In mathematics, an upper set (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...

only if every upper set is also open (or equivalently every lower set is also closed). Let A be a subset of X. The smallest upper set containing A is...

a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in...

well-ordered set S 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...

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...

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...

order – Mathematical phrase Directed set – Mathematical ordering with upper bounds Graded poset – partially ordered set equipped with a rank functionPages...

Lower Egypt, and ruled the Nile Delta, from Avaris. They chose Set, originally Upper Egypt's chief god, the god of foreigners and the god they found...

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}...

drive train is reconfigured to connect to the upper set of pedals, and the controls are moved to the upper handlebar area. Alternatively, a bicycle can...

P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have non-empty...

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...

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...

subset S of F is a base or basis for F if the upper set generated by S (i.e., the smallest upwards-closed set containing S) is equal to F. Since every filter...

{\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...

relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty subset S ⊆ X has a minimal...

related types, through the difference in spans between its wings – the upper set being of much greater span than the lower. In 1926, the Ministry of National...

For a preordered set P, any upper set O is Alexandrov-open. Inversely, a topology is Alexandrov if any intersection of open sets is open. Algebraic...

and upper semicontinuity and set-valued lower and upper semicontinuouty. An upper semicontinuous single-valued function is not necessarily upper semicontinuous...

has an upper bound. The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered sets without...

preorder ≤ on a set X, there is a unique Alexandrov topology on X for which the specialization preorder is ≤. The open sets are just the upper sets with respect...

The trading card game Magic: The Gathering has released a large number of sets since it was first published by Wizards of the Coast. After the 1993 release...

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 described in...

monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus...

are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and Yoneda's representability theorem generalizes Cayley's...

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...

dual concepts are upper and lower bounds, lower sets and upper sets, and ideals and filters. In topology, open sets and closed sets are dual concepts:...

In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton { a } {\displaystyle \{a\}}...