# Help with proof: The group of left quotients of a right reversible, cancellative semigroup is unique up to isomorphism.

I am trying to understand the proof that the group of left quotients of a right reversible, cancellative semigroup is unique up to isomorphism, which can be found in Clifford's $$`$$The Algebraic Theory of Semigroups':

Clifford, A. H.; Preston, G. B., The algebraic theory of semigroups. Vol. I., Mathematical Surveys. 7. Providence, R.I.: American Mathematical Society (AMS). xv, 224 p. (1961). ZBL0111.03403.

However some steps are skipped in the proof and I'm struggling to connect each step. Here's what I have so far:

We want to find an isomorphism between two groups of left quotients of S, call them $$(G,\cdot)$$ and $$(G', \circ)$$. Elements in these groups are of the form $$a^{-1}b$$ and $$a'^{-1}\circ b'$$ respectively, where $$a,b,a',b'\in S$$. Hence, we want an isomorphism mapping $$a^{-1}b\mapsto a^{-1}\circ b$$. We essentially have the form of isomorphism that we need - we now need to verify it is indeed an isomorphism.

Now, we can express any element of the form $$ab^{-1}$$ in $$G$$ as $$x^{-1}y$$, for some $$x,y\in S$$. This implies that $$xa=yb$$. Clifford then states that for $$ab^{-1}, cd^{-1}$$ in $$G$$, we have that $$ab^{-1}=cd^{-1}\iff (xa=yc \iff xb=yd)$$, for $$a,b,c,d\in S$$.

Problem (1): I'm not sure why this is relevant to showing that our map is an isomorphism, and

Problem (2): I'm not sure how to prove this statement.

Can anyone give any hints as to how to address these problems?

• Is $S$ a group or how are inverses defined ? Commented Oct 14, 2021 at 14:16
• @MaximilianJanisch, S is a semigroup which is right reversible and cancellative. Commented Oct 14, 2021 at 15:57
• @MaximilianJanisch I think the inverses are defined in terms of semigroups, i.e. two elements a and b are inverses of each other if aba=a and bab=b. I assume $S$ is an inverse semigroup, so we have a unique inverse for each element of $S$. Then, we use usual inverse notation. Commented Oct 14, 2021 at 16:57