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Partially ordered set equipped with a rank function
rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value. Graded posets play an important role
Graded_poset
Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which
Eulerian_poset
A strict Sperner poset is a graded poset in which all maximum antichains are rank levels. A strongly Sperner poset is a graded poset which is k-Sperner
Sperner property of a partially ordered set
Sperner_property_of_a_partially_ordered_set
Topics referred to by the same term
with several meanings Graded poset, a partially ordered set equipped with a rank function, sometimes called a ranked poset Graded vector space, a vector
Grade
Partially ordered set in Mathematics
a ranked poset is a partially ordered set in which one of the following (non-equivalent) conditions hold: it is a graded poset, or a poset with the property
Ranked_poset
This poset has a unique minimal element, zero orbit, and unique maximal element, the regular nilpotent orbit, but in general, it is not a graded poset. If
Nilpotent_orbit
Index of articles associated with the same name
areas of mathematics: Functionally graded elements are used in finite element analysis. A graded poset is a poset P {\displaystyle P} with a rank function
Graded_structure
Mathematical set with an ordering
Mathematical phrase Directed set – Mathematical ordering with upper bounds Graded poset – Partially ordered set equipped with a rank function Incidence algebra –
Partially_ordered_set
Visual depiction of a partially ordered set
linear time, if such a diagram exists. In particular, if the input poset is a graded poset, it is possible to determine in linear time whether there is a
Hasse_diagram
Construction in order theory
graded posets with unique minimal and maximal elements, preserving the property that the posets are Eulerian. The star product of two graded posets (
Star_product
differential poset, and in particular to be r-differential (where r is a positive integer), if it satisfies the following conditions: P is graded and locally
Differential_poset
Subset of incomparable elements
to mean strong antichain, a subset such that there is no element of the poset smaller than two distinct elements of the antichain.) A maximal antichain
Antichain
Set whose pairs have minima and maxima
lattice ( L , ≤ ) {\displaystyle (L,\leq )} is called graded, sometimes ranked (but see Ranked poset for an alternative meaning), if it can be equipped with
Lattice_(order)
Join-meet algebra on matroid flats
atomistic if every element is the supremum of some set of atoms. A poset is graded when it can be given a rank function r ( x ) {\displaystyle r(x)} mapping
Geometric_lattice
Group that admits a formal description in terms of reflections
for u as an initial segment. Indeed, the word length makes this into a graded poset. The Hasse diagrams corresponding to these orders are objects of study
Coxeter_group
3 , 3 , 1 ) {\displaystyle \textstyle (1,3,3,1)} . To an arbitrary graded poset P, Stanley associated a pair of polynomials f(P,x) and g(P,x). Their
H-vector
Set theory concept
theory – Subfield of mathematical logic Graded poset – Partially ordered set equipped with a rank function – a graded poset is analogous to a prewellordering
Prewellordering
Poset representing certain properties of a polytope
groups act transitively on the set of flags of the polytope. Eulerian poset Graded poset Regular polytope McMullen & Schulte 2002, p. 21-25 McMullen & Schulte
Abstract_polytope
Glossary of terms used in branch of mathematics
sets is open. Algebraic poset. A poset is algebraic if it has a base of compact elements. Antichain. An antichain is a poset in which no two elements
Glossary_of_order_theory
Branch of mathematics
attention to the logical importance of asymmetric relations." The term poset as an abbreviation for partially ordered set is attributed to Garrett Birkhoff
Order_theory
Geometric arrangements of points, foundational to Lie theory
-\alpha } is a nonnegative linear combination of simple roots. This poset is graded by deg ( ∑ α ∈ Δ λ α α ) = ∑ α ∈ Δ λ α {\textstyle \deg \left(\sum
Root_system
Pseudonym for a group of mathematicians
Stanley defined a Peck poset to be a graded partially ordered set that is rank symmetric, rank unimodal, and strongly Sperner. The posets in the original paper
G._W._Peck
Existence of certain infima or suprema of a given poset
existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special
Completeness_(order_theory)
Nonempty, upper-bounded, downward-closed subset
order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring
Ideal_(order_theory)
Mathematical system of orderings or sets
{\displaystyle {\bigl \{}\emptyset ,\{a\},\{a,b\},\{a,b,c\},\{a,b,c,d\}{\bigr \}}.} Poset antimatroids The lower sets of a finite partially ordered set form an antimatroid
Antimatroid
functor M : T → V e c K {\displaystyle M:T\to \mathbf {Vec} _{K}} from the poset category of T {\displaystyle T} to the category of vector spaces over K
Persistence_module
Special subset of a partially ordered set
filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and
Filter_(mathematics)
poset is a partially ordered set P such that for all x, y ∈ P, the interval [x, y] consists of finitely many elements. Given a locally finite poset P
Locally_finite_poset
Lattice formed by all integer partitions
which is indexed by the standard Young tableaux of shape p. The poset Y is graded: the minimal element is ∅, the unique partition of zero, and the partitions
Young's_lattice
Partially ordered set in which all subsets have both a supremum and infimum
generated from a given poset used in place of the set of generators considered above, then one speaks of a completion of the poset. The definition of the
Complete_lattice
Mathematical result or axiom on order relations
immediate corollary is Zorn's Lemma, that if every chain of a poset has an upper bound then the poset contains a maximal element, namely the upper bound of a
Hausdorff_maximal_principle
In mathematics, dimension of a ring
for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic
Krull_dimension
Subset of a preorder that contains all larger elements
The set of all lower sets of a given poset P {\displaystyle P} may be ordered by inclusion. The resulting poset, denoted J ( P ) {\displaystyle J(P)}
Upper_and_lower_sets
Equivalence of partially ordered sets
a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially
Order_isomorphism
Type of monotone function
{\displaystyle y\leq x} . If an order embedding exists from a poset S {\displaystyle S} to a poset T {\displaystyle T} , one says that S {\displaystyle S} can
Order_embedding
Graph linking pairs of comparable elements in a partial order
37–46, doi:10.1016/0012-365X(83)90019-5. Jung, H. A. (1978), "On a class of posets and the corresponding comparability graphs", Journal of Combinatorial Theory
Comparability_graph
Theorem on the largest antichain of sets
theorem for subsets of P ( E ) , {\displaystyle {\mathcal {P}}(E),} the poset of all subsets of E {\displaystyle E} . A chain is a subfamily { S 0 , S
Sperner's_theorem
Mathematical proposition equivalent to the axiom of choice
each chain in a poset P {\displaystyle P} has an upper bound, then P {\displaystyle P} has a maximal element. If each chain in a poset P {\displaystyle
Zorn's_lemma
three order relations a ≤ b ≥ c ≤ d is an example of a fence or zigzag poset; its Hasse diagram has the shape of the capital letter "N". It is not series-parallel
Series-parallel_partial_order
Generalization of the indicator function for classical sets in fuzzy logic
needed]; usually it is required that L {\displaystyle L} be at least a poset or lattice. The usual membership functions with values in [0, 1] are then
Membership function (mathematics)
Membership_function_(mathematics)
Order whose elements are all comparable
S2CID 38115497. Ganapathy, Jayanthi (1992). "Maximal Elements and Upper Bounds in Posets". Pi Mu Epsilon Journal. 9 (7): 462–464. ISSN 0031-952X. JSTOR 24340068
Total_order
Discrete math concept
> qi. The poset of partitions of n is linearly ordered (and is equivalent to lexicographical ordering) if and only if n ≤ 5. It is graded if and only
Dominance_order
Structure dual to a unital associative algebra
ε) is a coalgebra known as trigonometric coalgebra. For a locally finite poset P with set of intervals J, define the incidence coalgebra C with J as basis
Coalgebra
248-dimensional exceptional simple Lie group
"vector" representation then lies, not in this nonnegative-graded exterior algebra, but in the graded algebra of derivations over the exterior algebra; the
E8_(mathematics)
Mathematical concept regarding posets in (partial) order theory
Distributive Join and meet Partially ordered set Chain-complete Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice
Linked_set
Size of subsets in order theory
with m {\displaystyle m} elements are maximal. Thus the cofinality of this poset is n {\displaystyle n} choose m . {\displaystyle m.} A subset of the natural
Cofinality
Arithmetic operation
(1991), p. 75. Enderton (1977), p. 79. For a version that applies to any poset with the descending chain condition, see Bergman (2005), p. 100 Enderton
