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A set equipped with a total order is a totally ordered set; [5] the terms simply ordered set, [2] linearly ordered set, [3] [5] toset [6] and loset [7] [8] are also used. The term chain is sometimes defined as a synonym of totally ordered set , [ 5 ] but generally refers to a totally ordered subset of a given partially ordered set.
Total order A relation that is reflexive, antisymmetric, transitive and connected. [20] Strict total order A relation that is irreflexive, asymmetric, transitive and connected. Uniqueness properties: One-to-one [d] Injective and functional. For example, the green relation in the diagram is one-to-one, but the red, blue and black ones are not ...
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.
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that".
This area also includes one of order theory's most famous open problems, the 1/3–2/3 conjecture, which states that in any finite partially ordered set that is not totally ordered there exists a pair (,) of elements of for which the linear extensions of in which < number between 1/3 and 2/3 of the total number of linear extensions of . [11 ...
In Political science and Decision theory, order relations are typically used in the context of an agent's choice, for example the preferences of a voter over several political candidates. x ≺ y means that the voter prefers candidate y over candidate x. x ~ y means the voter is indifferent between candidates x and y.
In mathematics, a binary relation R ⊆ X×Y between two sets X and Y is total (or left total) if the source set X equals the domain {x : there is a y with xRy}. Conversely, R is called right total if Y equals the range {y : there is an x with xRy}. When f: X → Y is a function, the domain of f is all of X, hence f is a total relation.
This notion of "total" should not be confused with that of a total relation in the sense that for all there is a so that (see serial relation). Connectedness features prominently in the definition of total orders : a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected.