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### The Krull-Schmidt-Remak Theorem for Semigroups and Monoids

For finite groups, the Krull-Schmidt-Remak-Theorem holds, i.e. if $$H_1 \times H_2 \times \ldots \times H_k \cong G_1 \times G_2 \times \ldots \times G_l$$ where the $H_i, G_i$ could not be further ...
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### Is there a logarithmic size generating set for some classes of finite semigroups?

Following my question Why is the minimum size of a generating set for a finite group at most $\log_2 n$?, we know that finite groups have generating sets of size at most $\log_2 n$, and a similar ...
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### 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 ...
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### 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 ...
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### 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 $n$)?
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### 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 ...
Let $(G,\cdot)$ 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? ...
Possible Duplicate: A cyclic subsemigroup of a semigroup S that is a group My homework: An element $s^{i+k}$ on the cycle is idempotent iff $$s^{i+k} = s^{2i+2k} ,$$ or equivalently  ...