A type is LessThanComparable if it is ordered: it must be possible to compare two objects of that type using operator<, and operator< must be a strict weak ordering relation.
|X||A type that is a model of LessThanComparable|
|x, y, z||Object of type X|
Consider the relation !(x < y) && !(y < x). If this relation is transitive (that is, if !(x < y) && !(y < x) && !(y < z) && !(z < y) implies !(x < z) && !(z < x)), then it satisfies the mathematical definition of an equivalence relation. In this case, operator< is a strict weak ordering.
If operator< is a strict weak ordering, and if each equivalence class has only a single element, then operator< is a total ordering.
|Name||Expression||Type requirements||Return type|
|Less||x < y||Convertible to bool|
|Less||x < y||x and y are in the domain of <|
|Irreflexivity||x < x must be false.|
|Antisymmetry||x < y implies !(y < x) |
|Transitivity||x < y and y < z implies x < z |
 Only operator< is fundamental; the other inequality operators are essentially syntactic sugar.
 Antisymmetry is a theorem, not an axiom: it follows from irreflexivity and transitivity.
 Because of irreflexivity and transitivity, operator< always satisfies the definition of a partial ordering. The definition of a strict weak ordering is stricter, and the definition of a total ordering is stricter still.
Revised 05 December, 2006
|Copyright © 2000||Jeremy Siek, Univ.of Notre Dame (email@example.com)|
Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)