Gee Law’s Blog

A note on finitely generated/related/presented groups

Group presentation is a method to express groups as a quotient of a free group and a normal subgroup. In a presentation, there are generators and relators, and the presentation yields a ‘minimalist implementation’ of these generators and relations. Some cases of special interests are finitely generated/related/presented groups. From the definition, finite generation + finite relation is not linguistically equivalent to finite presentation. There has been some discussion on the relation of the three properties for several other algebraic structures. This entry proves that finite generation + finite relation imples finite presentation for groups.


理想(ideal)是出现在抽象代数(更特别地,环论)中的一类对象,其命名很有渊源。之前我只知道这个名字来源于一位数学家研究惟一分解时引入的“理想数”(该研究的思路和一般化最终形成了现代数学语言中“理想”的定义),但是具体的细节不是很明白;最近在阅读基于格(lattice)的密码学的内容,其中提到了环上的格(ring lattice),其中提到了理想格(环的一个子集,同时满足格和理想的要求),好奇心一发不可收拾,读了一些数学历史故事,对“理想数”加深了认识,记录下来以飨观众。文章最后讨论了“理想数”这个翻译的不达意之处。

Blogging my blog

A meta-blog that keeps the trace of my blog site. You will find the font size of this website less intimidating now. Plus, Google Plus share button has been removed.

PowerShell 6.1.0 (preview) messes verbs of directories, causing Previous Versions to open previous verions of a directory in PowerShell

Issue #6799 in PowerShell repository on GitHub raises the question why Previous Versions is opening previous versions of a directory in PowerShell, instead of File Explorer as expected. The reason is that PowerShell 6.1.0 (preview) installer creates a context menu to open the shell on any folder. However, it unfortunately names the verbs as open/runas, which is very dangerous since they are canonical and do not make much sense for a folder. Related is issue #7815, where shortcuts to a folder are opened in PowerShell.

用 Outlook 对象模型辅助学术日历管理


用 Outlook 对象模型辅助学术日历管理

Outlook 2016 GUI creates an appointment with extraneous attendee, causing and Outlook for iOS to show it as a meeting

Outlook 2016 GUI always creates an event with at least one attendee, which makes and Outlook for iOS have difficulty correctly interpreting the appointment. Instead, they think it is a meeting. However, if you create the event with Outlook object model, there is a good chance the event is created neatly. In addition, events created on or in Outlook for iOS are always neat.

Hosting a preview handler in WPF, correctly, part 2: interop

Continuing ‘Hosting a preview handler in WPF, part 1: UI and file associations’, we will do the major COM interop in this entry. Many people have failed to host the handler correctly because of careless implementation, so look carefully! Also, we will demonstrate some broken preview handlers, including ‘Adobe PDF Preview Handler for Vista’, ‘Microsoft Word previewer’ and ‘Microsoft PowerPoint previewer’.