Something similar to Kronecker basis theorem for semigroup

Question

I know for abelian group, there is a Kronecker decomposition theorem. It said any finite abelian group can be factored as direct sum of cyclic group of prime power order. I want to know is there anything similar with this for the abelian semigroup with identity? As for semigroup, we don't need the inversion, so it seems like it can also support some kind f factorization.

Answer 1

Not quite, but there is a weaker result. Let us start by a simple observation.

Lemma 1. Every finite commutative monoid is a quotient of the product of its monogenic submonoids.

Proof. Let $M$ be a commutative monoid and let $N$ be the product of its monogenic submonoids. Let $f : N \to M$ be the monoid morphism which maps each element of $N$ to the product of its coordinates. Then $f$ is clearly surjective and thus $M$ is a quotient of $N$.

It remains to describe the structure of finite monogenic monoids. A threshold monoid is a monoid of the following form, for some $t > 0$. It is defined on the set $\{0, 1, 2, \ldots, t\}$ by the (additive) operation $$ r + s = \min(r + s , t) $$

I let you verify the following result:

Lemma 2. Every finite monogenic monoid is a submonoid of the product of a threshold monoid by a cyclic group.

Recall that a monoid $M$ divides a monoid $N$ if $M$ is a quotient of a submonoid of $N$. One can show that division is a partial order on finite monoids (up to isomorphisms). The decomposition theorem for finite commutative monoids can now be stated as follows:

Theorem. Every finite commutative monoid divides a product of cyclic groups and threshold monoids.