [math.GN] 19 Jan 2024
Jan 19, 2024 · If, moreover, these intervals form a base for the topology of X, then X is called cyclically orderable [22]. (strictly cyclically orderable, in ...
[14]
Circular orders, ultra-homogeneous order structures and their ... - arXiv
Mar 17, 2018 · ... automorphism groups. ... When M(G), as a G-system, admits a circular order we say that G is intrinsically circularly ordered.
[15]
Dihedral Group -- from Wolfram MathWorld
The dihedral group D_n is the symmetry group of an n-sided regular polygon for n>1. The group order of D_n is 2n. Dihedral groups D_n are non-Abelian ...
[16]
[PDF] circular sorting - Math (Princeton)
Feb 11, 2025 · Circular sorting involves sorting labeled points on a circle by adjacent swaps, where an adjacent swap swaps labels of two adjacent vertices.
[17]
Circular Permutation -- from Wolfram MathWorld
The number of ways to arrange n distinct objects along a fixed (i.e., cannot be picked up out of the plane and turned over) circle is P_n=(n-1)!
[18]
[PDF] Voting on cyclic orders, group theory, and ballots
This is evident in that each group of four columns corresponds to a coset of (ABCD) in S4, which does not change the cyclic order that a given ballot favors, ...
[19]
[PDF] A Product Theorem in Free Groups
Sep 16, 2013 · A cyclic word is an equivalence class of this relation. That is, a cyclic word is a cyclically reduced word considered up to cyclic shifts.
[20]
Cyclically ordered groups
Rieger [1] did research on groupswhich admit a cyclical order. He showed that every cyclically ordered group can be obtained by a special construction from ...
[21]
[PDF] On monotone permutations of l-cyclically ordered sets - DML-CZ
The functions e and y belong to P(M)↑, x belongs to P(M)↓. Moreover ... [5] P. C. Fishburn, D. R. Woodall: Cyclic orders. Order 16 (1999), 149–164 ...
[22]
cyclic order in nLab
Sep 26, 2024 · A cyclic order is a way of arranging them 'on a circle' (with a chosen direction, say clockwise or counterclockwise).
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https://ncatlab.org/nlab/show/cyclic%2Border
[24]
Completion of a cyclically ordered group - EuDML
Čech, Bodové množiny, , Praha 1936. (1936); S. Černák, On the maximal Dedekind completion of a lattice ordered group, Math. ... cyclic order, extensions ...
[25]
[PDF] circularly ordered dynamical systems
Definition 2.8. We say that a compact S-system (X, τ) is circularly orderable if there exists a compatible circular order R on X such that ...
[26]
None
Below is a merged summary of the sections from "Elementary Topology Problem Textbook" by Viro et al., based on the provided summaries. To retain all information in a dense and organized manner, I will use a table in CSV format for each section, followed by a narrative summary where additional context or details are provided. The table will include page references, descriptions, and relevant theorems or problems, consolidating all mentions across the summaries.
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[PDF] Lorentzian dynamics and groups of circle diffeomorphisms
Sep 4, 2014 · diffeomorphism if f is a cyclic order preserving homeomorphism such that f and f−1 are ... that the light cone of g). We say that (M,g) is stably ...
[28]
[PDF] Topological manifolds
... manifold is a cyclic order on the set of its points such that the topology of this cyclic order (see 8′3) coincides with the topology of the 1-manifold.
[29]
[PDF] arXiv:1311.0499v1 [math.LO] 3 Nov 2013
Nov 3, 2013 · ,h) is not satisfied. Examples. (1) (G, R) is c-Archimedean iff ... cyclic order R: Let {gr,r ∈ L} be a family of representatives of L ...
[30]
[PDF] lexicographic product decompositions of cyclically ordered groups
If M1 is a nonempty subset of M, then we consider M1 to be cyclically ordered by the relation of cyclic order which is inherited from C. ... lexicographic product ...
[31]
Retracts of partially ordered sets - Cambridge University Press
Every dismantlable partially ordered set has the fixed point property. (Rival (1976)). Moreover, every retract of a dismantlable partially ordered set is.
[32]
Spanning retracts of a partially ordered set - ScienceDirect.com
Two general kinds of subsets of a partially ordered set P are always retracts of P: (1) every maximal chain of P is a retract; (2) in P, every isometric, ...
[33]
Cyclically ordered groups - Siberian Mathematical Journal
### Summary of Ladislav Rieger's Work on Cyclically Ordered Groups
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[PDF] arXiv:1609.04560v1 [math.DG] 15 Sep 2016
Sep 15, 2016 · Let H C PSL(2, R) be a Fuchsian Schottky group. ... partial cyclic order and some topological conditions on partially cyclically ordered.
[35]
On the number of circular orders on a group - ScienceDirect
Jun 15, 2018 · A circular order on a group G is defined by a left-invariant cyclic orientation cocycle c : G 3 → { ± 1 , 0 } satisfying a “nondegenerate ...
[36]
Extensions of partial cyclic orders and consecutive coordinate ...
Mar 27, 2018 · We show that the normalized volumes of these polytopes enumerate circular extensions of certain partial cyclic orders. Among other things this ...
[37]
Faradžev Read-type enumeration of non-isomorphic CC systems
Donald Knuth introduced abstract CC systems to represent configurations of points in a plane with a given orientation (clockwise or counterclockwise) of all ...
[38]
Cyclic orders and graphs of groups | Proceedings of the Edinburgh ...
Oct 21, 2024 · We examine a cyclic order on the directed edges of a tree whose vertices have cyclically ordered links. We use it to show that a graph of ...
[39]
https://www.annualreviews.org/content/journals/10.1146/annurev-psych-022824-043825
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[PDF] REVISITING MATHEMATICS EDUCATION - P4mriunismuh
... children sitting in a circle on the floor is struc- tured by its cyclic order -- “didn't you forget anybody'“? Or the children may count themselves, each ...
[41]
The development of children's understanding of cyclic aspects of time
Results show major progress in the representation of cyclic order and recurrence during the age period examined.
[42]
Psychological Models of Time: Arrows, Cycles and Spirals
A cyclical representation implies that events repeat themselves, leading to a significant similarity between past, present, and future, while a linear ...Missing: mental | Show results with:mental
[43]
Modeling Mental Spatial Reasoning About Cardinal Directions
May 16, 2014 · This article presents research into human mental spatial reasoning with orientation knowledge. In particular, we look at reasoning problems ...
[44]
Finding the answer in space: the mental whiteboard hypothesis on ...
Nov 25, 2014 · The mental whiteboard hypothesis proposes that serial order in working memory is based on spatial attention, using position markers as ...
[45]
The Circadian Brain and Cognition - Annual Reviews
Jan 17, 2025 · Circadian rhythms act directly on human cognition and indirectly through their fundamental influence on sleep/wake cycles.
[46]
CBC‐Clock Theory of Life – Integration of cellular circadian clocks ...
Aug 11, 2021 · Circadian clocks are inherently cognitive in nature. They participate in a cell's sentient awareness of the environment by anticipation of the ...
[47]
The Circularity of the Embodied Mind - Frontiers
Aug 11, 2020 · The aim of the paper is to explore the concept of circularity as a means of explaining the relation between the phenomenology of lived experience and the ...