dichotomy Noun
( dichotomies)
A separation or division into two; a distinction that results in such a division.
* 1989', Carole Pateman, ''6: Feminist Critiques of the Public/Private '''Dichotomy'' , ''The Disorder of Women: Democracy, Feminism, and Political Theory , page 118 ,
 The dichotomy' between the private and the public is central to almost two centuries of feminist writing and political struggle; it is, ultimately, what the feminist movement is all about. Although some feminists treat the ' dichotomy as a universal, transhistorical and transcultural feature of human existence, feminist criticism is primarily directed at the separation and opposition between the public and private spheres in liberal theory and practice.
* 2003 , Thérèse Encrenaz et al''., Storm Dunlop (translator), ''The Solar System'' [''Système Solaire ], page 232 ,
 The dichotomy between maria and highlands dominates lunar mineralogy.
* 2005 , S. P. Naidu, Public Administration: Concepts And Theories , page 55 ,
 Despite some contradictions found in the essay, its major emphasis is laid on the politicsadministration dichotomy' theory. It is largely devoted to the argument concerning the separability of politics and administration. The politicsadministration ' dichotomy initiated by Wilson was later elaborated by Frank J. Goodnow in his work, “Politics and Administration” (1900).
* 2008 , N. Gregory Mankiw, Principles of Economics , 6th Edition, page 723 ,
 All of this previous analysis was based on two related ideas: the classical dichotomy' and monetary neutrality. Recall that the classical ' dichotomy is the separation of variables into real variables (those that measure quantities or relative prices) and nominal variables (those measured in terms of money).
Such a division involving apparently incompatible or opposite principles; a duality.
(logic) The division of a class into two disjoint subclasses that are together comprehensive, as the division of man'' into ''white'' and ''not white .
* 2011 , Patrick J. Hurley, A Concise Introduction to Logic , page 162 ,
 But in the fallacy of false dichotomy , not only do the two alternatives fail to be jointly exhaustive, but they are not even likely. As a result, the disjunctive premise is false, or at least probably false.
* 2011 , Tomasz A. Gorarzd, Jacek Krzaczkowski, The Complexity of Problems Connected with TwoElement Algebras'', Pawe? M. Idziak, AndrzejWronski, ''Reports on Mathematical Logic: No. 46 , page 92 ,
 One can ask if for any algebra the considered problem is always in P or NPcomplete (P or coNPcomplete)? For example, the problem of the satisfiability of a system of polynomial equations over a group G is in P if G is abelian and NPcomplete otherwise ([7, 13]).
 One of the most widely known subclasses of NP which exhibits such a dichotomy', is the class of constraint satisfaction problems (CSP) on the set {0,1}, see [16]. Recently Bulatov proved the ' dichotomy for CSP on a threeelement set [3].
(biology, taxonomy) The division of a genus into two species; a division into two subordinate parts.
(astronomy) A phase of the moon when it appears half lit and half dark, as at the quadratures.
* 1854 , Edward Greswell, Origines Kalendariæ Italicæ: Tables of the Roman Calendar , Volume 1, page 261 ,
 The Ides of Januarius indeed, the preceding month, must have fallen on March 1 at midnight, two days before the first dichotomy of the mean new moon of that month, March 3 at midnight.
(biology) Successive division and subdivision; successive bifurcation, as of a stem of a plant or a vein of the body into two parts as it proceeds from its origin.
(biology) A fork (bifurcation) in a stem or vein.
* 1969 , J. F. Rigby, Permian Sphenopsids from Antarctica'', ''Geological Survey Professional Paper 613F , page F9 ,
 In one forked leaf there is a distinct vein dichotomy', and the leaf boundary commences 1.5 mm above the ' dichotomy .
* 2010 , V. Singh, P. C. Pande, D. K. Jain, Text Book Of Botany: Diversity Of Microbes And Cryptogams , 4th Edition, page 511 ,
 In most of the creeping species with dorsiventral stems (e.g., S. kraussiana'', ''S. laevigata'') roots arise at or close to the point of dichotomy'''; in species like ''S. rupestris'' and ''S. wallichii'' they arise at the point of '''dichotomy as well as other positions and in ''S. selaginoides'' and ''S. spinulosa they arise from knot like swellings present at the basal portion of the stem.
Synonyms
* (division into parts) partition, trichotomy
Derived terms
* dichotomic
* dichotomically
* dichotomise
* dichotomous
* false dichotomy
Related terms
* trichotomy
* polytomy
See also
* bifurcation
* bisection
* duality
* law of the excluded middle
* partition
( Webster 1913)

ultrafilter English
Noun
( en noun)
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 .
(mathematics, order theory) A filter (subset of a poset) that is maximal as a set with respect to the definition of (proper) (term).
 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.
Holonyms
* poset
Hypernyms
* filter
Hyponyms
* free ultrafilter
* principal ultrafilter
