Ordered Set

An ordered set - in order theory of mathematics - is an ambiguous term referring to a set that is either a partially ordered set or a totally ordered set. A set with a binary relation R on its elements that is reflexive (for all a in the set, aRa), antisymmetric (if aRb and bRa, then a = b) and transitive (if aRb and bRc, then aRc) is described as a partially ordered set or poset. If the binary relation is antisymmetric, transitive and also total (for all a and b in the set, aRb or bRa), 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.

In information theory, an ordered set is a non-data carrying set of bits as used in 8b/10b encoding.

Other articles related to "ordered set, ordered, set, sets":

Stable Theory - Unstable Theories
... speaking, a theory is unstable if one can use it to encode the ordered set of natural numbers ... Y=y1...yn defining a relation on Mn with an infinite totally ordered subset then the theory is unstable ... Any infinite totally ordered set has a subset isomorphic to either the positive or negative integers under the usual order, so one can assume the totally ordered subset is ordered like ...
Total Order
... In set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X ... A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain ... If X is totally ordered under ≤, then the following statements hold for all a, b and c in X If a ≤ b and b ≤ a then a = b (antisymmetry) If a ≤ b and b ≤ c then a ≤ c (transitivity) a ...
Maxima And Minima - In Relation To Sets
... Maxima and minima are more generally defined for sets ... In general, if an ordered set S has a greatest element m, m is a maximal element ... Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with respect to order induced by T, m is a least upper bound of S ...
Ordinal Optimization - Mathematical Foundations - Examples
... Standard examples of posets arising in mathematics include The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered ... The set of subsets of a given set (its power set) ordered by inclusion The set of subspaces of a vector space ordered by inclusion ... For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequence a precedes sequence b if every item in a precedes the corresponding item ...

Famous quotes containing the words set and/or ordered:

    Stories of law violations are weighed on a different set of scales in the Black mind than in the white. Petty crimes embarrass the community and many people wistfully wonder why Negroes don’t rob more banks, embezzle more funds and employ graft in the unions.... This ... appeals particularly to one who is unable to compete legally with his fellow citizens.
    Maya Angelou (b. 1928)

    But one sound always rose above the clamor of busy life and, no matter how much of a tintinnabulation, was never confused and, for a moment lifted everything into an ordered sphere: that of the bells.
    Johan Huizinga (1872–1945)