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Property of a partially ordered set
mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) is a fundamental property of the real numbers
Least-upper-bound_property
Greatest lower bound and least upper bound
equal to b. Consequently, the supremum is also referred to as the least upper bound (or LUB). The infimum is, in a precise sense, dual to the concept
Infimum_and_supremum
Nonexistence of gaps in the number line
least-upper-bound property states that every nonempty subset of real numbers having an upper bound (or bounded above) must have a least upper bound (or
Completeness of the real numbers
Completeness_of_the_real_numbers
the least upper bound property. It can be proved as follows: Let S be a non-empty subset of R ′ {\displaystyle \mathbb {R} '} and U be an upper bound for
Construction of the real numbers
Construction_of_the_real_numbers
Mathematics of real numbers and real functions
equivalent ways, one of which is the least upper bound property. This states that if a non-empty set of real numbers is bounded above, meaning that all of its
Real_analysis
In mathematics, a generalization of the real line
nonempty subset with an upper bound has a least upper bound in the set. More symbolically: S has the least upper bound property, and For each x in S and
Linear_continuum
Mathematical property of algebraic structures
numbers. The Archimedean property of real numbers holds also in constructive analysis, even though the least upper bound property may fail in that context
Archimedean_property
Method of construction of the real numbers
Dedekind cuts has the least-upper-bound property, i.e., every nonempty subset of it that has any upper bound has a least upper bound. Thus, constructing
Dedekind_cut
Term in mathematics
completeness property of the real numbers has several useful characterisations: The least-upper-bound property The greatest-lower-bound property The nested
Characterization (mathematics)
Characterization_(mathematics)
Partial order on matrices
The Loewner order does not have the least-upper-bound property, and therefore does not form a lattice. It is bounded: for any finite set S {\displaystyle
Loewner_order
Alternative decimal expansion of 1
completeness axiom, which states that every bounded sequence has a least upper bound. This least upper bound is one way to define infinite decimal expansions:
0.999...
Form of logic that allows quantification over predicates
second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has
Second-order_logic
Mathematical relation making a non-equal comparison
in P such that a < c < b. Least-upper-bound property: Every non-empty subset of P with an upper bound has a least upper bound (supremum) in P. If (F, +
Inequality_(mathematics)
Used to count, measure, and label
real numbers have an important but highly technical property called the least upper bound property. It can be shown that any complete, ordered field is
Number
Continuous real function on a closed interval has a maximum and a minimum
R : y = f(x) for some x ∈ [a,b]} is a bounded set. Hence, its least upper bound exists by least upper bound property of the real numbers. Let M = sup(f(x)) on [a
Extreme_value_theorem
All numbers between two given numbers
endpoint belongs to the interval. This is a consequence of the least-upper-bound property of the real numbers, which implies that if the elements of a non-empty
Interval_(mathematics)
Line formed by the real numbers
<, and this ordering is dense and has the least-upper-bound property. In addition to the above properties, the real line has no maximum or minimum element
Number_line
Branch of mathematics
classical analysis. Their completeness, often expressed by the least-upper-bound property, underlies basic results about limits, continuity, differentiation
Mathematical_analysis
Ordered field that does not satisfy the Archimedean property
field. Sometimes the term "complete" is used to mean that the least upper bound property holds, i.e. for Dedekind-completeness. There are no Dedekind-complete
Non-Archimedean_ordered_field
Rational number Irrational number Completeness of the real numbers Least-upper-bound property Real line Extended real number line Dedekind cut 0 1 0.999...
List_of_real_analysis_topics
Existence of certain infima or suprema of a given poset
motivation for considering completeness properties derives from the great importance of suprema (least upper bounds, joins, " ∨ {\displaystyle \vee }
Completeness_(order_theory)
total order. If M {\displaystyle M} has the least-upper-bound property (or greatest-lower-bound property) then the set of constraints also have it. The
Difference_bound_matrix
Bohemian polymath (1781–1848)
mathematical limit. Bolzano was the first to recognize the greatest lower bound property of the real numbers. Like several others of his day, he was skeptical[dubious
Bernard_Bolzano
Set whose pairs have minima and maxima
a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by
Lattice_(order)
List of scientists who are Christians
mathematical analysis, including the (ε, δ)-definition of limit, the least upper bound property of the real numbers, and the Bolzano–Weierstrass theorem. He also
List of Christians in science and technology
List_of_Christians_in_science_and_technology
Algebraic ring that need not have additive negative elements
monoid is a continuous monoid. That is, partially ordered with the least upper bound property, and for which addition and multiplication respect order and suprema
Semiring
German mathematician (1885–1955)
and moreover, that asking about the truth or falsity of the least upper bound property of the real numbers was as meaningful as asking about truth of
Hermann_Weyl
Concept in mathematics
real numbers. This example also demonstrates that the existence of a least upper bound (the number 0 in this case) does not imply the existence of a greatest
Greatest element and least element
Greatest_element_and_least_element
Unsolved geometry problem
planar set of diameter one. The diameter of a set by definition is the least upper bound of the distances between all pairs of points in the set. A shape covers
Lebesgue's universal covering problem
Lebesgue's_universal_covering_problem
Subset of a preorder that contains all larger elements
J(P)} has a least upper bound and a greatest lower bound), and indeed a distributive lattice (meaning that the two operations of least upper bound and greatest
Upper_and_lower_sets
First article on transfinite set theory
prove they exist. This principle (which is equivalent to the least upper bound property of the real numbers) comes from Dedekind's construction of the
Cantor's first set theory article
Cantor's_first_set_theory_article
Ranges of numbers contained in each other
a_{2},\dots \}} is bounded from above, where every b n {\displaystyle b_{n}} is an upper bound. This implies, that the least upper bound s = sup ( A ) {\displaystyle
Nested_intervals
Concept in philosophy and mathematics
divisibility does not imply gaplessness: the rationals do not enjoy the least upper bound property. That means that if one were to partition the rationals into two
Infinite_divisibility
Mathematical field
hence do not have the least upper bound property. We can address this by associating a sequence with the least upper bound of minimal complexity, analogously
Transseries
Branch of mathematics
fact, this upper bound is quite special: it is the smallest set that contains all of the sets. Hence, we have found the least upper bound of a set of
Order_theory
Collection of mathematical objects of finite size
is called an upper bound of S. The terms bounded from below and lower bound are similarly defined. A set S is bounded if it has both upper and lower bounds
Bounded_set
Mathematical proposition equivalent to the axiom of choice
following two properties: P {\displaystyle P} is nonempty; Every chain in P has an upper bound in P. Then P {\displaystyle P} has at least one maximal element
Zorn's_lemma
Partial order with joins
mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually
Semilattice
Limits ideals to be checked in order to determine the class number of a number field
In algebraic number theory, Minkowski's bound gives an upper bound of the norm of ideals to be checked in order to determine the class number of a number
Minkowski's_bound
Mathematical inequality
w)} an upper bound on g ( z ) {\displaystyle g(z)} for any choice of w ∈ W {\displaystyle w\in W} . Because the supremum is the least upper bound, sup z
Max–min_inequality
Number representing a continuous quantity
an upper bound in R {\displaystyle \mathbb {R} } has a least upper bound (a.k.a., supremum) in R {\displaystyle \mathbb {R} } . The last property applies
Real_number
Family of error-correcting codes that encode data in blocks
n-1} \over 2}\right\rfloor }{\binom {n}{i}}(q-1)^{i}\right]} The Singleton bound is that the sum of the rate and the relative distance of a block code cannot
Block_code
Glossary of terms used in branch of mathematics
a least element and a greatest element. Bounded complete. A poset is bounded complete if every of its subsets with some upper bound also has a least such
Glossary_of_order_theory
Relation between properties and composition of a compound
used to predict various properties of a composite material . It provides a theoretical upper- and lower-bound on properties such as the elastic modulus
Rule_of_mixtures
Join-meet algebra on matroid flats
y} have both a least upper bound, called the join or supremum, denoted by x ∨ y {\displaystyle x\vee y} , and a greatest lower bound, called the meet
Geometric_lattice
Scheduling technique in computer science
a higher bound. Kuo and Mok showed that for a task set made up of K harmonic task subsets (known as harmonic chains), the least upper bound test becomes:
