/ / / / / /

上一篇 下一篇 同標題 發表文章 文章列表

作者  emom (一刀) 站內  ALGEBRA
標題  Re: [問題] semigroup
時間  2010/10/13 Wed 19:57:02

※ 引述《kyod ( )》之銘言:
> (1)
> Is it true that a semigroup which has a left identity element
> and in which every element has a right inverse is a group ?
> 這題我的想法是想要找一個binary operation 然後有left identity
> and right inverse ,再利用群的性質(left identity = right identity)
>                                   (left inverse = right inverse)
> 製造出矛盾。

符號不好打,只回答 (1):

Let G be the set consisting all real 2 by 2 matrix A
s.t. the left-up entry of A is non-zero and both the left-down
and right-down entries of A are 0.
Under usual matrix multiplication, G is a semi-group.
Then any left identity has the form [ 1 k ] and
                                    [ 0 0 ]
for any [ a b ] in G, [ 1/a 0 ] is a right inverse.
        [ 0 0 ]       [  0  0 ]
If [ x y ] is a left inverse of [ 1 0 ],
   [ 0 0 ]                      [ 0 0 ]
then [ 1 k ] = [ x y ] [ 1 0 ] = [ x 0 ]
     [ 0 0 ]   [ 0 0 ] [ 0 0 ]   [ 0 0 ]
So k must be 0.
If [ x y ] is a left inverse of [ 1 1 ],
   [ 0 0 ]                      [ 0 0 ]
then [ 1 0 ] = [ x y ] [ 1 1 ] = [ x x ]
     [ 0 0 ]   [ 0 0 ] [ 0 0 ]   [ 0 0 ]
This leads to a contradiction.
Thus G is not a group.


--
發信站 [中央數學  織夢天堂 bbs.math.ncu.edu.tw]
  •FROM [emom 從 pc233.math.ncu.edu.tw 發表]

上一篇 下一篇 同標題 發表文章 文章列表