Countable composition closedness and integer-valued continuous functions in pointfree topology
For any archimedean$f$-ring $A$ with unit in whichbreak$awedge (1-a)leq 0$ for all $ain A$, the following are shown to be equivalent: 1. $A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all integer-valued continuous functions on some frame $L$. 2. $A$ is a homomorphic imag...
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| Format: | Article |
| Language: | English |
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Shahid Beheshti University
2013-12-01
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| Series: | Categories and General Algebraic Structures with Applications |
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| Online Access: | http://www.cgasa.ir/article_4262_73b32f9f16cd67536694bb804916b55f.pdf |
| _version_ | 1818821311786385408 |
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| author | Bernhard Banaschewski |
| author_facet | Bernhard Banaschewski |
| author_sort | Bernhard Banaschewski |
| collection | DOAJ |
| description | For any archimedean$f$-ring $A$ with unit in whichbreak$awedge (1-a)leq 0$ for all $ain A$, the following are shown to be equivalent: 1. $A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all integer-valued continuous functions on some frame $L$. 2. $A$ is a homomorphic image of the $l$-ring $C_{Bbb Z}(X)$ of all integer-valued continuous functions, in the usual sense, on some topological space $X$. 3. For any family $(a_n)_{nin omega}$ in $A$ there exists an $l$-ring homomorphism break$varphi :C_{Bbb Z}(Bbb Z^omega)rightarrow A$ such that $varphi(p_n)=a_n$ for the product projections break$p_n:{Bbb Z^omega}rightarrow Bbb Z$. This provides an integer-valued counterpart to a familiar result concerning real-valued continuous functions. |
| first_indexed | 2024-12-18T23:06:11Z |
| format | Article |
| id | doaj.art-9ca2a383409d4486ae016c56087047b7 |
| institution | Directory Open Access Journal |
| issn | 2345-5853 2345-5861 |
| language | English |
| last_indexed | 2024-12-18T23:06:11Z |
| publishDate | 2013-12-01 |
| publisher | Shahid Beheshti University |
| record_format | Article |
| series | Categories and General Algebraic Structures with Applications |
| spelling | doaj.art-9ca2a383409d4486ae016c56087047b72022-12-21T20:48:26ZengShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58532345-58612013-12-01111104262Countable composition closedness and integer-valued continuous functions in pointfree topologyBernhard Banaschewski0Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada.For any archimedean$f$-ring $A$ with unit in whichbreak$awedge (1-a)leq 0$ for all $ain A$, the following are shown to be equivalent: 1. $A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all integer-valued continuous functions on some frame $L$. 2. $A$ is a homomorphic image of the $l$-ring $C_{Bbb Z}(X)$ of all integer-valued continuous functions, in the usual sense, on some topological space $X$. 3. For any family $(a_n)_{nin omega}$ in $A$ there exists an $l$-ring homomorphism break$varphi :C_{Bbb Z}(Bbb Z^omega)rightarrow A$ such that $varphi(p_n)=a_n$ for the product projections break$p_n:{Bbb Z^omega}rightarrow Bbb Z$. This provides an integer-valued counterpart to a familiar result concerning real-valued continuous functions.http://www.cgasa.ir/article_4262_73b32f9f16cd67536694bb804916b55f.pdfFrames0-dimensional framesinteger-valued continuous functions on framesarchimedean ${mathbb Z}$-ringscountable $mathbb {Z}$-composition closedness |
| spellingShingle | Bernhard Banaschewski Countable composition closedness and integer-valued continuous functions in pointfree topology Categories and General Algebraic Structures with Applications Frames 0-dimensional frames integer-valued continuous functions on frames archimedean ${mathbb Z}$-rings countable $mathbb {Z}$-composition closedness |
| title | Countable composition closedness and integer-valued continuous functions in pointfree topology |
| title_full | Countable composition closedness and integer-valued continuous functions in pointfree topology |
| title_fullStr | Countable composition closedness and integer-valued continuous functions in pointfree topology |
| title_full_unstemmed | Countable composition closedness and integer-valued continuous functions in pointfree topology |
| title_short | Countable composition closedness and integer-valued continuous functions in pointfree topology |
| title_sort | countable composition closedness and integer valued continuous functions in pointfree topology |
| topic | Frames 0-dimensional frames integer-valued continuous functions on frames archimedean ${mathbb Z}$-rings countable $mathbb {Z}$-composition closedness |
| url | http://www.cgasa.ir/article_4262_73b32f9f16cd67536694bb804916b55f.pdf |
| work_keys_str_mv | AT bernhardbanaschewski countablecompositionclosednessandintegervaluedcontinuousfunctionsinpointfreetopology |