Minimal Presentations and factorizations

Minimal presentations¶

In mathematics many algebraic objects are given by means of (free) generators and some relations among them. For numerical semigroups it is sometimes useful to have representation of this form.

Let $S$ be a numerical semigroup minimally generated by $\{n_1,\dots,n_e\}$. Every element $s\in S$ admits an expression of the form $s=a_1n_1+\dots+ a_e n_e$, and for every $(a_1,\dots,a_e)\in \mathbb{N}^e$, the integer $a_1n_1+\dots+a_en_e\in S$. Thus, the following (monoid) morphism

is surjective. There is an isomorphism theorem for monoids, and consequently we have that $\mathbb{N}^e/\ker \varphi_S$ is isomorphic to $S$ as a monoid, where $\ker \phi_S=\{ (a,b)\in \mathbb{N}^e : \varphi_S(a)=\varphi_S(b)\}$. As a congruence, $\ker \varphi_S$ admints a system of generators, which is known as a presentation for $S$.

A minimal presentation for $S$ is just a minimal generating system of $\ker \varphi_S$.

Thus $\langle 2,3\rangle$ can be viewed as the monoid generated by $a$ and $b$, which fulfil the relation $3a=2b$.

Given $s\in S$, the set $\varphi_S^{-1}(s)$ corresponds to all the expressions that $s$ migh have in terms of the minimal generators of $S$. We will call this set, the set of factorizations of $s$, and we will denote it by $\mathsf{Z}(s)$.

A set ρ generates a congruence σ on $\mathbb{N}^e$ if for any pair $(a,b)\in \sigma$ there exists a chain $a_1,\dots,a_n$ such that

• $a_1=a$, $a_n=b$,
• for all $i$, there exists $c_i\in \mathbb{N}^e$ and $(\alpha_i,\beta_i)$ such that $(a_i,a_{i+1})=(\alpha_i+c_i,\beta_i+c_i)$ and either $(\alpha_i,\beta_i)\in \rho$ or $(\beta_i,\alpha_i)\in \rho$.

You can see the sequence $a_1,\dots,a_n$ as a chain of transformations from $a$ to $b$, and at each step we are applying a relation (or trade) chanbe $\alpha_i$ by $\beta_i$ (or viceversa).

For instance, in the above example $((10,0),(1,6))\in \ker\varphi_S$. We start with $(10,0)$ and apply the only minimal relation we have on our numerical semigroup: $((10,0),(10-3,2))\in \sigma$. If we do this a couple of times more, we obtain the chaing $(10,0)$, $(7,2)$, $(4,4)$, and $(1,6)$.

Assume that we want to check that $((5,1,0),(2,0,2))$ can be obtained from the minimal relations of $S$. Notice that these two factorizations have some common support, in fact $(5,1,0)=(3,1,0)+(2,0,0)$ and $(2,0,2)=(0,0,2)+(2,0,0)$. Thus, if we find a chain going from $(3,1,0)$ to $(0,0,2)$, then by adding $(2,0,0)$ to all the steps, we obtain a chain connecting our original factorizations. But we already have this chain, since $((0,0,2),(3,1,0))$ is already in our minimal presentation.

The idea can be extended to factorizations of larger elements in $S$. Whenever they have a common “factor”, remove it, and try to connect the new factorizations. This motivates the definition of $\mathcal{R}$-classes.

Let $X$ be a subset of $\mathbb{N}^e$. We say that $x,y\in X$ are connected if there exists a sequence $x_1,\dots,x_n\in X$ (for some $n\in\mathbb{N}$) such that

• $x_1=x$, $x_n=y$,
• for every $i$, $x_i$ and $x_{i+1}$ have common support.

The connected components of $X$ under this relation are called $R$-classes of $X$.

The idea is that whenever we are in the same $R$-class, we can connect any two factorizations with a chain where two consequtive elements have common support. For these two elements, we remove the common part, moving now to the factorizations of an element smaller than the original one. If these two new factorizations are in the same $R$-class, we repeat the process. This will end when we arrive at factorizations in different $R$-classes, and then we only have to connect all the chains that we found translated accordingly.

