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3
votes
1answer
74 views

Period of semigroup

Let $S$ be a finite semigroup of order $n$. Suppose that $S$ has index $m$ and period $r$, i.e. $S$ satisfies the identity $x^{m+r} = x^m$. Then it is quite easy to show that $m \leq n$. My question ...
4
votes
1answer
39 views

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 ...
0
votes
1answer
76 views

Subsemigroup of S_n(symmetric group) of largest size

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 $n$)?
0
votes
1answer
64 views

Name the digraph of these transformation subsemigroups

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 ...
6
votes
5answers
848 views

Is there an idempotent element in a finite semigroup?

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? ...