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In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two
Series-parallel_partial_order
Generalised alphabetical order
The result is a partial order. If A {\displaystyle A} and B {\displaystyle B} are each totally ordered, then the result is a total order as well. The lexicographical
Lexicographic_order
On chains and antichains in partial orders
needed to cover all elements. This number is called the width of the partial order. The theorem is named for the mathematician Robert P. Dilworth, who
Dilworth's_theorem
Mathematical set with an ordering
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
Partially_ordered_set
Recursively-formed graph with two terminal vertices
graph Cograph Hanner polytope Series-parallel partial order Eppstein, David (1992). "Parallel recognition of series–parallel graphs" (PDF). Information and
Series–parallel_graph
Types of electrical circuits
(electrical circuits) Resistance distance Series-parallel duality Series-parallel partial order Series and parallel springs Topology (electrical circuits)
Series_and_parallel_circuits
Mathematical ranking of a set
corresponding (non-strict) total order. The reflexive closure of a strict weak ordering is a type of series-parallel partial order. The number of distinct weak
Weak_ordering
Order whose elements are all comparable
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 ≤
Total_order
Mathematical concept for comparing objects
)} , the set of natural numbers with standard ordering, is a well partial order (in fact, a well-order). However, ( Z , ≤ ) {\displaystyle (\mathbb {Z}
Well-quasi-ordering
Topics referred to by the same term
order, in partial order theory Series–parallel graph in graph theory Series–parallel networks problem, a combinatorial problem about series–parallel graphs
Series-parallel
Reflexive and transitive binary relation
relations and (non-strict) partial orders. Both of these are special cases of a preorder: an antisymmetric preorder is a partial order, and a symmetric preorder
Preorder
Order-preserving mathematical function
{\displaystyle \leq } denote the partial order relation of any partially ordered set, a monotone function, also called isotone, or order-preserving, satisfies the
Monotonic_function
Set whose pairs have minima and maxima
requirement that the meet and join semilattices define the same partial order. An order-theoretic lattice gives rise to the two binary operations ∨ {\displaystyle
Lattice_(order)
Well-quasi-ordering of finite trees
reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion). In
Kruskal's_tree_theorem
Branch of mathematics
a and b in P, we have that: a ≤ b or b ≤ a. A partial order with this property is called a total order. These orders can also be called linear orders
Order_theory
Nonempty, upper-bounded, downward-closed subset
and definitions such as "ideal", "order ideal", "Frink ideal", or "partial order ideal" mean one another. Ideals were introduced by Marshall H. Stone
Ideal_(order_theory)
Visual depiction of a partially ordered set
endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order. Hasse diagrams are named after Helmut Hasse (1898–1979); according
Hasse_diagram
Graph formed by complementation and disjoint union
clique-width at most 2. A cograph is a comparability graph of a series-parallel partial order. A cograph is a permutation graph of a separable permutation
Cograph
Special type of lattice
construct a topological space with an additional partial order on its points, yielding a (completely order-separated) ordered Stone space (or Priestley space)
Distributive_lattice
Class of mathematical orderings
the set of possible order types is uncountable. Tree (set theory), generalization Ordinal number Well-founded set Well partial order Prewellordering Directed
Well-order
Construction in order theory
B} , respectively, the product order (also called the coordinatewise order or componentwise order) is a partial order ≤ {\displaystyle \leq } on the Cartesian
Product_order
Isomorphism type of ordered sets
especially in set theory and order 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 incomparable elements
define the height of a partial order to be the maximum cardinality of a chain. Mirsky's theorem states that in any partial order of finite height, the
Antichain
Data structure for permutations
somewhat simpler than the corresponding operations on PQ trees. Series-parallel partial order Booth, Kellogg S.; Lueker, George S. (1976). "Testing for the
PQ_tree
Algebraic object with an ordered structure
higher-order, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings. A field
Ordered_field
Property of a relation on a set
linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that
Connected_relation
Equivalence of partially ordered sets
the order) it would follow that x ≤ y {\displaystyle x\leq y} and y ≤ x {\displaystyle y\leq x} , implying by the definition of a partial order that
Order_isomorphism
Lattice formed by all integer partitions
group. In Young's theory, the objects now called Young diagrams and the partial order on them played a key, even decisive, role. Young's lattice prominently
Young's_lattice
Mathematical ordering of a partial order
order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order
Linear_extension
Set theory concept
theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The
Club_set
Type of monotone function
In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered
Order_embedding
Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive
Reflexive_closure
Partially ordered set in which all subsets have both a supremum and infimum
Both order theory and universal algebra study them as a special class of lattices. Complete lattices must not be confused with complete partial orders
Complete_lattice
Well-founded relation Ordinal number Well-quasi-ordering Semilattice Lattice (Directed) complete partial order, (d)cpo Bounded complete Complete lattice Knaster–Tarski
List_of_order_theory_topics
Derivative of a function with multiple variables
First-order partial derivatives: ∂ f ∂ x = f x ′ = ∂ x f . {\displaystyle {\frac {\partial f}{\partial x}}=f'_{x}=\partial _{x}f.} Second-order partial derivatives:
Partial_derivative
Mathematical operation
semigroup with involution. The composition of (partial) functions (that is, functional relations) is again a (partial) function. If R {\displaystyle R} and S
Composition_of_relations
Alternative mathematical ordering
a cyclic order if it is cyclic, asymmetric, transitive, and connected. Dropping the "connected" requirement results in a partial cyclic order. A set with
Cyclic_order
Binary relation over a set and itself