Let us illustrate this with an example.

We know that $((7,0,0),(0,0,3))\in \ker\varphi_S$, and we want to see how can we obtain this pair from the minimal relations of $S$. Both $(7,0,0)$ and $(0,0,3)$ are in the same $R$-class. We can, instance, connect them with the sequence $(7,0,0)$, $(3,1,1)$, $(0,0,3)$. By transitivity, if we find a chain of trades going from $(7,0,0)$ to $(3,1,1))$ and another from $(3,1,1)$ to $(0,0,3)$, by joining them we will find a chain from $(7,0,0)$ to $(0,0,3)$.

By removing the common part of $(7,0,0)$ and $(3,1,1)$, we obtain $(4,0,0)$ and $(0,1,1)$, and the pair $((4,0,0),(0,1,1))$ is in our minimal presentation. We do the same with $(3,1,1)$ and $(0,0,3)$, obtaining $(3,1,0)$ and $(0,0,2)$, and $((0,0,2),(3,1,0))$ is in our minimal presentation. Let $\sigma=\ker\varphi_S$. Then $(7,0,0)=((4,0,0)+(3,0,0))\sigma ((0,1,1)+(3,0,0))=(3,1,1)\sigma ((3,1,0)+(0,0,1))\sigma ((0,0,2)+(0,0,1))=(0,0,3)$.

In this construction, elements with more than one $R$-class are crucial. These elements are called Betti elements (or degrees) of $S$.

It is not hard to prove that if $s\in S$ is a Betti element, then $s=n_i+w$ with $i>2$ and $w\in \operatorname{Ap}(S,n_1)$. Thus the number of Betti elements is finite. As a consequence of this, all minimal presentations have the same cardinality, since we only need relations “connecting” different $R$-classes of the factorizations of the Betti elements of $s$. This also provides a way do determine all minimal relations of a numerical semigroup (up to symmetry).

Minimal presentations of numerical semigroups with embedding dimension two have cardinality one. Thos for embedding dimension three have cardinality two or three. There are numerical semigroups with embedding dimension four with arbitrarily large minimal presentations.

Minimal presentations and binomial ideals¶

Let $S$ be a numerical semigroup minimally generated by $\{n_1,\dots,n_e\}$. Let $K$ be a field and $K[x_1,\dots,x_e]$ be the polynomial ring on the variables $x_1,\dots, x_e$ with coefficients in $K$.

Let $t$ be another unknown. We can define the subring $K[S]=K[t^s :s \in S]=K[t^{n_1},dots,t^{n_e}]\subseteq K[t]$, which is known as the semigroup ring of $S$. Let $\psi_S:K[x_1,\dots,x_e]\to K[t]$ be the unique ring homomorphism determined by $\psi_S(x_i)=t^{n_i}$.

For $a=(a_1,\dots,a_e)\in \mathbb{N}^e$, write $x^a=x_1^{a_1}\cdots x_e^{a_e}$. Clearly, $\psi_S(x^a)=t^{\phi_S(a)}$, and consequently whenever $(a,b)\in \ker \psi_S$, we have that $x^a-x^b\in \ker \psi_S$. It is not hard to prove that

and that a set ρ generates $\ker \phi_S$ as a congruence if and only if $\{ x^a-x^b : (a,b)\in \rho\}$ generates $\ker\psi_S$ as an ideal.

In order to compute $\ker \psi_S$ we can use elimination in the following way.

We identify each variable to $t$ to the power of the corresponding minimal generator.

Now, we eliminate the variable $t$.

Next, we extract the exponents of the binomials obtained.

The resulting set does not have to be minimal in general. We can detect those factorizations that correspond to Betti elements.

