If $G$ is a finite semi-group and $\forall x,y,z \in G: xy=yz \implies x=z$ then $G$ is an Abelian group
If $G$ is a finite semi-group and $\forall x,y,z \in G: xy=yz \implies x=z$ then $G$ is an Abelian group. I have no idea where to start. I'm stuck! I can't prove even the existence of the identity ...
For $n\geq 3$, Is $A_n$ (the alternating group) is the proper subsemigroup of largest size of $S_n$ (the symmetric group of degree $n$)?
I am trying to track down the name of this structure and some references. You take all members of the transformation semigroup on $n$ elements, $T_n$. For two members $x, y\in T_n$: if $x$ is in the ...
Let $(G,.)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ...