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A strict total order on a set is a strict partial order on in which any two distinct elements are comparable. That is, a strict total order is a binary relation < {\displaystyle <} on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} :
For the sorting to be unique, these two are restricted to a total order and a strict total order, respectively. Sorting n-tuples (depending on context also called e.g. records consisting of fields) can be done based on one or more of its components. More generally objects can be sorted based on a property.
A strict weak order that is trichotomous is called a strict total order. [14] The total preorder which is the inverse of its complement is in this case a total order . For a strict weak order < {\displaystyle \,<\,} another associated reflexive relation is its reflexive closure , a (non-strict) partial order ≤ . {\displaystyle \,\leq .}
The usual strict total order on N, "less than" (denoted by "<"), can be defined in terms of addition via the rule x < y ↔ ∃z (Sz + x = y). Equivalently, we get a definitional conservative extension of Q by taking "<" as primitive and adding this rule as an eighth axiom; this system is termed " Robinson arithmetic R " in Boolos, Burgess ...
Partial order – an antisymmetric preorder; Total preorder – a connected (formerly called total) preorder; Equivalence relation – a symmetric preorder; Strict weak ordering – a strict partial order in which incomparability is an equivalence relation; Total ordering – a connected (total), antisymmetric, and transitive relation
For example, "divides " is a partial, but not a total order on natural numbers, " <" is a strict total order on , and "is parallel to " is an equivalence relation on the set of all lines in the Euclidean plane.
A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).
If f <. g, then s can dominate t only if one of s's subterms does. If f. > g, then s dominates t if s dominates each of t's subterms. If f = g, then the immediate subterms of s and t need to be compared recursively. Depending on the particular method, different variations of path orderings exist. [2] [3] The latter variations include: