# ordering relation

﻿
ordering relation
A partial ordering on a set is a relation < that is transitive and reflexive and antisymmetric. That is, (i) x < y & y < zx < z ; (ii) x < x ; (iii) x < y & y < xx = y . If we add (iv) that at least one of x < y, x = y, and y < x holds (the relation is connected, or, all elements of the set are comparable), then the ordering is a total ordering (intuitively, the elements can be arranged along a straight line); otherwise it is a partial ordering. A well-ordering is an ordering such that every non-empty subset of the set contains a minimal element, that is, some element m such that there is no x ? m in the set such that x < m. A well-ordering on a set A is a linear ordering with the property that every nonempty subset of A has a minimal element.

Philosophy dictionary. . 2011.

### Look at other dictionaries:

• partial ordering relation — noun A partial order …   Wiktionary

• total ordering relation — noun A total order …   Wiktionary

• relation — Philosophically relations are interesting because of the historic prejudice, given its most forceful expression by Leibniz, that they are somehow ‘unreal’ compared to the intrinsic, monadic properties of things. A way of putting the idea is that… …   Philosophy dictionary

• ordering relations — relation between sizes in mathematics …   English contemporary dictionary

• Well-quasi-ordering — In mathematics, specifically order theory, a well quasi ordering or wqo is a well founded quasi ordering with an additional restriction on sequences that there is no infinite sequence x i with x i ot le x j for all i < j . Motivation We can use… …   Wikipedia

• linear ordering — See ordering relation …   Philosophy dictionary

• partial ordering — See ordering relation …   Philosophy dictionary

• well-ordering — See ordering relation …   Philosophy dictionary

• Binary relation — Relation (mathematics) redirects here. For a more general notion of relation, see Finitary relation. For a more combinatorial viewpoint, see Theory of relations. In mathematics, a binary relation on a set A is a collection of ordered pairs of… …   Wikipedia

• Strict weak ordering — The 13 possible strict weak orderings on a set of three elements {a, b, c}. The only partially ordered sets are coloured, while totally ordered ones are in black. Two orderings are shown as connected by an edge if they differ by a single… …   Wikipedia

### Share the article and excerpts

##### Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.