total order: Definition from Answers.com

Wikipedia: total order

In mathematics, a total order, linear order, simple order, or (non-strict) ordering on a set X is any binary relation on X that is antisymmetric, transitive, and total. This means that if we denote one such relation by ≤ then the following statements hold for all a, b and c in X:

if a ≤ b and b ≤ a then a = b (antisymmetry)

if a ≤ b and b ≤ c then a ≤ c (transitivity)

a ≤ b or b ≤ a (totality or completeness)

A set paired with an associated total order on it is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain.

A relation's property of "totality" can be described this way: that any pair of elements in the set are mutually comparable under the relation.

Notice that the totality condition implies reflexivity, that is, a ≤ a. Thus a total order is also a partial order, that is, a binary relation which is reflexive, antisymmetric and transitive. A total order can also be defined as a partial order that is "total", that is satisfies the "totality" condition.

Alternatively, one may define a totally ordered set as a particular kind of lattice, namely one in which we have

$\{a\vee b, a\wedge b\} = \{a, b\}$ for all a, b.

We then write a ≤ b if and only if $a = a\wedge b$ . It follows that a totally ordered set is a distributive lattice.^[1]

Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being maps which respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b).

A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.

Strict total order

For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) relation <, called a strict total order, which can equivalently be defined in two ways:

Properties:

The relation is transitive: a < b and b < c implies a < c.
The relation is trichotomous: exactly one of a < b, b < a and a = b is true.
The relation is a strict weak order, where the associated equivalence is equality.

We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤ can equivalently be defined in two ways:

a ≤ b if and only if a < b or a = b
a ≤ b if and only if not b < a

Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.

We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whether we are talking about the non-strict or the strict total order.

Examples

The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc.

Any subset of a totally ordered set, with the restriction of the order on the whole set.

Any partially ordered set X where every two elements are comparable (i.e. if a,b are members of X either a ≤ b or b ≤ a or both).

Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders).

If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on X by setting x₁ < x₂ if and only if f(x₁) < f(x₂).

The lexicographical order on the Cartesian product of a set of totally ordered sets indexed by an ordinal, is itself a total order. For example, any set of words ordered alphabetically is a totally ordered set, viewed as a subset of a Cartesian product of a countable number of copies of a set formed by adding the space symbol to the alphabet (and defining a space to be less than any letter).

The set of real numbers ordered by the usual less than (<) or greater than (>) relations is totally ordered, hence also the subsets of natural numbers, integers, and rational numbers. Each of these can be shown to be the unique (to within isomorphism) smallest example of a totally ordered set with a certain property, (a total order A is the smallest with a certain property if whenever B has the property, there is an order isomorphism from A to a subset of B):
- The natural numbers comprise the smallest totally ordered set with no upper bound.
- The integers comprise the smallest totally ordered set with neither an upper nor a lower bound.
- The rational numbers comprise the smallest totally ordered set with no upper or lower bound, which is dense in the sense that for every a and b such that a < b there is a c such that a < c < b.
- The real numbers comprise the smallest unbounded connected totally ordered set. (See below for the definition of the topology.)

Further concepts

Order topology

For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, the order topology.

When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology on N induced by < and the order topology on N induced by > (in this case they happen to be identical but will not in general).

The order topology induced by a total order may be shown to be hereditarily normal.

Completeness

A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upper bound. For example, the set of real numbers is complete but the set of rational numbers is not.

There are a number of results relating properties of the order topology to the completeness of X:

If the order topology on X is connected, X is complete.
X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is two points a and b in X with a < b such that no c satisfies a < c < b.)
X is complete if and only if every bounded set that is closed in the order topology is compact.

A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervals of real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line). There are order-preserving homeomorphisms between these examples.

Chains

While from a definition point of view, chain is merely a synonym for totally ordered set the term is usually used to describe a totally ordered subset of some partial order. Thus the reals would probably be described as a totally ordered set. However, if we were to consider all subsets of the integers partially ordered by inclusion then the totally ordered set under inclusion { I_n : n is a natural number} defined in an above example would frequently be called a chain.

The preferential use of chain to refer to a totally ordered subset of a partial order likely stems from the important role such totally ordered subsets play in Zorn's lemma.

Finite total orders

A simple counting argument will verify that any finite totally-ordered set (and hence any subset thereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observing that every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphic to an initial segment of the natural numbers ordered by <. In other words a total order on a set with k elements induces a bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with order type ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).

Contrast with a partial order, which lacks the third condition. An example of a partial order is the happened-before relation.

Orders on the Cartesian product of totally ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product of two totally ordered sets are:

Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order.
(a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order.
(a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of the corresponding strict total orders). This is also a partial order.

All three can similarly be defined for the Cartesian product of more than two sets.

Applied to the vector space Rⁿ, each of these make it an ordered vector space.

A real function of n real variables defined on a subset of Rⁿ defines a strict weak order and a corresponding total preorder on that subset.

Notes

^ Distributive lattice/Proofs

References

George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN 0-7167-0442-0
John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4

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