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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Brandt semigroup ： ウィキペディア英語版 Brandt semigroup In mathematics, Brandt semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideals and which are also inverse semigroups. They are built in the same way as completely 0-simple semigroups: Let ''G'' be a group and $I,\; J$ be non-empty sets. Define a matrix $P$ of dimension $|I|\backslash times\; |J|$ with entries in $G^0=G\; \backslash cup\; \backslash .$ Then, it can be shown that every 0-simple semigroup is of the form $S=(I\backslash times\; G^0\backslash times\; J)$ with the operation $(i,a,j)$ *(k,b,n)=(i,a p_ b,n). As Brandt semigroups are also inverse semigroups, the construction is more specialized and in fact, I = J (Howie 1995). Thus, a Brandt semigroup has the form $S=(I\backslash times\; G^0\backslash times\; I)$ with the operation $(i,a,j)$ *(k,b,n)=(i,a p_ b,n). Moreover, the matrix $P$ is diagonal with only the identity element e of the group G in its diagonal. ==Remarks== 1) The idempotents have the form (i,e,i) where e is the identity of G 2) There are equivalent way to define the Brandt semigroup. Here is another one: ac=bc≠0 or ca=cb≠0 ⇒ a=b ab≠0 and bc≠0 ⇒ abc≠0 If ''a'' ≠ 0 then there is unique ''x'',''y'',''z'' for which ''xa'' = ''a'', ''ay'' = ''a'', ''za'' = ''y''. For all idempotents ''e'' and ''f'' nonzero, ''eSf'' ≠ 0 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Brandt semigroup」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.