# Semi-Groups

• Feb 22nd 2006, 04:18 AM
topsquark
Semi-Groups
I've got one of my own:

Let G be a semi-group. (That means G is a set with an associative binary operation on it.) Assume further that in G:
1) There exists a left identity e such that ea=a for all a in G and
2) For all a in G there exists a right inverse $a^{-1}$ such that $aa^{-1}=e$.

The question: Is G a group?

My answer is "no" but I don't have anything like a proof. What I'm stuck on is the uniqueness of e...I can prove that if there is a d such that da=a for all a in G then $d^{-1}=e^{-1}=e$, but I can't prove d=e or d is not equal to e...all I have is there doesn't appear to be a contradiction in assuming that more than one left identity exists. What would really be nice is if there were an example showing that G is not a group. (Or, alternately, a proof that it is.)

Any suggestions? Thanks!

-Dan
• Feb 22nd 2006, 02:55 PM
ThePerfectHacker
Quote:

Originally Posted by topsquark
I've got one of my own:

Let G be a semi-group. (That means G is a set with an associative binary operation on it.) Assume further that in G:
1) There exists a left identity e such that ea=a for all a in G and
2) For all a in G there exists a right inverse $a^{-1}$ such that $aa^{-1}=e$.

The question: Is G a group?

My answer is "no" but I don't have anything like a proof. What I'm stuck on is the uniqueness of e...I can prove that if there is a d such that da=a for all a in G then $d^{-1}=e^{-1}=e$, but I can't prove d=e or d is not equal to e...all I have is there doesn't appear to be a contradiction in assuming that more than one left identity exists. What would really be nice is if there were an example showing that G is not a group. (Or, alternately, a proof that it is.)

Any suggestions? Thanks!

-Dan

G is not a group. Consider this if $ex=x$ for all $x$ implied that $ex=xe$ then why then is it that we define one of the properties of a group such that $ex=xe=x$? Because there are cases where it is not. I cannot think of a case but semi-groups are hardly ever studied.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://mathhelpforum.com/advanced-algebra/1966-semi-groups-print.html", "fetch_time": 1505965434000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2017-39/segments/1505818687606.9/warc/CC-MAIN-20170921025857-20170921045857-00254.warc.gz", "warc_record_offset": 216295076, "warc_record_length": 3125, "token_count": 577, "char_count": 2056, "score": 3.515625, "int_score": 4, "crawl": "CC-MAIN-2017-39", "snapshot_type": "longest", "language": "en", "language_score": 0.9541124701499939, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00045-of-00064.parquet"}