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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...

in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences,...

{\displaystyle \leq } if, and only if, ≤ {\displaystyle \leq } is a total order. Another way of defining a partial order, found in computer science,...

every nonzero real number a. The total order is denoted a < b . {\displaystyle a<b.} being that it is a total order means two properties: given two real...

equivalent common definitions of an ordered field. The definition of total order appeared first historically and is a first-order axiomatization of the...

strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if,...

regions according to the mathematical structure it has. In the case of total order, as for memory addresses, these are simply chunks. Like the hierarchical...

is a strict well ordering if and only if it is a well-founded strict total order. The distinction between strict and non-strict well orders is often ignored...

In mathematics, a partial order or total order < on a set X {\displaystyle X} is said to be dense if, for all x {\displaystyle x} and y {\displaystyle...

space X is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and...

partitions (partitions of the elements into disjoint subsets, together with a total order on the subsets). In many cases another representation called a preferential...

corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order. In set theory, a set x is called a well-founded...

as a weak order or total preorder of objects. It is not necessarily a total order of objects because two different objects can have the same ranking. The...

✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering...

Szpilrajn in 1930, states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements...

asymmetric, and transitive. Total order A relation that is reflexive, antisymmetric, transitive and connected. Strict total order A relation that is irreflexive...

asymmetric, and transitive. A total order is a relation that is reflexive, antisymmetric, transitive and connected. A strict total order is a relation that is...

a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic...

✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering...

algorithm may be used to convert a total order into a sequence in this way. A linear extension of a partial order is a total order that is compatible with it...