ordering

Webster's Dictionary (GCIDE)

1.To put in order; to reduce to a methodical arrangement; to arrange in a series, or with reference to an end. Hence, to regulate; to dispose; to direct; to rule. [1913 Webster]

To him that ordereth his conversation aright.— Ps. 1. 23.[1913 Webster]

Warriors old with ordered spear and shield.— Milton.[1913 Webster]

2.To give an order to; to command; as, to order troops to advance. [1913 Webster]

3.To give an order for; to secure by an order; as, to order a carriage; to order groceries. [1913 Webster]

4.(Eccl.) To admit to holy orders; to ordain; to receive into the ranks of the ministry. [1913 Webster]

These ordered folk be especially titled to God.— Chaucer.[1913 Webster]

Persons presented to be ordered deacons.— Bk. of Com. Prayer.[1913 Webster]

Order arms (Mil.), the command at which a rifle is brought to a position with its butt resting on the ground; also, the position taken at such a command. [1913 Webster]

Free On-line Dictionary of Computing

<mathematics> (Informally, "order", "ordering") A binary relation R that is a pre-order (i.e. it is reflexive (x R x) and transitive (x R y R z => x R z)) and antisymmetric (x R y R x => x = y).

The order 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)