Addition
Mathematical ordering with upper bounds
required explicitly. A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if its downward closure
Directed_set
Sets whose elements have degrees of membership
been provided by Gottwald, S. (2010). "An early approach toward graded identity and graded membership in set theory". Fuzzy Sets and Systems. 161 (18): 2369–2379
Fuzzy_set
Mathematical ordering of a partial order
infinite poset N × N {\displaystyle \mathbb {N} \times \mathbb {N} } . Similarly, standard Young tableaux can be considered as linear extensions of a poset corresponding
Linear_extension
Mathematical result on order relations
to this poset. Zorn's lemma states that a partial order in which every chain has an upper bound has a maximal element. A chain in this poset is a set
Szpilrajn_extension_theorem
Mathematical property of subsets in order theory
sets ("posets") is reflexive: every poset is cofinal in itself. It is also transitive: if B {\displaystyle B} is a cofinal subset of a poset A , {\displaystyle
Cofinal_(mathematics)
Analysis of datasets using techniques from topology
distance. In fact, the interleaving distance is the terminal object in a poset category of stable metrics on multidimensional persistence modules in a
Topological_data_analysis
Special type of lattice
distributive lattice is isomorphic to the lattice of lower sets of the poset of its join-prime (equivalently: join-irreducible) elements. This establishes
Distributive_lattice
Construction in order theory
Kim, Hee Sik (1998), "4.2 Product Order and Lexicographic Order", Basic Posets, World Scientific, pp. 64–78, ISBN 9789810235895 Sudhir R. Ghorpade; Balmohan
Product_order
completion Ideal completion Way-below relation Continuous poset Continuous lattice Algebraic poset Scott domain Algebraic lattice Scott information system
List_of_order_theory_topics
Set theory concept
-complete proper filter on the set κ {\displaystyle \kappa } ; that is, on the poset ( ℘ ( κ ) , ⊆ ) {\displaystyle (\wp (\kappa ),\subseteq )} . If κ {\displaystyle
Club_set
Graded lattice with modular maximal chain
lattice requires the condition of (2), but relaxes the requirement of gradedness. A group is supersolvable if and only if its lattice of subgroups is supersolvable
Supersolvable_lattice
Concept in order theory
element with another element is the other element. Thus every pair in this poset has both a meet and a join and the poset can be classified as a lattice.
Join_and_meet
Partial order with joins
in terms of the existence of suitable Galois connections between related posets—an approach of special interest for category theoretic investigations of
Semilattice
Ideals in a Boolean algebra can be extended to prime ideals
set. If the considered partially ordered set (poset) has binary suprema (a.k.a. joins), as do the posets within this article, then this is equivalently
Boolean_prime_ideal_theorem
Term in the mathematical area of order theory
sets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other. The importance of this simple
Duality_(order_theory)
been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be Preorders
List of order structures in mathematics
List_of_order_structures_in_mathematics
Partial order in lattice theory
particular via Topkis' Theorem. Given a lattice X {\displaystyle X} , a poset Θ {\displaystyle \Theta } , a constraint correspondence D : Θ ⇉ X {\displaystyle
Strong_set_order
Class of mathematical orderings
An initial segment, determined by an element x {\displaystyle x} of a poset X {\displaystyle X} , is a subset of form { y ∈ X ∣ y < x } {\displaystyle
Well-order
Mathematical relation inside orderings
supersedes or succeeds x {\displaystyle x} in terms of their respective poset's order relation. When x ⋖ y {\displaystyle x\lessdot y} , it is said that
Covering_relation
include enumeration of P-partitions, permutations, tableaux, chains of posets, reduced decompositions in finite Coxeter groups (via Stanley symmetric
Quasisymmetric_function
History of maths
surjection followed by an injection. Examples are the ordinal α considered as a poset and hence a category. The opposite R° of a Reedy category R is also a Reedy
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
pelâk plaque plastique پلاستیک plâstik / pelâstik plastic pochette پوشت pošet breast pocket handkerchief point پوئن poan score, point pommade پماد pomâd
List of French loanwords in Persian
List_of_French_loanwords_in_Persian
travel, tourism, insurance
GRADED POSET
GRADED POSET
Girl/Female
Latin American English Irish
Grace.
