Somewhere Colin M. Campbell noted:



> 
> If $A$ is a semigroup defined as $$A=Sg(\pi)=\langle a\_1,\cdots, a\_d\mid u\_1=v\_1,\cdots,u\_e=v\_e\rangle $$ then the same generators with the same relations can also be interpreted as the presentation of the following group:
>  $$A^\*=Gp(\pi)=\langle a\_1,\cdots, a\_d\mid u\_1=v\_1,\cdots,u\_e=v\_e\rangle $$
> 
> 
> 


So he consider a semigroup first and then its possible analogous group form. I am working on this ideas and asking if we can do converse direction? I mean can we start with a (for example) finite group such as $$D\_8=\langle x,y\mid x^2=y^4=(xy)^2=1\rangle$$ and then construct a finite semigroup accordingly? I see, I can write $x^2=1$ as $x^3=x$ without any problems but I want to be sure about doing this job. Sorry if this question seems ridiculous for you. Thanks for your comments and your time.


**Edit:** By saying `corresponding semigroup`, I mean one possible related semigroup.



Using group presentation for its corresponding semigroup?
You can do this in some examples. For example in braid groups or, more generally, Artin groups, you can just interpret the group presentation immediately as a semigroup (or monoid) presentation. In general, in that situation, the semigroup defined will not embed into the group. An obvious example is $\langle x \mid x^4=x^2 \rangle$, which defines a semigroup of order $3$ and a group of order $2$. It turns out that, for Artin groups, the semigroup (or monoid) does embed into the group, but that is a nontrivial result.


If $1$ occurs in the presentation, as in your example, then you can still interpret it as a monoid presentation, and it is easy to see that the monoid defined by $\langle x,y \mid x^2 = y^4 = (xy)^2 = 1 \rangle$ is the same as the group. That is essentially because the first two relations tell you that $x$ and $y$ have inverses.


But, in general, the relations of a group presentation, such as the surface group $\langle w,x,y,z \mid wxw^{-1}x^{-1}yzy^{-1}z^{-1} \rangle$ will contain inverses of generators, and I don't see any way that you can make a semigroup or monoid out of that, unless you want to adjoin the inverses as extra generators.


