Presentation of groups and positive expressions 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?

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.
