Width and height of numerical semigroups

Every numerical semigroup is partially ordered by addition. A first step towards numerical semigroup theory is to classify the types of partial orders that they entail. The most basic properties of a poset are the size of maximal antichains and chains: the width and the height of the poset. These two properties will be determined for numerical semigroups.

Theorem. let $S$ be a numerical semigroup. Then $S$ has infinite height.

Proof. let $S$ be a numerical semigroup, then $S$ is idempotent-free. Therefore, let $x \in S$ then the semigroup generated by $(x)$ is equal to $\mathbb{Z}_+$ which forms an infinite chain as a poset. Therefore, $S$ has an infinite chain, so it is of infinite height. $\square$

The maximal chains of a numerical semigroup all have order type $\omega$, same as the order type of $\mathbb{Z}_+$ so the chain problem is completely solved for numerical semigroups. The determination of maximal antichains and the width of a numerical semigroup is a bit more involved because they can take on multiple values.

Theorem. let $S$ be a numerical semigroup. Then the width of $S$ is the multiplicity $m(S)$.

Proof. let $S$ be a numerical semigroup with $m(S)$ its multiplicity. Let $(m(S))$ be the subsemigroup of $\mathbb{N}$ that is generated by $m(S)$ then $(m(S))$ has width $m(S)$ with maximal antichains consisting of any maximal set of representatives modulo $m(S)$. Consider the preorder induced by $m(S)$ acting on $S$ then it is an induced suborder of $(m(S))$ acting on $\mathbb{N}$ and so must have width at most $m(S)$. The order type of $S$ is an order extension of the action preorder of $(m(S))$ acting on $S$, so its width must be less then $m(S)$.

This establishes an upper bound on the width of $S$. In order to get the lower bound, consider that every numerical semigroup is cofinite. Therefore, $S$ must have a maximal system of representatives modulo $m(S)$ because if it did not there would be an infinite set of terms modulo some value of $m(S)$ missing from $S$. Therefore, the width is greater then or equal to $m(S)$. Finally, because $m(S) \leq width(S) \leq m(S)$ we have that $width(S) = m(S)$. $\square$

Isomorphism types of commutative J-trivial semigroups of a given order type are partially ordered pointwise. Semilattices are always the least commutative J-trivial semigroup of a given order type, because they produce the least upper bound. Commutative J-trivial semigroups other then semilattices arise from producing non-minimal upper bounds on partial orders.

Proposition. a numerical semigroup is an upper bound producing function for a finite width and infinite height partial order.

The classification of the width and the height of a numerical semigroup is just the first step towards describing the order type of the semigroup. Of course, it is not enough to describe a commutative J-trivial semigroup by its order type because there are many isomorphism types of commutative J-trivial semigroups on a given partial order, but this is just a small part of the theory.

Definition. the lattice of numerical semigroups $L$ is the set of all cofinite submonoids of $\mathbb{N}$. $L$ is a completely join closed and finite intersection closed sublattice of $Sub(\mathbb{N})$.

It is not hard to see that $m : L \to \mathbb{Z}_+$ is antitone, so that equivalently the width of a numerical semigroup is antitone much like in the case of partial orders. Again, much like in the case of partial orders every numerical semigroup has a linear extension. Except in this case, there is only one linear extension of a numerical semigroup: $\mathbb{N}$.