Length based factorization invariants¶

Let $S$ be minimally generated by $\{n_1,\dots,n_e\}$. Recall that the set of factorizations of an element $s\in S$ is $\mathsf{Z}(s)=\varphi_S^{-1}(s)$, that is,

Sets of lengths of factorizations¶

The length of a factorization $z=(z_1,\dots,z_e)$ is the number of minimal generators involved in it, that is, $|z|=z_1+\dots+z_e$. We define the set of lenghts of the factorizations of $s$ as

Delta sets¶

One can arrange the lengths of the factorizations of an element $s$ in the following way $\mathsf{L}(s)=\{l_1<\dots<l_t\}$. The Delta set of $s$ is then defined as $\Delta(s)=\{l_2-l_1,l_3-l_2,\dots, l_t-l_{t-1}\}$.

The Delta set of $S$ is the union of all the delta sets of its elements.

Recall that a minimal presentation of $s$ was

And that one can “walk” from a factorization to any other factorization of the same element by using this minimal relations.

So it comes as no surprise that $\Delta(S)=\{2\}$. In fact, it is not difficult to prove $\min(\Delta(S))=\gcd\{ |a-b| : (a,b)\in \sigma\}$, with σ any minimal presentation of $S$, and that $\max(\Delta(S))=\max\{\Delta(b) : b\in \operatorname{Betti}(S)\}$.

The idea behing the minimum is that any possible “jump” will be a linear combination of the “jumps” in the minimal relations. As for the maximum, “jumps” are preserved under translations, and thus the largest “jump” will be achieved between two factorizations with no commom support and in a Betti element.

The structure of $\Delta(S)$ is very well known when $S$ has embedding dimension two or three, and even when it is generated by an arithmetice sequence.

The elasticity¶

The elasticity of the factorizations of an element $s$ in a numerical semigroup $S$ is ratio between the largest length and smallest lenght of its factorizations, that is,

The elasticity of the semigroup $S$ is defined as

For numerical semigroups, this supremum becomes a maximum, and the elasticity is attained at the element $n_1 n_e$ (the product of the smallest generator times the largest generator).

Distance based factorization invariants¶

Given $x=(x_1,\dots,x_e)$ and $y=(y_1,\dots,y_e)$ in $\mathbb{N}^e$, their “common part” is

and the distance between $x$ and $y$ as

We have already used DotEliahouGraph, the labels of the edges are the distances between de factorizations they connect.

The catenary degree¶

In addition, DotFactorizationGraph draws a minimum spanning tree.

In particular, this means that we can go from any factorization of 100 in $S$ to any other factorization of the same element by using a path such that two consecutive nodes are at a distance of at most four. This is precisely the idea behind the concept of catenary degree.

Let $z$ and $z'$ be two factorizations of $s$. An $N$-chain joining $z$ and $z'$ is a sequence $z_1,\dots,z_t$ of factorizations of $s$ such that $\operatorname{d}(z_i,z_{i+1})\le N$. The catenary degree of $s$, denoted $\operatorname{c}(s)$, in the minimum $N$ such that for any two factorizations of $s$ there exists an $N$-chain connecting them.

The catenary degree of $S$ is

This supremum is a maximum.

Recall that by using the minimal relations of $S$ we can find a path joining any two different factorizations of an element. Thus, it is not hard to prove that the maximum of the catenary degree of $S$ is attained at one of its Betti elements.

The tame degree¶

The catenary degree measures the minimum distance needed to find paths connecting any two factorizations of an element in the semigroup in such a way that every step in the path is withing that minimum distance.

The tame degree intends to measure a radius $r$ in which for any factorization $z$ of $s\in S$, with $s-n_i\in S$, you will find another factorization $z'$ such that $\operatorname{d}(z,z')\le r$ and $z_i'\neq 0$.

Let $s\in S$ such that $s-n_i\in S$ for some $i\in\{1,\dots,e\}$. Define $\mathsf{Z}_i(s)=\{z : z\in \mathsf{Z}(s), z_i\neq 0\}$, which is nonempty as $s-n_i\in S$. Set

and

The tame degree of $S$ is defined as

Usually the tame degre is not attained at the Betti elements.

It can be shown that the tame degree of $S$ is attained at an element $s\in S$ such that $s$ has a factorization in $\operatorname{Minimals}_\le (\mathsf{Z}(n_i+S))$ for some $i\in\{1,\dots,e\}$. Elements having this property are of the form $n_i+w$ with $w\in \operatorname{Ap}(S,n_j)$, more specifically, there exists $i,j\in \{1,\dots,e\}$ such that $s-n_i,s-n_j\in S$ and $s-(n_i+n_j)\not\in S$.

For $s\in S$, the Rosales graph $G_s$ is defined as follows. The vertices of $G_s$ are the minimal generators $n_i$ such that $s-n_i\in S$, and $n_in_j$ is an edge whenever $s-(n_i+n_j)\in S$.

The number of connected components of $G_s$ coincides with the set of $R$-classes of the set of factorizations of $s$. Thus the catenary degree is attained in an $s$ with $G_s$ not connected, and the tame degree in an $s$ with $G_s$ not complete.

Primality¶

Recall that $S$ induces an order over the integers $a\le_S b$ if $b-a\in S$. If $a$ and $b$ are in $S$, then we say that $a$ divides $b$ if $b-a\in S$. In this way, minimal generators (irreducibles, atoms, primitive elements) are those not having proper divisors. A natural question arises: are there “prime” elements in a numerical semigroup. The ones to be candidates to be prime are the minimal generators of the semigroup.

Let $S$ be minimally generated by $\{n_1,\dots,n_e\}$. Consider the set $\operatorname{Minimals}_\le (\mathsf{Z}(n_i+S))$. Notice that $S\setminus(n_i+S)=\operatorname{Ap}(S,n_i)$, and so the set $\mathsf{Z}(n_i+S)=\mathbb{N^e}\setminus \mathsf{Z}(\operatorname{Ap}(S,n_i))$. Let $z \in \operatorname{Minimals}_\le (\mathsf{Z}(n_i+S))$, $z\neq e_i$. Then $n_i$ divides $\varphi_S(z)$ and cannot divide any of its “factors”. Hence, $n_i$ cannot be prime.

Let $s\in S$. The ω-primality of $S$, $\omega(S,s)$, is defined as the least integer $N$ such that whenever $s$ divides $a_1+\dots+a_n$ for some $a_1,\dots,a_n\in S$, then $s$ divides $a_{i_1}+\dots+a_{i_N}$ for some $\{i_1,\dots,i_N\}\subseteq \{1,\dots,n\}$. By an argument similar to the one given above,

Observe that if $z\in \operatorname{Minimals}_\le(\mathsf{Z}(s+S))$, then $\varphi(z)=s+t$ for some $t\in S$. Let $i$ be such $z_i\neq 0$. Then $z-z_i\not\in \mathsf{Z}(s+S)$, and thus $s+t-n_i\not\in s+S$, which means that $t\in \operatorname{Ap}(S,n_i)$. This bounds the search for cmputing $\operatorname{Minimals}_\le(\mathsf{Z}(s+S))$.

There is an alternative way to compute the set $\operatorname{Minimals}_\le(\mathsf{Z}(s+S))$. Observe that $x\in \mathsf{Z}(s+S)$ if an only if $n_1x_1+\dots+x_en_e=s+n_1t_1+\dots+n_et_e$ for some $t_1,\dots,t_e\in \mathbb{N}$. So we can solve the problem by looking at the nonnegative integer solutions (in $\mathbb{N}^{2e}$ to the equation

and then project onto the first $e$ coordinates.

The omega primality of $S$ is defined as the maximum of the omega primalities of its minimal generators.

It can be shown that

Divisors¶

Let $s$ be an element in a numerical semigroup $S$. The set of divisors of $s$ is

The Feng-Rao distance of $s$ is defined as

Let $c$ be the conductor of $S$ and let $g$ be its genus. It can be shown that for $s\ge 2c-1$,

The generalized Feng-Rao distance is defined as

where $\operatorname{D}(s_1,\dots,s_r)=\bigcup_{i=1}^r \operatorname{D}(s_i)$. It can be shown that for $s\ge 2c-1$,

The constant $E(S,r)$ is known as the $r$th Feng-Rao number. For $r=2$,

The definition of the Apéry set for elements not in $S$ is the same as the one given above.