Boy/Male
Gaelic
noble.
Surname or Lastname
English
English : variant of Grace.
Boy/Male
Australian, Gaelic, Irish
Noble; Renowned
Surname or Lastname
Swedish
Swedish : unexplained.German : unexplained.English : unexplained.
Surname or Lastname
English
English : patronymic from Grave 1.French : topographic name from the plural of Old French grave ‘gravel’ (see Grave).
Surname or Lastname
Northern Irish
Northern Irish : reduced form of McGlade.English : topographic name for someone who lived in a glade, Middle English glade.English : from an Old English personal name Glæd.German (also Gläde) : nickname for a handsome man, from Middle Low German glad(de) ‘smooth’, ‘shining’.
Surname or Lastname
English
English : from Old French grateor, gratour, gratier ‘one who grates’, hence possibly an occupational name for a furbisher.German (Gräter) : see Graeter.
Surname or Lastname
English
English : unexplained.Possibly an Americanized form of German Grauer.Alternatively, perhaps a respelling of French Gruyer, an occupational name from Old French gruier ‘forester’.
Surname or Lastname
English
English : occupational name for an engraver, from Old English grafere, græfere ‘engraver’, ‘sculptor’ (Old French graveur). It is possible that the name was also an occupational name for a miner, from Old English grafan ‘to dig’.German (also Gräver) : variant of Graber.
Surname or Lastname
English
English : variant of Greeley.Possibly an Americanized form of German Greulich.
Girl/Female
German, Teutonic
Guarded
Boy/Male
American, British, English
Gray-haired; Son of the Gray Family; Son of Gregory
Male
English
English surname transferred to forename use, from an Anglicized form of Irish Gaelic Ó Bradain, BRADEN means "descendant of Bradán," hence "salmon."
Surname or Lastname
English
English : nickname from Middle English, Old French grace ‘charm’, ‘pleasantness’ (Latin gratia).English : from the female personal name Grace, which was popular in the Middle Ages. This seems in the first instance to have been from a Germanic element grīs ‘gray’ (see Grice 1), but was soon associated by folk etymology with the Latin word meaning ‘charm’.
Surname or Lastname
English
English : variant of Gladden.
Girl/Female
American, Arabic, Australian, British, Chinese, Christian, Danish, English, French, German, Gujarati, Indian, Irish, Jamaican, Latin, Muslim, Portuguese, Swedish
Mercy; God's Favor; Grace; Grace of God; Kindness; Thanks; Love; Favour; Blessing; Charm; Good will
Boy/Male
Muslim
One who drives a boat
Surname or Lastname
English (East Anglia)
English (East Anglia) : perhaps a habitational name from a house bearing the sign of a bunch of grapes. The vocabulary word is attested from the 13th century (at first in the compound wingrape), and comes from Old French grape, which is probably related to a Germanic element meaning ‘hook’.
Surname or Lastname
English
English : metonymic occupational name for a gardener, from Old Anglo-Norman French gardin ‘garden’. Compare Gardener.Americanized form of French Desjardins.
GRADED POSET
GRADED POSET
GRADED POSET
GRADED POSET
GRADED POSET
GRADED POSET
GRADED POSET
travel, tourism, insurance