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Pseudoideal
In the theory of partially ordered sets, a pseudoideal is a subset characterized by a bounding operator LU. Basic definitions LU(''A'') is the set of all lower bounds of the set of all upper bounds of the subset ''A'' of a partially ordered set. A subset ''I'' of a partially ordered set (''P'', ≤) is a Doyle pseudoideal, if the following condition holds: For every finite subset ''S'' of ''P'' that has a supremum in ''P'', if S\subseteq I then \operatorname{LU}(S)\subseteq I. A subset ''I'' of a partially ordered set (''P'', ≤) is a pseudoideal, if the following condition holds: For every subset ''S'' of ''P'' having at most two elements that has a supremum in ''P'', if ''S'' \subseteq ''I'' then LU(''S'') \subseteq ''I''. Remarks #Every Frink ideal ''I'' is a Doyle pseudoideal. #A subset ''I'' of a lattice (''P'', ≤) is a Doyle pseudoideal if and only if it is a lower set that is closed under finite joins ( suprema). Related notions *Frink ideal Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frink Ideal
In mathematics, a Frink ideal, introduced by Orrin Frink, is a certain kind of subset of a partially ordered set. Basic definitions LU(''A'') is the set of all common lower bounds of the set of all common upper bounds of the subset ''A'' of a partially ordered set. A subset ''I'' of a partially ordered set (''P'', ≤) is a Frink ideal, if the following condition holds: For every finite subset ''S'' of ''I'', we have LU(''S'') \subseteq ''I''. A subset ''I'' of a partially ordered set (''P'', ≤) is a normal ideal or a cut if LU(''I'') \subseteq ''I''. Remarks #Every Frink ideal ''I'' is a lower set. #A subset ''I'' of a lattice (''P'', ≤) is a Frink ideal if and only if it is a lower set that is closed under finite joins (suprema). # Every normal ideal is a Frink ideal. Related notions *pseudoideal *Doyle pseudoideal References * *{{cite journal, author=Niederle, Josef, title=Ideals in ordered sets, journal=Rendiconti del Circolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) 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 indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lower Bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that . Every subset of the natural numbers has a lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that . Every subset of the natural numbers has a lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suprema
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set P is the least element in P that is greater than or equal to each element of S, if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and maxim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rendiconti Del Circolo Matematico Di Palermo
The Circolo Matematico di Palermo (Mathematical Circle of Palermo) is an Italian mathematical society, founded in Palermo by Sicilian geometer Giovanni B. Guccia in 1884.The Mathematical Circle of Palermo . Retrieved 2011-06-19. It began accepting foreign members in 1888, and by the time of Guccia's death in 1914 it had become the foremost international mathematical society, with approximately one thousand members. However, subsequently to that time it declined in influence. Publications ''Rendiconti del Circolo Matemat ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
