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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} :
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 ...
Several types of orders can be defined from numerical data on the items of the order: a total order results from attaching distinct real numbers to each item and using the numerical comparisons to order the items; instead, if distinct items are allowed to have equal numerical scores, one obtains a strict weak ordering.
Total order. A total order T is a partial order in which, for each x and y in T, we have x ≤ y or y ≤ x. Total orders are also called linear orders or chains. Total relation. Synonym for Connected relation. Transitive relation. A relation R on a set X is transitive, if x R y and y R z imply x R z, for all elements x, y, z in X. Transitive ...
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the ...
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A strict partial order, also called strict order, [citation needed] is a relation that is irreflexive, antisymmetric, and transitive. A total order, also called linear order, simple order, or chain, is a relation that is reflexive, antisymmetric, transitive and connected. [15]