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What (who) is partial ordering - definition
SET ORDERED BY A TRANSITIVE, ANTISYMMETRIC, AND REFLEXIVE BINARY RELATION
PartialOrderedSet; PartialOrder; Partial order; Poset; Partial ordering relation; Partial ordering; Partially ordered; Strict order; Partially ordered sets; Ordered n-tuple; Strict partial ordering; Strict partial order; Poset category; Ordered collection; Non-strict order; Ordered set; Strict ordering; Interval (partial order); Ordinal sum; Partial Order; Partially-ordered set
!['''Fig.2''' [[Commutative diagram]] about the connections between strict/non-strict relations and their duals, via the operations of reflexive closure (''cls''), irreflexive kernel (''ker''), and converse relation (''cnv''). Each relation is depicted by its [[logical matrix]] for the poset whose [[Hasse diagram]] is depicted in the center. For example <math>3 \not\leq 4</math> so row 3, column 4 of the bottom left matrix is empty.](https://commons.wikimedia.org/wiki/Special:FilePath/PartialOrders redundencies svg.svg?width=200 "'''Fig.2''' [[Commutative diagram]] about the connections between strict/non-strict relations and their duals, via the operations of reflexive closure (''cls''), irreflexive kernel (''ker''), and converse relation (''cnv''). Each relation is depicted by its [[logical matrix]] for the poset whose [[Hasse diagram]] is depicted in the center. For example <math>3 \not\leq 4</math> so row 3, column 4 of the bottom left matrix is empty.")
partial ordering
A relation R is a partial ordering if it is a pre-order
(i.e. it is reflexive (x R x) and transitive (x R y R z =>
x R z)) and it is also antisymmetric (x R y R x => x = y).
The ordering is partial, rather than total, because there may
exist elements x and y for which neither x R y nor y R x.
In domain theory, if D is a set of values including the
undefined value (bottom) then we can define a partial
ordering relation <= on D by
x <= y if x = bottom or x = y.
The constructed set D x D contains the very undefined element,
(bottom, bottom) and the not so undefined elements, (x,
bottom) and (bottom, x). The partial ordering on D x D is
then
(x1,y1) <= (x2,y2) if x1 <= x2 and y1 <= y2.
The partial ordering on D -> D is defined by
f <= g if f(x) <= g(x) for all x in D.
(No f x is more defined than g x.)
A lattice is a partial ordering where all finite subsets
have a least upper bound and a greatest lower bound.
("<=" is written in LaTeX as sqsubseteq).
(1995-02-03)
partially ordered set
A set with a partial ordering.
Partial derivative
DERIVATIVE OF A FUNCTION OF SEVERAL VARIABLES WITH RESPECT TO ONE VARIABLE, WITH THE OTHERS HELD CONSTANT
Partial Derivatives; Partial derivatives; Partial differentiation; Partial derivation; Mixed partial derivatives; Mixed derivatives; Partial Derivative; Mixed partial derivative; Partial differential; Partial symbol; Partial differentiation; Del (∂); Cross derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
Wikipedia
Partially ordered set
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, transitive and antisymmetric. A partially ordered set (poset for short) is a set on which a partial order is defined.