Rate-monotonic_scheduling
atomistic (not to be confused with atomic) if every element is the least upper bound of a set of atoms. The linear order with three elements is not atomistic
Atom_(order_theory)
Branch of mathematics relating to posets
converge, i.e. in orders in which all directed sets have a least upper bound. This property defines the class of directed-complete partial orders, or dcpo
Domain_theory
opened and the coming item is placed inside this new bin. Next-Fit is a bounded space algorithm - it requires only one partially-filled bin to be open
Next-fit_bin_packing
Matrix property in linear algebra
linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. The
Restricted_isometry_property
Fragment of metric temporal logic
lower bound of each interval is 0 or the upper bound is infinity. Similarly we denote by L0 (respectively, L∞) the subset of L such that the lower bound of
Metric interval temporal logic
Metric_interval_temporal_logic
unit), Least element (minimum, bottom, zero) Maximal element, minimal element Upper bound Least upper bound (supremum, join) Greatest lower bound (infimum
List_of_order_theory_topics
Mathematical set containing no elements
)=+\infty .} That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive
Empty_set
Smallest unit of a chemical element
of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other
Atom
Collection of subsets covering all t-element subsets
where the Schönheim lower bound gives density asymptotic to v / 4 {\displaystyle v/4} while the Erdős–Spencer upper bound gives density asymptotic to
Covering_design
Problem in computer science
a randomized upper bound of O ( log 1.5 n ) {\displaystyle O(\log ^{1.5}n)} and subsequently gave a randomized amortized upper bound of O ( log n
List-labeling_problem
Exponentially decreasing bounds on tail distributions of random variables
In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function
Chernoff_bound
Number of independent rational basis points with infinite order
one can obtain an upper bound of 2.3 {\displaystyle 2.3} for the average rank. Heath-Brown showed that one can obtain an upper bound of 2 {\displaystyle
Rank_of_an_elliptic_curve
Topic in computer science
general notions of property testing in the context of graphs, we say a tester for graph property P should distinguish with at least two-thirds probability
Property_testing
Order whose elements are all comparable
the real numbers a property of the relation ≤ is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum)
Total_order
Class of mathematical orderings
contains for every subset T with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of T in S. If ≤ is a non-strict
Well-order
a partially ordered set is bounded complete if all of its subsets that have some upper bound also have a least upper bound. Such a partial order can also
Bounded_complete_poset
Type of binary search tree
Once we find an upper bound on the performance of the tango tree, we can divide them to bound the competitive ratio. To find a lower bound on the work done
Tango_tree
Branch of mathematical combinatorics
original structure be in order to ensure that at least one of the pieces has a given interesting property? This idea can be defined as partition regularity
Ramsey_theory
Theorem in order and lattice theory
transfinite induction: f α+1 = f (f α) and f γ for a limit ordinal γ is the least upper bound of the f β for all β ordinals less than γ. The dual theorem holds
Knaster–Tarski_theorem
Concept in probability theory
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some
Markov's_inequality
Property of geometry, also used to generalize the notion of "distance" in metric spaces
between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result
Triangle_inequality
Algorithm
satisfy this property, then B* will identify an optimal path to the goal state. To back up the intervals within the tree, a parent's upper bound is set to
B*
Partially ordered set in which all subsets have both a supremum and infimum
(meet). A conditionally complete lattice satisfies at least one of these properties for bounded and nonempty subsets. For comparison, in a general lattice
Complete_lattice
Min-max theorem in graph theory
that the family F consisting of all cycles has the Erdős–Pósa property, with bounding function f(k) = Θ(k log k). Robertson and Seymour (1986) generalized
Erdős–Pósa_theorem
non-negative integers is also known as a bound function, because in this case it provides a trivial upper bound on the number of iterations of a loop before
Loop_variant
Geometry of the location of polynomial roots
of the polynomial. Some of these geometrical properties are related to a single polynomial, such as upper bounds on the absolute values of the roots, which
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Unsolved geometry problem about planar regions
Norwood, Poole & Laidacker (1992) gave weaker upper bounds. In the convex case, Wang (2006) improved an upper bound to 0.270911861. Khandhawit, Pagonakis &
Moser's_worm_problem
Concept in theoretical computer science
\uparrow } 5. 47176870 is an upper bound for space(5), because S(5) = 47176870 () and S(n) ≥ space(n). 4098 is an upper bound for num(5), because Σ(5) =
Busy_beaver
Mathematical set with an ordering
not have a least element, but any prime number is a minimal element for it. In this poset, 60 is an upper bound (though not a least upper bound) of the subset
Partially_ordered_set
Method in combinatorics
there are few containers), and we can upper bound their size in an essentially optimal way using some simple properties of the hypergraph. We recall the following
Container_method
Points with no three in a line
of this upper bound in the Lean theorem prover. As of March 2023, there is no exponential improvement to Ellenberg and Gijswijt's upper bound. Jiang showed
Cap_set
Statement in mathematical combinatorics
An upper bound for R(r, s) can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first exponential lower bound was
Ramsey's_theorem
Mathematical analysis
analysis does not prove the least-upper-bound principle, i.e. that any subset of the real line R would have a least upper bound (or supremum), possibly infinite
Constructive_analysis
Mathematical optimization concept
the coefficients of x in the constraints are at least cT. This linear combination gives us an upper bound on the objective. The variables y of the dual
Dual_linear_program
Regular graph with girth more than twice its diameter
− 1 ( d − 1 ) i , {\displaystyle 1+d\sum _{i=0}^{k-1}(d-1)^{i},} an upper bound on the largest possible number of vertices in any graph with this degree
Moore_graph
Number denoting a graph's closeness to a tree
said to have bounded local treewidth, or the diameter-treewidth property, if the treewidth of the graphs in the family is upper bounded by a function
Treewidth
Describes approximate behavior of a function
description of a function in terms of big O notation only provides an upper bound on the growth rate of the function. Associated with big O notation are
Big_O_notation
Infinite cardinal number
useful properties of the set ω 1 {\displaystyle \omega _{1}} : Any countable subset of ω 1 {\displaystyle \omega _{1}} has an upper bound in ω 1 {\displaystyle
Aleph_number
Type of statistical probability
It may also be of interest to derive a 95% upper confidence bound for the median air lead level. Such a bound for μ {\displaystyle \mu } is given by X ¯
Tolerance_interval
Problem in game theory
the other countries. This implies an upper bound on the utility of each coalition, called the aspiration upper bound. There is at most one welfare-distribution
Fair_river_sharing
z_{R}} . Therefore, x ∗ ∈ X R {\displaystyle x^{*}\in X_{R}} provides an upper bound on z R {\displaystyle z_{R}} . If in addition to the previous assumptions
Relaxation_(approximation)
is said to have the bounded variability property if and only if γ ′ {\displaystyle \gamma '} has the bounded variability property. A set of signal is
Signal_(model_checking)
Partially ordered vector space, ordered as a lattice
is the least upper bound or supremum (resp. greater lower bound or infimum) of S {\displaystyle S} if it is an upper bound (resp. a lower bound) of S {\displaystyle
Riesz_space
Fixed-point theorem
non-empty poset that is chain complete, meaning each chain has a least upper bound, and f : X → X {\displaystyle f:X\to X} is a function such that f
Bourbaki–Witt_theorem
Number of intersection points of algebraic curves and hypersurfaces
provides only an upper bound of the number of points, which is almost always reached. This bound is often referred to as the Bézout bound. Bézout's theorem
Bézout's_theorem
Subset of Euclidean space is compact if and only if it is closed and bounded
the Heine–Borel property. A metric space ( X , d ) {\displaystyle (X,d)} is said to have the Heine–Borel property if each closed, bounded set in X {\displaystyle
Heine–Borel_theorem
Nature reserve in California, United States
The Upper Las Virgenes Canyon Open Space Preserve is a large open space nature preserve owned and operated by the Santa Monica Mountains Conservancy spanning
Upper Las Virgenes Canyon Open Space Preserve
Upper_Las_Virgenes_Canyon_Open_Space_Preserve
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
application of B − 1 {\textstyle B^{-1}} . Minkowski's theorem gives an upper bound for the length of the shortest nonzero vector. This result has applications
Minkowski's_theorem
Proposition on the domination number of Cartesian products of graphs
dominate 15 of the 16 vertices. Therefore, at least four vertices are required to dominate the entire graph, the bound given by Vizing's conjecture. It is possible
Vizing's_conjecture
Theorem in mathematics
that x 1 < ⋯ < x n , {\displaystyle x_{1}<\cdots <x_{n},} then: The upper bound in (1) is attained only for permutations σ {\displaystyle \sigma } that
Rearrangement_inequality
Unsolved problem in mathematics
MR 2978700. Burek, Dominik; Żmija, Błażej (2019). "A new upper bound for numbers with the Lehmer property and its application to repunit numbers". International
Lehmer's_totient_problem
Shape that blocks all lines of sight
The shortest opaque set for any bounded convex set in the plane has length at most the perimeter of the set, and at least half the perimeter. For the square
Opaque_set
Mathematical ordering with upper bounds
filtered set) is a preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set A {\displaystyle A}
Directed_set
Mathematical result or axiom on order relations
that if every chain of a poset has an upper bound then the poset contains a maximal element, namely the upper bound of a maximal chain. The Hausdorff maximal
Hausdorff_maximal_principle
LEAST UPPER-BOUND-PROPERTY
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LEAST UPPER-BOUND-PROPERTY
LEAST UPPER-BOUND-PROPERTY
LEAST UPPER-BOUND-PROPERTY
LEAST UPPER-BOUND-PROPERTY
LEAST UPPER-BOUND-PROPERTY
LEAST UPPER-BOUND-PROPERTY
LEAST UPPER-BOUND-PROPERTY