Subset of a preorder that contains all larger elements
A Hasse diagram of the divisors of , ordered by the relation is divisor of, with the upper set colored green. The white sets form the lower set
In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X)[1] of a partially ordered set is a subset with the following property: if s is in S and if x in X is larger than s (that is, if ), then x is in S. In other words, this means that any x element of X that is to some element of S is necessarily also an element of S.
The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset S of X with the property that any element x of X that is to some element of S is necessarily also an element of S.
In mathematics, an upperset (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set ( X , ≤ ) {\displaystyle...
mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K that is greater than...
than any other lower bound). Similarly, an upper bound of a subset S {\displaystyle S} of a partially ordered set ( P , ≤ ) {\displaystyle (P,\leq )} is an...
definition. A directed set is a set A {\displaystyle A} with a preorder such that every finite subset of A {\displaystyle A} has an upper bound. In this definition...
descriptions as a fallback Directed set – Mathematical ordering with upper bounds Graded poset – partially ordered set equipped with a rank function, sometimes...
non-empty totally ordered set with no upper bound. The integers form an initial non-empty totally ordered set with neither an upper nor a lower bound. The...
{\displaystyle X} x ∈ X {\displaystyle x\in X} The upper contour set of x {\displaystyle x} is the set of all y {\displaystyle y} that are related to x...
A somewhat weaker notion of density is the upper Banach density d ∗ ( A ) {\displaystyle d^{*}(A)} of a set A ⊆ N . {\displaystyle A\subseteq \mathbb {N}...
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...
terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory (Pawlak 1991)...
Formally, the upper closure of a set S in a poset P is given by the set {x in P | there is some y in S with y ≤ x}. A set that is equal to its upper closure...
The Upper Deck Company, LLC (colloquially as Upper Deck and Upper Deck Authenticated, Ltd. in the UK), founded in 1988, is a private company primarily...
has an upper bound. The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered sets without...
monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus...
associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X {\displaystyle X} and...
Upper class in modern societies is the social class composed of people who hold the highest social status, usually are the wealthiest members of class...
P is called Scott-open if it is an upperset and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have non-empty...
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...
ordered data set, split this data set exactly in half. The lower quartile value is the median of the lower half of the data. The upper quartile value...
with an upper triangular matrix is not necessarily triangular either. The set of unitriangular matrices forms a Lie group. The set of strictly upper (or lower)...
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