Definition·D052
Total order
A partial order in which any two elements are comparable.
A partial order
on
is a total order (or linear order) when any two elements are comparable:
The pair
is then a totally ordered set, or chain.
In words
For any a and b, if both lie in A then they compare one way or the other.
Rests onA02
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Comparability together with reflexivity, antisymmetry and transitivity makes exactly one of
with
,
, or
with
hold for each pair, the abstract form of trichotomy. The order on the naturals is the prototype. A total order in which every nonempty subset has a least element is a well order; totality alone does not secure this, as the integers (once built) are totally ordered yet have no least element.