In order theory, a continuous poset is a partially ordered set in which every element is the directed supremum of elements approximating it. Let a , b...

elements x, y of X, at least one of x R y or y R x holds. Continuous poset. A poset is continuous if it has a base, i.e. a subset B of P such that every...

branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered...

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function...

basis is also called a continuous ω-cpo (or continuous dcpo). Note that complete partial order is never used to mean a poset in which all subsets have...

of a poset for obtaining these directed sets, then the poset is even algebraic. Both concepts can be applied to lattices as follows: A continuous lattice...

partially ordered sets. It states that if X is a non-empty chain complete poset, and f : X → X {\displaystyle f:X\to X} such that f ( x ) ≥ x {\displaystyle...

Dedekind completion Ideal completion Way-below relation Continuous poset Continuous lattice Algebraic poset Scott domain Algebraic lattice Scott information...

functions between topological spaces Scott continuity, for functions between posets Continuity (set theory), for functions between ordinals Continuity (category...

(transitivity). A set with a partial order on it is called a partially ordered set, poset, or just ordered set if the intended meaning is clear. By checking these...

directed join. When Q {\displaystyle Q} is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of...

element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element...

topological space whose partially ordered set of open subsets is a continuous poset. Equivalently, X {\displaystyle X} is core-compact if it is exponentiable...

{\displaystyle X} contained in Y {\displaystyle Y} form a poset under inclusion. A maximal element of this poset is called a convex component of Y . {\displaystyle...

order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring...

preserves arbitrary meets and joins. Both L and its dual order Lop are continuous posets.[citation needed] Direct products of [0,1], i.e. sets of all functions...

partially ordered set (poset) to itself is the fixed point which is less than each other fixed point, according to the order of the poset. A function need not...

cardinals * An operation that takes a forcing poset and a name for a forcing poset and produces a new forcing poset. ∞ The class of all ordinals, or at least...

identity in an enriched category. Since R ∗ {\displaystyle R^{*}} is a poset, all diagrams that are required for an enriched category commute automatically...

particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories...

\mathbb {R} \mid 0\leq t\leq 1\}} is the ordered unit interval, a continuous chain poset. More geometrically, we may list the elements P = { x 1 , … , x...

–) to the functor T(–, x) for each x in the lattice [0, 1] taken as a poset category. In the standard semantics of t-norm based fuzzy logics, where...

any continuous strictly increasing function from ordinals to ordinals has one (and indeed many) fixed points. Every closure operator on a poset has many...

algebraic poset. Since C is also a lattice, it is often referred to as an algebraic lattice in this context. Conversely, if C is an algebraic poset, then...

orders also inherit some properties of the underlying posets. For instance if A and B are continuous lattices, then so is the set of functions A → B with...

almost-disjointly generated by the unit ball is the cofactors. The coproduct of a poset category is the join operation. The coproduct construction given above is...

length formula for binary trees using the hook walk in 1989. Proctor gave a poset generalization (see below). The hook length formula can be understood intuitively...

In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete...

required explicitly. A directed subset of a poset is not required to be downward closed; a subset of a poset is directed if and only if its downward closure...

we can equivalently define a frame to be a cocomplete cartesian closed poset. The system of all open sets of a given topological space ordered by inclusion...