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In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } 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} :
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 ...
A partial order with this property is called a total order. These orders can also be called linear orders or chains. While many familiar orders are linear, the subset order on sets provides an example where this is not the case. Another example is given by the divisibility (or "is-a-factor-of") relation |.
Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. An alphabetical list of many notions of order theory can be found in the order theory glossary.
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 < another associated reflexive relation is its reflexive closure, a (non-strict) partial order .
Of particular importance are relations that satisfy certain combinations of properties. A partial order is a relation that is reflexive, antisymmetric, and transitive, [3] an equivalence relation is a relation that is reflexive, symmetric, and transitive, [4] a function is a relation that is right-unique and left-total (see below). [5] [6]
In order theory, the Szpilrajn extension theorem (also called the order-extension principle), proved by Edward Szpilrajn in 1930, [1] states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair ...
Lexicographical order, an ordering method on sequences analogous to alphabetical order on words; List of order topics, list of order theory topics; Order theory, study of various binary relations known as orders; Order topology, a topology of total order for totally ordered sets; Ordinal numbers, numbers assigned to sets based on their set ...