), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders.
That is, a strict total order is a binary relation
that can be defined in two equivalent ways: Conversely, the reflexive closure of a strict total order
This high number of nested levels of sets explains the usefulness of the term.
A common example of the use of chain for referring to totally ordered subsets is Zorn's lemma which asserts that, if every chain in a partially ordered set X has an upper bound in X, then X contains at least one maximal element.
[13] Zorn's lemma is commonly used with X being a set of subsets; in this case, the upper bound is obtained by proving that the union of the elements of a chain in X is in X.
This is the way that is generally used to prove that a vector space has Hamel bases and that a ring has maximal ideals.
[15] For example, an order is well founded if it has the descending chain condition.
The dimension of a space is often defined or characterized as the maximal length of chains of subspaces.
For example, the dimension of a vector space is the maximal length of chains of linear subspaces, and the Krull dimension of a commutative ring is the maximal length of chains of prime ideals.
Another example is the use of "chain" as a synonym for a walk in a graph.
One may define a totally ordered set as a particular kind of lattice, namely one in which we have We then write a ≤ b if and only if
Hence a totally ordered set is a distributive lattice.
A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset thereof) has a least element.
In other words, a total order on a set with k elements induces a bijection with the first k natural numbers.
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).
In other words, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions.
For example, over the real numbers a property of the relation ≤ is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ≤ to the rational numbers.
are pairwise disjoint, then the natural total order on
is defined by The first-order theory of total orders is decidable, i.e. there is an algorithm for deciding which first-order statements hold for all total orders.
Using interpretability in S2S, the monadic second-order theory of countable total orders is also decidable.
All three can similarly be defined for the Cartesian product of more than two sets.
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.
All definitions tacitly require the homogeneous relation
A term's definition may require additional properties that are not listed in this table.
A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order.
There are only a few nontrivial structures that are (interdefinable as) reducts of a total order.
Forgetting the orientation results in a betweenness relation.
Forgetting the location of the ends results in a cyclic order.
Forgetting both data results in a separation relation.