linear preorder or weak order, is a relation that is reflexive, transitive, and connected. A partial order, also called order,[citation needed] is a relation
Homogeneous_relation
Type of logical relation
is all of X, hence f is a total relation. On the other hand, if f is a partial function, then the domain may be a proper subset of X, in which case f
Total_relation
Certain topology in mathematics
is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given
Order_topology
Type of ordering of a set
In mathematics, a partial order or total order < on a set X {\displaystyle X} is said to be dense if, for all x {\displaystyle x} and y {\displaystyle
Dense_order
Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive
Locally_finite_poset
Characterizes the height of any finite partially ordered set
maximum cardinality of a chain, a totally ordered subset of the given partial order. For instance, in the set of positive integers from 1 to N, ordered
Mirsky's_theorem
Type of topology in mathematics
opposite convention also exists.) The following dictionary holds between order-theoretic notions and topological notions: Open sets are upper sets, Closed
Alexandrov_topology
Type of binary relation
In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order, then
Well-founded_relation
Mathematical property of subsets in order theory
x\in X.} The superset relation ⊇ {\displaystyle \,\supseteq \,} is a partial order on N x {\displaystyle {\mathcal {N}}_{x}} : explicitly, for any sets
Cofinal_(mathematics)
Special subset of a partially ordered set
preordering to associated partial ordering. Historically, filters generalized to order-theoretic lattices before arbitrary partial orders. In the case of
Filter_(mathematics)
Size of subsets in order theory
In 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
Cofinality
Term in the mathematical area of order theory
order isomorphism. Since partial orders are antisymmetric, the only ones that are self-dual are the equivalence relations (but the notion of partial order
Duality_(order_theory)
Property of elements related by inequalities
comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing
Comparability
Smallest transitive relation containing a given binary relation
acyclic graph (DAG) is the reachability relation of the DAG and a strict partial order. The transitive closure of an undirected graph produces a cluster graph
Transitive_closure
Relationship between elements of two sets
asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so
Binary_relation
Algebraic structure modeling logical operations
relation ≤ defined by a ≤ b if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet a ∧ b and the
Boolean_algebra_(structure)
algebra) Free Boolean algebra Monadic Boolean algebra De Morgan algebra First-order logic Heyting algebra Lindenbaum–Tarski algebra Skew Boolean algebra Algebraic
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Existence of certain infima or suprema of a given poset
directed subsets of a poset have a supremum, then the order is a directed-complete partial order (dcpo). These are especially important in domain theory
Completeness_(order_theory)
Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive
Symmetric_closure
Bound lattice in which every element has a complement
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every
Complemented_lattice
Concept in order theory
directed meet or directed infimum. Let A {\displaystyle A} be a set with a partial order ≤ , {\displaystyle \,\leq ,\,} and let x , y ∈ A . {\displaystyle x
Join_and_meet
Mathematical ordering with upper bounds
\supseteq ,\,} define partial orders on any given family of sets. A non-empty family of sets is a directed set with respect to the partial order ⊇ {\displaystyle
Directed_set
Subset of a preorder that contains all larger elements
upper set containing all minimal elements of Y . {\displaystyle Y.} For partial orders satisfying the descending chain condition, antichains and upper
Upper_and_lower_sets
Partially ordered topological space
space X {\displaystyle X} equipped with a closed partial order ≤ {\displaystyle \leq } , i.e. a partial order whose graph { ( x , y ) ∈ X 2 ∣ x ≤ y } {\displaystyle
Partially_ordered_space
Mathematical result or axiom on order relations
collection of sets, the relation "is a proper subset of" is a strict partial order on A. Suppose that A is the collection of all circular regions (interiors
Hausdorff_maximal_principle
Reversal of the order of elements of a binary relation
transitive, connected, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its converse
Converse_relation
Electronic filter circuitry used in loudspeakers
the individual filters are connected in series, and a driver or driver combination is connected in parallel with each filter. To understand the signal
Audio_crossover
Graph linking pairs of comparable elements in a partial order
and order theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability
Comparability_graph
Mathematical measure for partial orders
The partial orders of order dimension two include the series-parallel partial orders (Valdes, Tarjan & Lawler 1982). They are exactly the partial orders
Order_dimension
Numerical ordering with a margin of error
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
Semiorder
ordered sets (or posets), orderings in which some pairs are comparable and others might not be Preorders, a generalization of partial orders allowing ties
List of order structures in mathematics
List_of_order_structures_in_mathematics
Mathematical proposition equivalent to the axiom of choice
The word "partial" is meant to indicate that not every pair of elements of a partially ordered set is required to be comparable under the order relation
Zorn's_lemma
Mathematical result on order relations
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
Szpilrajn_extension_theorem
analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X {\displaystyle X} that has a partial order ≤ {\displaystyle
Topological_vector_lattice
Mathematical operation
composition of relations but is robust to refinement of the relations into (partial) functions or injective relations. Unlike ordinary composition of relations
Demonic_composition
involving Bessel functions or the positivity of Cesàro means of certain Jacobi series. Such functions occur in other areas of mathematics such as probability
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
then a ≤ b or b ≤ a (downward totality). While between partial orders it is usual to consider order-preserving functions, the most important type of functions
Prefix_order
Mathematical relation inside orderings
to graphically express the partial order by means of the Hasse diagram. Let X {\displaystyle X} be a set with a partial order ≤ {\displaystyle \leq } .
