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...
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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...
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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...
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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...
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Well-order (redirect from Well-ordered set)
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...
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Infimum and supremum (redirect from Least upper bound)
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...
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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...
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Total order (redirect from Total ordered set)
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...
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order – Mathematical phrase Directed set – Mathematical ordering with upper bounds Graded poset – partially ordered set equipped with a rank functionPages...
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Natural density (redirect from Set density)
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}...
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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...
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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...
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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...
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{\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...
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Semilattice (redirect from Upper semi-lattice)
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...
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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...
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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...
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Well-founded relation (redirect from Well-founded set)
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...
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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...
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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...
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Semi-continuity (redirect from Upper semi-continuous)
and upper semicontinuity and set-valued lower and upper semicontinuouty. An upper semicontinuous single-valued function is not necessarily upper semicontinuous...
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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...
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Preorder (redirect from Preordered set)
has an upper bound. The definition of convergence via nets is important in topology, where preorders cannot be replaced by partially ordered sets without...
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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...
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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...
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monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus...
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Order theory (section Partially ordered sets)
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...
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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:...
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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...
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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\}}...
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