Crossword clues for preorder
The Collaborative International Dictionary
Preorder \Pre*or"der\, v. t.
To order to arrange beforehand; to foreordain.
--Sir W.
Hamilton.
Wiktionary
(context computing theory English) Of a tree traversal, recursively visiting the root before the left and right subtrees. n. (context set theory English) A binary relation that is reflexive and transitive. v
(context transitive English) To order (goods) in advance, before they are available.
Wikipedia
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder.
The name 'preorder' comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they're neither necessarily anti-symmetric nor symmetric. Because a preorder is a binary relation, the symbol ≤ can be used as the notational device for the relation. However, because they are not necessarily anti-symmetric, some of the ordinary intuition associated to the symbol ≤ may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, is not always useful or worthwhile, depending on the problem domain being studied.
In words, when a ≤ b, one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or $\lesssim$ is used instead of ≤.
To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices. The converse is not true: most directed graphs are neither reflexive nor transitive. In general, the corresponding graphs may contain cycles. A preorder that is antisymmetric no longer has cycles; it is a partial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder's corresponding directed graph may have many disconnected components.
The term preorder may refer to:
In mathematics:
- Preorder, a reflexive, transitive relation
- Preorder field, a field of sets structure on a set with preorder
- Preordered field, a field with a preorder
- Preordering, a vertex ordering from a tree or other graph traversal; see Depth-first search#Vertex orderings
In marketing:
- Pre-order – an order placed for an item which has not yet been released.
Usage examples of "preorder".
And there I stood, in the bakery surrounded by the warmth and moisture of the last of the preorders, waiting.
This was matched by an American snafu when the Tenth Air Force Headquarters sent an antiaircraft battery on the first transports instead of the infantry and food preordered by Merrill and Hunter.
But Tay was shielded by his Druid fire, the magic set in place at the moment of his release, preordered to protect him from the garden's rage.
Her progress brought her through the shopping area to the quay, where she greeted the fishermen and bought two snapper and a wriggling lobster that had been preordered by the Quarter Deck Hotel.
Show preorders soared to record levels, and within six weeks Celestial had sold the entire first year's projected production, sending Goldstein to the Asian Rim on a quest for more capacity.