Covering_relation
Glossary of terms used in branch of mathematics
articles: completeness properties of partial orders distributivity laws of order theory In the following, partial orders will usually just be denoted by
Glossary_of_order_theory
Algebraic structure used in logic
it is the Heyting algebra of truth values of the intuitionistic higher-order logic induced by the topos. More generally, the set of subobjects of any
Heyting_algebra
a quasi-order.[clarification needed] A partial ranking ≤ ′ {\displaystyle \leq '} of Q {\displaystyle Q} is a well-founded partial ordering of Q {\displaystyle
Better-quasi-ordering
Generalization of the concept of subsequence to the case of nets
A function h : I → A {\displaystyle h:I\to A} is monotone, order-preserving, and an order homomorphism if whenever i ≤ j {\displaystyle i\leq j} then
Subnet_(mathematics)
Type of graph in mathematics
root. The word "multitree" has also been used to refer to a series–parallel partial order, or to other structures formed by combining multiple trees.
Multitree
Partially ordered vector space, ordered as a lattice
vector lattice whose preorder is a partial order. Equivalently, it is an ordered vector space for which the ordering is a lattice. Note that many authors
Riesz_space
axiom, this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little
Specialization_preorder
of an Eulerian poset with a top element, obtained by reversing the partial order, is Eulerian. Richard Stanley defined the toric h-vector of a ranked
Eulerian_poset
Ideals in a Boolean algebra can be extended to prime ideals
distributive lattices and maximal ideals (of order theory). This article focuses on prime ideal theorems from order theory. Although the various prime ideal
Boolean_prime_ideal_theorem
Banach space with a compatible structure of a lattice
X}\|f(x)\|_{Y}{\text{.}}} Then 𝒞(X,Y) is a Banach lattice under the pointwise partial order: f ≤ g ⇔ ( ∀ x ∈ X ) ( f ( x ) ≤ g ( x ) ) . {\displaystyle {f\leq g}\Leftrightarrow
Banach_lattice
Video game series
versions for Windows, and Game Boy Advance. The fourth game in the series, Driver: Parallel Lines, was released 14 March 2006 for PlayStation 2 and Xbox in
Driver_(series)
Construction in order theory
\{{\widehat {1}}\})\cup (Q\setminus \{{\widehat {0}}\})} . We define the partial order ≤ P ∗ Q {\displaystyle \leq _{P*Q}} by x ≤ y {\displaystyle x\leq y}
Star_product
Vector space with a partial order
partially ordered vector space is a real vector space equipped with a partial order that is compatible with the vector space operations. Given a vector
Ordered_vector_space
Localized dielectric breakdown under high voltage stress
capacitance of the void. The parallel capacitor represents the remaining unvoided capacitance of the sample. Whenever partial discharge is initiated, high
Partial_discharge
There are equally many countable order types and real numbers
theory and order theory, the Cantor–Bernstein theorem states that the cardinality of the second type class, the class of countable order types, equals
Cantor–Bernstein_theorem
In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also
Locally_convex_vector_lattice
Uniqueness of countable dense linear orders
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear
Cantor's_isomorphism_theorem
Laver's theorem, in order theory, states that order embeddability of countable total orders is a well-quasi-ordering. That is, for every infinite sequence
Laver's_theorem
Mathematical theorem
derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate function f ( x 1
Symmetry of second derivatives
Symmetry_of_second_derivatives
In mathematics, specifically in order theory and functional analysis, a normed lattice is a topological vector lattice that is also a normed space whose
Normed_vector_lattice
Unsolved problem on partial orders
which each element is incomparable to at most six others, series-parallel partial orders, partial orders whose Hasse diagram is N-free, semiorders, and polytrees
1/3–2/3_conjecture
Partially ordered set equipped with a rank function
ordering, meaning that for all x and y in the order, if x < y then ρ(x) < ρ(y), and The rank is consistent with the covering relation of the ordering
Graded_poset
series-parallel partial orders may also be characterized by four-element forbidden suborders. Every permutation defines a partial order whose order dimension
Separable_permutation
Set theory concept
induces a wellordering on the quotient X / ∼ . {\displaystyle X/{\sim }.} The order-type of this induced wellordering is an ordinal, referred to as the length
Prewellordering
and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) X that has a partial order ≤ making
Ordered topological vector space
Ordered_topological_vector_space
SERIES PARALLEL-PARTIAL-ORDER
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