Ordered set

Encyclopedia

In order theory

in mathematics, a set with a binary relation

R on its elements that is reflexive

(for all

(if

(if

or poset. If the binary relation is antisymmetric, transitive and also total (for all

In information theory

, an

.

Order theory

Order theory is a branch of mathematics which investigates our 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 article introduces the field and gives some basic definitions...

in mathematics, a set with a binary relation

Binary relation

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...

R on its elements that is reflexive

Reflexive relation

In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...

(for all

*a*in the set,*a*R*a*), antisymmetricAntisymmetric relation

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...

(if

*a*R*b*and*b*R*a*, then*a*=*b*) and transitiveTransitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....

(if

*a*R*b*and*b*R*c*, then*a*R*c*) is described as a partially ordered setPartially ordered set

In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...

or poset. If the binary relation is antisymmetric, transitive and also total (for all

*a*and*b*in the set,*a*R*b*or*b*R*a*), then the set is a totally ordered set. If every non-empty subset has a least element, then the set is a**well-ordered set**

.Well-order

In mathematics, a well-order relation on a set S is a strict total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded strict total order...

In information theory

Information theory

Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...

, an

**ordered set**is a non-data carrying set of bits as used in 8b/10b encoding8B/10B encoding

In telecommunications, 8b/10b is a line code that maps 8-bit symbols to 10-bit symbols to achieve DC-balance and bounded disparity, and yet provide enough state changes to allow reasonable clock recovery. This means that the difference between the count of 1s and 0s in a string of at least 20 bits...

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