| 词条 | Product order |
| 释义 |
In mathematics, given two ordered sets A and B, one can induce a partial ordering on the Cartesian product {{nowrap|A × B}}. Given two pairs (a1,b1) and (a2,b2) in {{nowrap|A × B}}, one sets {{nowrap|(a1,b1) ≤ (a2,b2)}} if and only if {{nowrap|a1 ≤ a2}} and {{nowrap|b1 ≤ b2}}. This ordering is called the product order,[1][2][3][4] or alternatively the coordinatewise order,[3][3][4] or even the componentwise order.[2][5] Another possible ordering on {{nowrap|A × B}} is the lexicographical order. Unlike the latter, the product order of two totally ordered sets is not total. For example, the pairs {{nowrap|(0, 1)}} and {{nowrap|(1, 0)}} are incomparable in the product order of {{nowrap|0 < 1}} with itself. The lexicographic order of totally ordered sets is however a linear extension of their product order. In general, the product order is a subrelation of the lexicographic order.[6] The Cartesian product with product order is the categorical product in the category of partially ordered sets with monotone functions.[5] The product order generalizes to arbitrary (possibly infinitary) Cartesian products. Furthermore, given a set A, the product order over the Cartesian product {{nowrap|∏A{0, 1} }} can be identified with the inclusion ordering of subsets of A.[7] The notion applies equally well to preorders. The product order is also the categorical product in a number of richer categories, including lattices and Boolean algebras.[5] References1. ^{{citation | last1 = Neggers | first1 = J. | last2 = Kim | first2 = Hee Sik | contribution = 4.2 Product Order and Lexicographic Order | isbn = 9789810235895 | pages = 64–78 | publisher = World Scientific | title = Basic Posets | url = https://books.google.com/books?id=-ip3-wejeR8C&pg=PA64 | year = 1998}} 2. ^1 {{cite book|author1=Sudhir R. Ghorpade|author2=Balmohan V. Limaye|title=A Course in Multivariable Calculus and Analysis|year=2010|publisher=Springer |isbn=978-1-4419-1621-1|page=5}} 3. ^Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002, p. 18 4. ^{{cite book|author1=Alexander Shen|author2=Nikolai Konstantinovich Vereshchagin|title=Basic Set Theory|year=2002|publisher=American Mathematical Soc.|isbn=978-0-8218-2731-4|page=43}} 5. ^1 2 {{cite book|author=Paul Taylor|title=Practical Foundations of Mathematics|year=1999|publisher=Cambridge University Press|isbn=978-0-521-63107-5|pages=144–145 and 216}} 6. ^1 2 {{cite book|author=Egbert Harzheim|title=Ordered Sets|year=2006|publisher=Springer|isbn=978-0-387-24222-4|pages=86–88}} 7. ^1 {{cite book|author=Victor W. Marek|title=Introduction to Mathematics of Satisfiability|year=2009|publisher=CRC Press|isbn=978-1-4398-0174-1|page=17}} See also
1 : Order theory |
| 随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。