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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...

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...

"On finite solvable groups in which normality is a transitive relation". Journal of Group Theory. 3 (2). doi:10.1515/jgth.2000.012. ISSN 1433-5883. Archived...

ordered Bell numbers count partitions that have been equipped with a total order. Their alternative name, the Fubini numbers, comes from a connection...

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...

absence of a total order relation for n > 1 {\displaystyle n>1} . In this case, various solutions can be "incomparable" as there is no total order relation...

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...

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

Order in mathematics may refer to: Total order and partial order, a binary relation generalizing the usual ordering of numbers and of words in a dictionary...

complete graph is a comparability graph, the comparability graph of a total order. All acyclic orientations of a complete graph are transitive. Every bipartite...

than or equal to (≤) the keys in the node's children, according to some total order. Heaps where the parent key is greater than or equal to (≥) the child...

sort items of any type for which a "less-than" relation (formally, a total order) is defined. It is a comparison-based sort since elements a and b are...

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...

to obey the 1/3–2/3 conjecture: in any finite semiorder that is not a total order, there exists a pair of elements x {\displaystyle x} and y {\displaystyle...

buy include the ability to track the price of a product, compare the total order cost at online or nearby stores, and access product reviews and related...

suprema/infima. Total order – Order whose elements are all comparable G. Markowsky and B.K. Rosen. Bases for chain-complete posets IBM Journal of Research...

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

these sets, the index set T {\displaystyle T} can be another set with a total order or a more general set, such as the Cartesian plane R 2 {\displaystyle...

linearization of X {\displaystyle X} is an extension of the partial order into a total order. It is easy to verify that o ( X ) {\displaystyle o(X)} is an upper bound...

types of partial orders are specified by adding further axioms, such as: Total order: For every a and b in P, a ≤ b or b ≤ a . Dense order: For all a and...