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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} :
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 ...
Otherwise the operator selects a boundary sub-relation described in terms of its logical matrix: is the side diagonal if is an upper right triangular linear order or strict order. fringe ( R ) {\displaystyle \operatorname {fringe} (R)} is the block fringe if R {\displaystyle R} is irreflexive ( R ⊆ I ¯ {\displaystyle R\subseteq {\bar ...
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 .}
If different items have different sort key values then this defines a unique order of the items. Workers sorting parcels in a postal facility. A standard order is often called ascending (corresponding to the fact that the standard order of numbers is ascending, i.e. A to Z, 0 to 9), the reverse order descending (Z to A, 9 to 0).
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
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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).