# Proving two elements of a set are equal based on a two-sided identity

Say I have a set S w/ an associative binary operation *: S x S -> S and a two-sided identity e, and let $s\varepsilon S$. Let $\widetilde{s}$L and $\widetilde{s}$R be elements of S such that $\widetilde{s}$L * s = e = s * $\widetilde{s}$R

How can I prove that $\widetilde{s}$L = $\widetilde{s}$R ?

Since the left and right identity are equal to e which is a two-sided identity, that proves that the left and right are equal just by the definition of a two-sided identity right? Is there some more in-depth proof I'm not seeing here?

• Does this hold for all $s \in S$, or do $\tilde{s}_L$ and $\tilde{s}_R$ depend on $s$? (Note: this was typed as Does this hold for all $s \in S$, or do $\tilde{s}_L$ and $\tilde{s}_R$ depend on $s$?) – pjs36 Sep 8 '15 at 22:35
• I thought they were the same thing. I'll be honest, I don't know what the squiggly line over the s stands for. But I figured since s is an element in S then $\tilde{s}_L$ and $\tilde{s}_R$ are as well. – pfinferno Sep 8 '15 at 23:10

$$\tilde s_R=e*\tilde s_R=(\tilde s_L*s)*\tilde s_R=\tilde s_L*(s*\tilde s_R)=\tilde s_L*e=\tilde s_L\;.$$
• @pfinferno: I'd suggest that when you use notation in a question that you don't understand, you should point that out -- it would make it easier to understand where your problems lie, and what sort of answer might be helpful. I don't think this tilde is a standard notational convention. I suspect it's being used to mark inverses. In any case $\tilde s_L$ and $\tilde s_R$ are left and right inverses of $s$, respectively, and the statement being proved is that with a two-sided identity left and right inverses are equal. – joriki Sep 9 '15 at 15:21