### Crossword clues for preorder

##### The Collaborative International Dictionary

**Preorder**

Preorder \Pre*or"der\, v. t.
To order to arrange beforehand; to foreordain.

--Sir W.
Hamilton.

##### Wiktionary

**preorder**

(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

**Preorder**

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.

**Preorder (disambiguation)**

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.