different between poset vs ultrafilter

English

Etymology

Abbreviation of partially ordered set.

Noun

poset (plural posets)

1. (set theory, order theory) A partially ordered set.
   • 1973, Barbara L. Osofsky, Homological Dimensions of Modules, American Mathematical Society, ?ISBN, page 76,

     42. Definition. A poset (partially ordered set) (X, ?) (usually written just X) is a set X together with a transitive, antisymmetric relation ? on X.

     43. Definition. A linearly ordered set or chain is a poset (X, ?), such that ?a, b ? X, either a ? b or b ? a or a = b.

   • 1998, Yuri A. Drozd, Representations of bisected posets and reflection functors, Idun Reiten, Sverre O. Smalø, Øyvind Solberg (editors), Algebras and Modules II, American Mathematical Society (for Canadian Mathematical Society), page 153,

     We construct a complete set of reflection functors for the representations of posets and prove that they really have the usual properties. In particular, when the poset is of finite representation type, all of its indecomposable representations can be obtained from some "trivial" ones via relations. To define such reflection functors, a wider class of matrix problem is introduced, called "representations of bisected posets".

   • 1999, Manfred Stern, Semimodular Lattices: Theory and Applications, Cambridge University Press, page 189,

     The combinatorial interest in posets is largely due to two unoriented graphs associated with a given poset: the comparability graph (which we shall not consider here) and the covering graph.

Synonyms

• See also Thesaurus:partially ordered set

Further reading

• Hasse diagram on Wikipedia.Wikipedia
• Lattice (order) on Wikipedia.Wikipedia

Anagrams

• ETOPS, Poets, Potes, T pose, T-pose, Topes, e-stop, estop, pesto, poets, potes, septo-, stoep, stope, topes

Serbo-Croatian

Alternative forms

• (Ijekavian): p?sjet
• (Montenegrin): p??et

Pronunciation

• IPA^(key): /pôset/
• Hyphenation: po?set

Noun

p?set m (Cyrillic spelling ??????)

1. visit

Declension

English

Etymology

ultra- +? filter

Noun

ultrafilter (plural ultrafilters)

1. A device that performs ultrafiltration.
2. (mathematics, of a set, whose subsets are partially ordered by inclusion) A proper filter which has a law of dichotomy for complements.

   A proper filter of a set has these three properties: (1) it does not contain the empty set, (2) if it contains a subset then it contains all supersets of that subset, and (3) if it contains a pair of subsets then it also contains their intersection. To make it into an ultrafilter it must be made as large as possible without including the empty set. That can be prevented by not allowing any pair of disjoint sets to be both included. If, given a pair of complementary subsets, one of them is prevented from being included, then all subsets of it should be prevented from being included as well, by the second rule. That takes care of all subsets disjoint from the other complementary subset, which should then be included, in order to make the filter approach maximality, i.e., turn it into an ultrafilter.

3. (mathematics, order theory) A filter (subset of a poset) that is maximal as a set with respect to the definition of proper filter.

   An ultrafilter is maximal in the sense that if any other element of the poset not already in it were added to it, one could deduce (from the laws which define the filter, and the given ordering relation, i.e., the structure of the poset) that the resulting filter must be improper; i.e., it must contain all the elements of the poset.

Hypernyms

• filter

Hyponyms

• free ultrafilter
• principal ultrafilter

Holonyms

• poset

Translations

Verb

ultrafilter (third-person singular simple present ultrafilters, present participle ultrafiltering, simple past and past participle ultrafiltered)

1. (transitive) To filter by ultrafiltration.

Further reading

• ultrafilter on Wikipedia.Wikipedia