弱空间式Locale和Locale的反射性

【作者】 孙向荣；

【导师】 贺伟；

【作者基本信息】 南京师范大学 ， 基础数学， 2007， 博士

【摘要】 本文对locale范畴的乘积保持性以及反射性进行了研究，得到的主要结果如下：1．引入弱空间式locale的定义，证明弱空间式locale范畴WSloc为范畴Loc的余反射满子范畴，弱空间式locale的乘积是弱空间式的。给出了弱空间式locale的一组等价刻画。特别地，证明了一个locale A是空间式的当且仅当它的核映射localeN(A)是弱空间式的；一个空间式locale的每一个子locale都是空间式的当且仅当它的每一个子locale是弱空间式的。最后，证明了弱空间式性在定向函子下保持不变。2．在locale中引入正则元，完全正则元以及零维元的概念，构造性的给出了任意locale的正则反射，完全正则反射以及零维反射的描述。对于满足‘(?)’关系插入性的locale，证明了弱正则元与正则元等价，给出其更好形式的正则反射构造。进一步，利用平稳(flat)子locale的扩张引理给出了locale的紧正则反射，紧完全正则反射以及紧零维反射的构造性描述。3．对于正规locale，给出其Banaschewski-Mulvey形式的紧正则反射，并证明其与Johnstone形式的Wallman紧化一致，从而推广了Johnstone在文献[]中的主要结果。进一步讨论其与前一章得到的紧正则反射之间的关系，证明了A的理想格上的正则元就是A的正则理想，说明了几种正则紧反射构造的等价性。4．对于满足‘(?)’关系插入性的locale，运用前一章得到的紧正则反射，证明了满足插入性的可紧化locale的紧正则反射保持和反射连通性。因正规locale满足插入性，推广了B.Banascewski的相关结果，部分解决了文献[21]的一个问题。5．引入限制反射子范畴的定义。证明了：若范畴B是A在范畴C上的限制反射子范畴，范畴C是B的反射子范畴，则范畴C是A的反射子范畴，即给出更弱形式的反射子范畴的复合定理。更多还原

【Abstract】 In this dissertation, we investigate some properties on locales. The main results are summarized as follows.1. The concept of weakly spatial locale is presented. We show that the category WSLoc of weakly spatial locales is a coreflective subcategory of the category Loc and the localic product of weakly spatial locale is still weakly spatial. Moreover, we show that a locale A is spatial if and only if the nucleus locale N(A) is weakly spatial. Every sublocale of a locale A is spatial if and only if every sublocale of A is weakly spatial. Also we prove that the weekly spatiality is preserved by directed functors.2. By introducing regular elements, completely regular elements and zero-dimensiomal elements, we give constructive descriptions of the completely regular reflection and the zero-dimensional reflection of locales in much simple form. For a locale A, which satisfies the subdivisibility property, we show that the locale R(A) of regular elements is the regular reflection of A. Furthermore, we obtain the compact regular reflection, the completely regular reflection and the compact zero-dimensional reflection of locales by the extension lemma of flat sulocale.3. For normal locales, we prove that the Banaschewski-Mulvey’s compact regular reflection construction of locales is isomorphisc to the Johnstone Wallman com-pcactification of locales. We show that a subfit semi-normal locale is normal, but the converse is not true in general. Furthermore, we generalized the main result in[76].4. By applying the compact regular reflection construction of locales given in the former chapter, we show that the compact regular reflection preserves and reflects connectedness for compactifiable locales, which satisfies the subdivisibility property.5. we give a weakly form of the composition theorem of reflectors.更多还原