For a group $G=〈S|R〉$, $S,R$ are both finite. A positive element $g\in G$ is an element of $G$ that can be written as a finite product of elements of $S$ only. A positive expression of $g$ is a word $w$ with elements of $S$ its alphabet and only contain elements of $S$ and $w$ reduce to $g$ after applying relations in $R$. A basic positive replacement of $G$ is a relation $w_1=w_2$, with $w_1w_2^{-1}$ or $w_2w_1^{-1}$ a relation in $R$ and $w_1,w_2$ are positive expressions of the same positive element.

Is it true that any positive expression $w$ for any positive element $g$ can be deformed into another positive expression $w'$ of $g$ only by finite applications of basic positive replacements?

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    $\begingroup$ The word problem for groups is known to be undecidable. $\endgroup$ – Michael Burr Jul 13 '15 at 4:28
  • $\begingroup$ It looks like all of your basic replacements are invertible, so if $w$ reduces to $g$ and $w'$ reduces to $g$, you might be able to start with $w$, reduce to $g$ and then work backwards to $w'$. $\endgroup$ – Michael Burr Jul 13 '15 at 4:30
  • $\begingroup$ Thank you very much! But practically in my application relations are restricted to $ab=ba$ and $ab=bca$ types. This reduction to $g$ is exactly what I look for.. $\endgroup$ – Ying Zhou Jul 13 '15 at 5:15
  • $\begingroup$ If you have a specific case, it might be good to post that as a different question. Although the definition of a positive expression is somewhat unusual, I think that your statement may be true in general because the steps are invertible. $\endgroup$ – Michael Burr Jul 13 '15 at 5:17

It is not clear from your description whether you consider the identity element to be positive. Since it is the empty product of elements of $S$, I am going to assume that it is positive.

Consider the presentation $\langle x,y \mid y^2=1, yxy=x^2 \rangle$. As a group presentation, it defines a group of order $6$ (the dihedral group). But as a monoid presentation it defines a monoid of order $8$, with elements $\{ 1,x,x^2,x^3,y,xy,yx,xyx\}$. So the equations $x^3=1$ and $xyx=y$ are true in the group, but not in the monoid.

But the only basic positive replacements are $y^2 \leftrightarrow 1$ and $yxy \leftrightarrow x^2$, which are valid in the monoid, so the positive relation $x^3=1$ in the group cannot be carried out using basic positive replacements.

  • $\begingroup$ I don't understand your comment. Do you have some problem with the example? The group defined by that presentation is definitely $S_3$. $\endgroup$ – Derek Holt Jul 13 '15 at 21:30
  • $\begingroup$ I get it now..Take $G\rightarrow Inn(G)$ and I can prove $x^3=1$ if $G$ is a group, otherwise we only get $x^4=x$. $\endgroup$ – Ying Zhou Jul 14 '15 at 16:09

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