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Group with a cyclic order respected by the group operation
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_group
Mathematical group that can be generated as the set of powers of a single element
a cyclically ordered group is cyclic. A metacyclic group is a group containing a cyclic normal subgroup whose quotient is also cyclic. These groups include
Cyclic_group
Alternative mathematical ordering
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
Cyclic_order
Group with translationally invariant total order
announced in 2020. Cyclically ordered group Hahn embedding theorem Partially ordered group Deroin, Navas & Rivas 2014, 1.1.1. For abelian groups see Def. 6 in
Linearly_ordered_group
Group with a compatible partial order
integrally closed. Cyclically ordered group – Group with a cyclic order respected by the group operation Linearly ordered group – Group with translationally
Partially_ordered_group
Algebraic object with an ordered structure
orderings. Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic
Ordered_field
In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group G {\displaystyle
Free-by-cyclic_group
Type of group in mathematics
finitely generated abelian groups. The subgroups of a primary cyclic group are linearly ordered by inclusion. The only other groups that have this property
Primary_cyclic_group
Vector space with a partial order
In mathematics, an ordered vector space or partially ordered vector space is a real vector space equipped with a partial order that is compatible with
Ordered_vector_space
Order whose elements are all comparable
set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term
Total_order
Topics referred to by the same term
with a cyclic order Necklace (combinatorics), an equivalence classes of cyclically ordered sequences of symbols modulo certain symmetries Cyclic (mathematics)
Cycle
Commutative group (mathematics)
defined for any ordered pair of elements of A, that the result is well-defined, and that the result belongs to A. A group in which the group operation is
Abelian_group
cyclic. in the study of infinite groups, a Z-group is a group which possesses a very general form of central series. in the study of ordered groups,
Z-group
free group with ordered basis [ x1, …, xn ] is generated by the following 4 elementary Nielsen transformations: Switch x1 and x2 Cyclically permute x1, x2
Automorphism group of a free group
Automorphism_group_of_a_free_group
Class of mathematical orderings
well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible
Well-order
Type of classification in algebra
mathematics, an Archimedean group is a linearly ordered group for which the Archimedean property holds: every two positive group elements are bounded by integer
Archimedean_group
Number that permute or shift cyclically when multiplied by another number
or shift cyclically when they are multiplied by another integer n {\displaystyle n} . Examples are: 142857 × 3 = 428571 (shifts cyclically one place
Transposable_integer
In mathematics, the cyclic category or cycle category or category of cycles is a category of finite cyclically ordered sets and degree-1 maps between them
Cyclic_category
Mathematical term in group theory
Prüfer group. The Prüfer p-group is the unique infinite p-group that is locally cyclic (every finite set of elements generates a cyclic group). As seen
Prüfer_group
specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are
List of order structures in mathematics
List_of_order_structures_in_mathematics
Mathematical concept
groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups
Direct_product_of_groups
Type of group in abstract algebra
symmetric group of {1, 2, ..., n} are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not
Symmetric_group
Sporadic simple group
an ordered pair of coordinates, a dodecad, and a dodecad together with a single coordinate. There is a natural connection between the Mathieu groups and
Mathieu_group_M24
Mathematical set with an ordering
Nested set collection Order polytope Ordered field – Algebraic object with an ordered structure Ordered group – Group with a compatible partial orderPages
Partially_ordered_set
subgroups of a group is the lattice defined by its subgroups, partially ordered by set inclusion. locally cyclic group A group is locally cyclic if every finitely
Glossary_of_group_theory
Special subset of a partially ordered set
mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear
Filter_(mathematics)
Partially ordered vector space, ordered as a lattice
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice
Riesz_space
Integer whose multiples are digit rotations
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the
Cyclic_number
Group with series of normal subgroups where all factors are cyclic
mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability
Supersolvable_group
Certain topology in mathematics
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,
Order_topology
Well-quasi-ordering of finite trees
states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. A finitary application
Kruskal's_tree_theorem
Special type of lattice
z in L: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty finite joins
Distributive_lattice
Order-preserving mathematical function
mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose
Monotonic_function
Mathematical ranking of a set
generalization of totally ordered sets (rankings without ties) and are in turn generalized by (strictly) partially ordered sets and preorders. There are
Weak_ordering
Set with associative invertible operation
the three axioms. Formally, a group is an ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the
Group_(mathematics)
Number in {..., –2, –1, 0, 1, 2, ...}
\mathbb {Z} } under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to Z {\displaystyle \mathbb {Z}
