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...

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...

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...

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...

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...

descriptions as a fallback Directed set – Mathematical ordering with upper bounds Graded poset – partially ordered set equipped with a rank functionPages...

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...

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...

{\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...

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...

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...

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...

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

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

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:...

The Upper Deck Company, LLC (colloquially as Upper Deck and Upper Deck Authenticated, Ltd. in the UK) is a private company primarily known for producing...

attributed by some metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse"...

mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set A may be indexed or labeled...

between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships...

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...

mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P {\textstyle P} is a certain subset of P , {\displaystyle P...

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...

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...

subset of any ordered set, every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of...

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...

Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally...