# Semigroups: Do the following 4 equations imply x=y?

• June 1st 2010, 03:32 PM
kiwijoey
Semigroups: Do the following 4 equations imply x=y?
xayxxa = xayxxayxya = yayxyaxyya = yaxyyaxyxa = xaxy
• June 1st 2010, 03:34 PM
kiwijoey
Why didn't that post correctly?Equation 1: xayxxa = xayxEquation 2: xayxya = yayxEquation 3: yaxyya = yaxyEquation 4: yaxyxa = xaxy
• June 2nd 2010, 12:37 AM
Swlabr
Quote:

Originally Posted by kiwijoey
Why didn't that post correctly?Equation 1: xayxxa = xayxEquation 2: xayxya = yayxEquation 3: yaxyya = yaxyEquation 4: yaxyxa = xaxy

Well...no. Every relation is of the form $u=v$ where $|u| = 6$ and $|v|=4$. As you are in a semigroup there is no way of getting a relation of the form $u=v$ where $|u|=1=|v|$ as you cannot cancel anything.

What do you mean by the element $a$?