Integer
Subset of a group that forms a group itself
element a of H generates a finite cyclic subgroup of H, say of order n, and then the inverse of a is an−1. If the group operation is instead denoted by
Subgroup
Visual depiction of a partially ordered set
represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ( S , ≤ ) {\displaystyle
Hasse_diagram
Number representing a continuous quantity
largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group. Ordered fields
Real_number
Nonabelian group in algebraic group theory
that cyclically rotates i, j, and k. One can show that the binary tetrahedral group is isomorphic to the special linear group SL(2,3) – the group of all
Binary_tetrahedral_group
On chains and antichains in partial orders
combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of incomparable elements equals the
Dilworth's_theorem
Nonempty, upper-bounded, downward-closed subset
mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion
Ideal_(order_theory)
Subset of incomparable elements
partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a finite partially ordered set
Antichain
Greek and Marshall Islands containership company
issuing 13.3 million shares 2011 Company continues to expand counter cyclically its fleet, taking delivery of nine second hand vessels and contracting
Costamare
Type of ordering of a set
covering relation is empty. The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real
Dense_order
Group whose operation is a composition of braids
permutation action of the symmetric group on n {\displaystyle n} strands operating on the indices of coordinates. That is, an ordered n {\displaystyle n} -tuple
Braid_group
Branch of mathematics
for partially ordered set is attributed to Garrett Birkhoff in the second edition of his influential book Lattice Theory. Causal sets Cyclic order Hierarchy
Order_theory
Size of subsets in order theory
mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. Formally
Cofinality
System for naming organic chemical compounds
substituents in order of citation (for example: in a cyclic ring with only bromine and chlorine functional groups, alphabetically bromo- is cited before chloro-
IUPAC nomenclature of organic chemistry
IUPAC_nomenclature_of_organic_chemistry
Group that is a topological space with continuous group operations
certain sense assembled from a system of finite groups Ordered topological vector space Topological abelian group Topological field – Algebraic structure with
Topological_group
Mathematical version of an order change
linear order, or the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings)
Permutation
Mathematical concept regarding posets in (partial) order theory
In mathematics, an upwards linked set A is a subset of a partially ordered set, P, in which any two of elements A have a common upper bound in P. Similarly
Linked_set
analysis) – Topology of an ordered vector space Ordered field – Algebraic object with an ordered structure Ordered group – Group with a compatible partial
Ordered topological vector space
Ordered_topological_vector_space
Equivalence of partially ordered sets
partially ordered groups (po-groups) ( G , ≤ G ) {\displaystyle (G,\leq _{G})} and ( H , ≤ H ) {\displaystyle (H,\leq _{H})} , an isomorphism of po-groups from
Order_isomorphism
Isomorphism type of ordered sets
In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there
Order_type
Subset of a preorder that contains all larger elements
In mathematics, an upper set S {\displaystyle S} of a partially ordered set X {\displaystyle X} is a subset such that if s is in S and if x in X is larger
Upper_and_lower_sets
minimal condition. A group can be Noetherian but not Artinian, such as the infinite cyclic group, and unlike for rings, a group can be Artinian but not
Subgroup_series
Number n where n and totient(n) are coprime
definition is that a number n is cyclic if and only if any group of order n is cyclic. Any prime number is clearly cyclic. All cyclic numbers are square-free.
Cyclic_number_(group_theory)
In mathematics, especially order theory, a prefix ordered set generalizes the intuitive concept of a tree by introducing the possibility of continuous
Prefix_order
Term in the mathematical area of order theory
mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This
Duality_(order_theory)
Construction in order theory
order if both A {\displaystyle A} and B {\displaystyle B} are totally ordered. However the product order of two total orders is not in general total;
Product_order
Ordered arrangement of atoms, ions, or molecules in a crystalline material
structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic
Crystal_structure
Quantale Partially ordered monoid Ordered group Archimedean property Ordered ring Ordered field Artinian ring Noetherian Linearly ordered group Monomial order
List_of_order_theory_topics
Mathematical ordering with upper bounds
generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets (contrast partially ordered sets, which need not be directed)
Directed_set
Group that has only one element
being the trivial group { e } {\displaystyle \{\mathrm {e} \}} and the group G {\displaystyle G} itself. The trivial group is cyclic of order 1 {\displaystyle
Trivial_group
Partially ordered set equipped with a rank function
mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all
Graded_poset
odd). Then the poset of cells of L, ordered by the inclusion of their closures, is Eulerian. Let W be a Coxeter group with Bruhat order. Then (W,≤) is an
Eulerian_poset
Mathematical result or axiom on order relations
It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset, where "maximal" is with respect
Hausdorff_maximal_principle
Set whose pairs have minima and maxima
subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called
Lattice_(order)
Number of orderings allowing ties
In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the weak orderings on a set of n {\displaystyle n} elements
Ordered_Bell_number
Algebraic structure with addition, multiplication, and division
do not form a group, since a group must have at least one element. Every finite subgroup of the multiplicative group of a field is cyclic (see Root of
Field_(mathematics)
Partially ordered set in which all subsets have both a supremum and infimum
the trivial group {e} is the minimum subgroup of G, while the maximum subgroup is the group G itself. The ideals of a ring, when ordered by inclusion
Complete_lattice
Mathematical property of subsets in order theory
partially ordered set A {\displaystyle A} admits a totally ordered cofinal subset, then we can find a subset B {\displaystyle B} that is well-ordered and cofinal
Cofinal_(mathematics)
Banach space with a compatible structure of a lattice
that is complete Normed vector lattice Riesz space – Partially ordered vector space, ordered as a lattice Lattice (order) – Set whose pairs have minima and
Banach_lattice
Mathematical ordering of a partial order
partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. A partial order is a
Linear_extension
British rock musician and songwriter (1946–1991)
of Elton John. Shortly before his own death in November 1991, Mercury ordered that a watercolour by John's favourite artist, the 19th-century English
Freddie_Mercury
Type of mathematical object
variables, with relations describing an ordered pair of mutually inverse matrices. For any positive integer n, the group μn of nth roots of unity is the kernel
Group_scheme
There are equally many countable order types and real numbers
can have no higher cardinality. Plotkin, J. M., ed. (2005). Hausdorff on Ordered Sets. History of Mathematics. Vol. 25. American Mathematical Society. p
Cantor–Bernstein_theorem
Mathematical relation inside orderings
mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that
Covering_relation
Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz
Reflexive_closure
Characterizes the height of any finite partially ordered set
combinatorics, Mirsky's theorem characterizes the height of any finite partially ordered set in terms of a partition of the order into a minimum number of antichains
Mirsky's_theorem
Branch of music theory
be ordered cyclically so that the series of intervals between successive notes is the same read forward or backward. For instance, in the cyclical ordering
Set_theory_(music)
Mathematical operation
matrix, assuming rows (top to bottom) and columns (left to right) are ordered alphabetically: ( 1 0 0 0 1 0 0 0 1 1 1 1 ) . {\displaystyle
Composition_of_relations
Theorem in order theory
Miller, E. W. (1940), "Concerning similarity transformations of linearly ordered sets", Bulletin of the American Mathematical Society, 46 (4): 322–326,
Dushnik–Miller_theorem
Topology of an ordered vector space
specifically in order theory and functional analysis, the order topology of an ordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} is the finest locally
Order topology (functional analysis)
Order_topology_(functional_analysis)
Construction in order theory
Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz
Star_product
Existence of certain infima or suprema of a given poset
assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real
Completeness_(order_theory)
order is also important for identifying suitable topologies on partially ordered sets, as is done in order theory. Consider any topological space X {\displaystyle
Specialization_preorder
possesses a neighborhood base at the origin consisting of solid sets. Ordered vector lattices have important applications in spectral theory. If X {\displaystyle
Topological_vector_lattice
Mathematical result on order relations
{\displaystyle R} on a set X {\displaystyle X} is formally defined as a set of ordered pairs ( x , y ) {\displaystyle (x,y)} of elements of X , {\displaystyle
Szpilrajn_extension_theorem
Bound lattice in which every element has a complement
ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice
Complemented_lattice
Elements taken to zero by a homomorphism
abelian groups can be considered a particular kind of module kernel when the underlying ring is the integers. Let G {\displaystyle G} be the cyclic group on
Kernel_(algebra)
Concept in order theory
order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least upper bound) of S , {\displaystyle
Join_and_meet
Reversal of the order of elements of a binary relation
Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relations as sets), and actually an involutive
Converse_relation
Type of monotone function
kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute
Order_embedding
Type of logical relation
Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz
Total_relation
Graph linking pairs of comparable elements in a partial order
comparable to each other in a partial order. For any strict partially ordered set (S,<), the comparability graph of (S, <) is the graph (S, ⊥) of which
Comparability_graph
Lattice formed by all integer partitions
lattice is a lattice (and hence also a partially ordered set) Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers
Young's_lattice
Property of elements related by inequalities
x{\cancel {\overset {<}{\underset {>}{=}}}}y} is true. A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn
Comparability
a well-quasi-ordering. That is, for every infinite sequence of totally-ordered countable sets, there exists an order embedding from an earlier member
Laver's_theorem
CYCLICALLY ORDERED-GROUP
CYCLICALLY ORDERED-GROUP
Girl/Female
African, Arabic, Muslim
Well-ordered; Well-arranged
Boy/Male
Muslim
Ordered, Pasted, Appointed
Girl/Female
Indian
Well-arranged, Well-ordered
Male
Arthurian
, a son of Lot; traitor to Arthur.
Boy/Male
English Arthurian Legend
Brave.
Boy/Male
Indian
Ordered, Pasted, Appointed
Male
English
Old English Arthurian legend name of a Knight of the Round Table who was the illegitimate son and traitor of King Arthur, possibly MORDRED means "sea counsel." He was brother (or half-brother) to Agravain, Gaheris, Gareth, and Gawain, and noted for having crowned himself and married Guinevere while Arthur was waging war on Emperor Lucius of Rome. He was killed by Arthur at the Battle of Camlann.Â
Boy/Male
Indian
Responsibility; Ordered
Boy/Male
African, Indian, Sanskrit
Clear Spoken Person; Ordered
Girl/Female
Muslim
Well-arranged, Well-ordered
Boy/Male
Hindu
Orderly
Girl/Female
Shakespearean
The Tragedy of Macbeth' Lady Macduff, wife to Macduff, murdered on Macbeth's orders.
Boy/Male
Arabic, Australian, Muslim
Ordered; Appointed
Boy/Male
Tamil
Orderly
Boy/Male
Tamil
Mitanshu | மீதாஂஷà¯Â
Bordered, Friendly element
Mitanshu | மீதாஂஷà¯Â
Boy/Male
Hindu, Indian, Telugu
Bordered; Friendly Element
Boy/Male
American, British, Christian, English
Brave; Brave Counselor
Girl/Female
Greek
Murdered Agamemnon.
Surname or Lastname
English (Lancashire)
English (Lancashire) : habitational name from a place in Lancashire, called Ormerod, from the Old Norse personal name Ormr (see Orme 1) or Ormarr (a compound of orm ‘serpent’ + herr ‘army’) + Old English rod ‘clearing’.
Girl/Female
English, Peruvian
Plaster; Powdered
CYCLICALLY ORDERED-GROUP
CYCLICALLY ORDERED-GROUP
Boy/Male
Hindu
Sea
Surname or Lastname
English (Derbyshire)
English (Derbyshire) : variant of Orton.
Boy/Male
Indian, Punjabi, Sikh
Imbued in the Lord's Love
Girl/Female
Muslim
Glow of Moon, Light of the Moon
Boy/Male
Hindu, Indian
Love to God
Boy/Male
Biblical
An assembly.
Girl/Female
Sikh
Girl/Female
Arabic, Australian, Malaysian, Muslim, Parsi
Brilliant Beauty
Boy/Male
Muslim/Islamic
Desiring willing
Boy/Male
Gujarati, Hindu, Indian, Malayalam, Marathi, Oriya, Telugu
Light of the World
CYCLICALLY ORDERED-GROUP
CYCLICALLY ORDERED-GROUP
CYCLICALLY ORDERED-GROUP
CYCLICALLY ORDERED-GROUP
CYCLICALLY ORDERED-GROUP
n.
To give an order to; to command; as, to order troops to advance.
a.
Of or pertaining to a cycle or circle; moving in cycles; as, cyclical time.
a.
Observant of order, authority, or rule; hence, obedient; quiet; peaceable; not unruly; as, orderly children; an orderly community.
v. i.
To give orders; to issue commands.
n.
Right arrangement; a normal, correct, or fit condition; as, the house is in order; the machinery is out of order.
n.
One who puts in order, arranges, methodizes, or regulates.
a.
Alt. of Cyclical
a.
Covered or adorned with osiers; as, osiered banks.
n.
One who gives orders.
a.
Conformed to order; in order; regular; as, an orderly course or plan.
n.
A noncommissioned officer or soldier who attends a superior officer to carry his orders, or to render other service.
n.
To admit to holy orders; to ordain; to receive into the ranks of the ministry.
n.
To give an order for; to secure by an order; as, to order a carriage; to order groceries.
a.
Being on duty; keeping order; conveying orders.
a.
Performed in good or established order; well-regulated.
n.
A number of things or persons arranged in a fixed or suitable place, or relative position; a rank; a row; a grade; especially, a rank or class in society; a group or division of men in the same social or other position; also, a distinct character, kind, or sort; as, the higher or lower orders of society; talent of a high order.
a.
Well-ordered; orderly; regular; methodical.
imp. & p. p.
of Order
adv.
In a cynical manner.
adv.
According to due order; regularly; methodically; duly.