# Coefficient Rings of Numerical Semigroup Algebras

I-Chiau Huang  Institute of Mathematics, Academia Sinica, 6F, Astronomy-
Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan,
R.O.C<EMAIL_ADDRESS>and  Raheleh Jafari  Mosaheb Institute of
Mathematics, Kharazmi University, and School of Mathematics, Institute for
Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
<EMAIL_ADDRESS>
###### Abstract.

Numerical semigroup rings are investigated from the relative viewpoint. It is
known that algebraic properties such as singularities of a numerical semigroup
ring are properties of a flat numerical semigroup algebra. In this paper, we
show that arithmetic and set-theoretic properties of a numerical semigroup
ring are properties of an equi-gcd numerical semigroup algebra.

###### Key words and phrases:

coefficient ring, Frobenius monomial, gap monomial, irreducibility, numerical
semigroup algebra, radical, symmetry.

###### 2010 Mathematics Subject Classification:

13B02, 20M25

Raheleh Jafari was in part supported by a grant from IPM (No. 99130112).

##  1\. Introduction

A numerical semigroup $S$ is a monoid generated by finitely many positive
rational numbers. In the literature, it is often assumed that
$S\subset{\mathbb{N}}$ and $\gcd(S)=1$. With these assumptions, the numerical
semigroup ring is defined as the subring

(1) $\kappa[\\![\mathbf{u}^{S}]\\!]:=\Big{\\{}\sum_{s\in
S}a_{s}\mathbf{u}^{s}\colon a_{s}\in\kappa\Big{\\}}$

of the power series ring $\kappa[\\![\mathbf{u}]\\!]$ over a field $\kappa$.
In addition to $R=\kappa[\\![\mathbf{u}^{S}]\\!]$, we also use the notation
$S=\log_{\mathbf{u}}R$ to indicate the correspondence between logarithms and
exponents of semigroups and rings. Because of the correspondence, one may work
either on the logarithmic side or the exponential side of the theory. In this
paper, we choose to stay in the exponential side in compliance with the
previous study of ring theoretic properties such as Cohen-Macaulayness,
Gorensteiness and complete intersection [5]. The reader preferring semigroups
can switch freely to the logarithmic side. Our relative viewpoint on
arithmetic and set-theoretic properties such as symmetry, irreducibility and
Cohen-Macaulay type suggests further developments for numerical semigroups.

In the literature, the numerical semigroup ring (1) is commonly denoted as
$\kappa[\\![S]\\!]$, or $\kappa[\\![s_{1},\ldots,s_{n}]\\!]$ if $S$ is
generated by $s_{1},\ldots,s_{n}$. Although the common notation leads to no
misunderstanding in the classical situation, the appearance of rings such as
${\mathbb{Q}}[\\![3,5,7]\\!]$ are vague in distinguishing elements $3,5,7$ in
the underlying field $\mathbb{Q}$ from exponents $3,5,7$ of monomials
$\mathbf{u}^{3},\mathbf{u}^{5},\mathbf{u}^{7}$. To emphasize the relative
nature, our notation is more handy in adding monomials to a numerical
semigroup ring. For instance, given a numerical semigroup ring $R$ in the
variable $\mathbf{u}$, we may add a monomial $\mathbf{u}^{t}$ to $R$ and
obtain $R[\\![\mathbf{u}^{t}]\\!]$.

The assumptions $S\subset{\mathbb{N}}$ and $\gcd(S)=1$ for a numerical
semigroup $S$ are not essential for the construction of
$\kappa[\\![\mathbf{u}^{S}]\\!]$. For an arbitrary numerical semigroup $S$, we
may still define a numerical semigroup ring $\kappa[\\![\mathbf{u}^{S}]\\!]$
as (1) allowing for monomials in $\mathbf{u}$ to have rational exponents.
Given a positive rational number $t$, the numerical semigroup ring
$\kappa[\\![\mathbf{v}^{tS}]\\!]$ is isomorphic to
$\kappa[\\![\mathbf{u}^{S}]\\!]$ via the symbolized relation
$\mathbf{v}^{t}=\mathbf{u}$. For a numerical semigroup $S$, there is a unique
positive rational number $t$ such that $tS\subset{\mathbb{N}}$ and
$\gcd(tS)=1$. Therefore any numerical semigroup ring is isomorphic to one
constructed from a numerical semigroup $S$ with the conditions
$S\subset{\mathbb{N}}$ and $\gcd(S)=1$.

A numerical semigroup ring has various algebra structures. It is an algebra
over the underlying field, as well as over other subrings. We consider the
category of numerical semigroup rings. A morphism in the category gives rise
to an algebra. As seen in [4, 5], such a relative viewpoint sheds light to the
algebraic properties of numerical semigroup rings. In this paper, we show that
arithmetic and set-theoretic properties of a numerical semigroup ring are also
relative in nature. To be precise, we consider numerical semigroups $S\subset
S^{\prime}$. The numerical semigroup ring
$R^{\prime}:=\kappa[\\![\mathbf{u}^{S^{\prime}}]\\!]$ is an algebra over the
numerical semigroup ring $R:=\kappa[\\![\mathbf{u}^{S}]\\!]$. We denote the
algebra $R^{\prime}$ together with the coefficient ring $R$ by $R^{\prime}/R$
and call it a numerical semigroup algebra. A numerical semigroup algebra and
its coefficient ring can be represented using different variables. In this
paper, we use the same variable for a numerical semigroup algebra and its
coefficient ring.

In the classical case that $S\subset{\mathbb{N}}$ and $\gcd(S)=1$, there
naturally arise two numerical semigroup algebras:

  * •

The ring $\kappa[\\![\mathbf{u}^{S}]\\!]$ is an algebra over a Noether
normalization $\kappa[\\![\mathbf{u}^{s}]\\!]$, where $0\neq s\in S$.

  * •

The ring $\kappa[\\![\mathbf{u}^{S}]\\!]$ serves also as a coefficient ring
for the algebra $\kappa[\\![\mathbf{u}]\\!]$.

Singularities such as Cohen-Macaulayness, Gorensteiness and complete
intersection of the ring $\kappa[\\![\mathbf{u}^{S}]\\!]$ are in fact
properties of the flat algebra
$\kappa[\\![\mathbf{u}^{S}]\\!]/\kappa[\\![\mathbf{u}^{s}]\\!]$. We may
replace the power series ring $\kappa[\\![\mathbf{u}^{s}]\\!]$ by an arbitrary
numerical semigroup ring $R$ and consider singularities of a numerical
semigroup algebra $R^{\prime\prime}/R$ and its subalgebra $R^{\prime}/R$. See
[4, 5] for investigations emphasizing algebras over a fixed coefficient ring.
The purpose of this paper is to clarify notions of the ring
$\kappa[\\![\mathbf{u}^{S}]\\!]$ that are in fact notions of the algebra
$\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$. We will replace
the power series ring $\kappa[\\![\mathbf{u}]\\!]$ by an arbitrary numerical
semigroup ring $R^{\prime\prime}$ and regard it as an algebra over various
coefficient rings. To emphasize the role of coefficient rings for the fixed
$R^{\prime\prime}$, we call a coefficient ring $R^{\prime}$ containing another
coefficient ring $R$ an extension of $R$. For the case
$R^{\prime\prime}=\kappa[\\![\mathbf{u}]\\!]$, the semigroup
$\log_{\mathbf{u}}R^{\prime}$ is called an oversemigroup in [15] and also
called an extension of $\log_{\mathbf{u}}R$ in [8].

For a numerical semigroup $S\subset{\mathbb{N}}$ with $\gcd(S)=1$, there are
many results concerning Frobenius numbers, pseudo-Frobenius numbers, gaps,
symmetry, pseudo-symmetry, almost symmetry, irreducibility and Cohen-Macaulay
type. On the way to exhibiting the relative nature of these invariants and
notions, new notions appear. Let $R^{\prime}/R$ be a numerical semigroup
algebra in the variable $\mathbf{u}$. Resembling
$\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$, we choose an
integer $n$ such that $n\log_{\mathbf{u}}R^{\prime}\subset{\mathbb{N}}$. The
algebra $R^{\prime}/R$ is called equi-gcd if
$\gcd(n\log_{\mathbf{u}}R)=\gcd(n\log_{\mathbf{u}}R^{\prime})$. This condition
is independent of the choice of $n$. In particular,
$\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$ is equi-gcd.
Arithmetic and set-theoretic properties of the semigroup $S$ often show up
inside ${\mathbb{N}}$. We will prove that certain results in [3, 6, 7, 8, 9,
10, 13, 17] for numerical semigroups are special cases of general phenomenons
for coefficient rings of equi-gcd numerical semigroup algebras. Along this
line, generalizations of more results for numerical semigroups are expected.

Beyond the classical theory of numerical semigroups, there are new phenomenons
in our framework. For example, the numerical semigroup ring
$R=k[\\![\mathbf{u}^{3},\mathbf{u}^{7},\mathbf{u}^{8}]\\!]$ is not symmetric
as seen in Kunz’s work [6]. Adding the monomial $\mathbf{u}^{5}$, we obtain a
larger numerical semigroup ring $R^{\prime}=R[\\![\mathbf{u}^{5}]\\!]$, which
is still not symmetric. We will see that the coefficient ring of the algebra
$R^{\prime}/R$ is symmetric. In this example, a symmetric phenomenon arises
from two non-symmetric numerical semigroup rings.

This paper is organized as follows: For an equi-gcd numerical semigroup
algebra, we will define Frobenius monomials, pseudo-Frobenius monomials, gap
monomials and sporadic monomials in Section 2 as generalizations of
corresponding invariants for numerical semigroups. In the classical case, the
number of gap monomials has an upper bound in terms of the numbers of pseudo-
Frobenius monomials and sporadic monomials [3]. For a general equi-gcd
numerical semigroup algebra, besides an analogous upper bound, we will also
give a lower bound for the number of gap monomials in terms of the numbers of
Frobenius monomials and sporadic monomials. In this section, Kunz’s criterion
for a numerical semigroup ring to be Gorenstein [6] is generalized.

The next two sections are about symmetry, irreducibility and their connection.
Section 3 is devoted to the study of the irreducibility of coefficient rings
of equi-gcd numerical semigroup algebras. It is known that irreducible
numerical semigroups are either symmetric or pseudo-symmetric [10]. As a main
conclusion of this section, we generalize this result to equi-gcd numerical
semigroup algebras. Pseudo-Frobenius numbers of numerical semigroups can be
read from Apéry sets. We present an analogous way to obtain pseudo-Frobenius
monomials in Section 4, which follows a characterization of almost symmetric
semigroup algebras.

In the last two sections, we will see classes of extensions. Radicals of
coefficient rings are introduced in Section 5 to generalize the classical
result that a numerical semigroup is one over $n$ of infinitely many symmetric
numerical semigroups [17]. Finally, in Section 6, we investigate when all
extensions of a coefficient ring are intersections of radicals. The classical
version of this problem was studied for numerical semigroups in [8].
Fundamental gap monomials emerge as in the classical case.

##  2\. Frobenius and Pseudo-Frobenius Monomials

In this section, we consider a numerical semigroup algebra $R^{\prime}/R$ in
the variable $\mathbf{u}$. Recall that the conductor of $R^{\prime}/R$ is the
ideal $\mathfrak{f}(R^{\prime}/R):=\\{a\in R\colon aR^{\prime}\subset R\\}$ of
$R^{\prime}$. Conductors are generated by monomials and satisfy the following
properties.

  * •

$\mathfrak{f}(R^{\prime}/R)=R^{\prime}$ if and only if $R^{\prime}=R$.

  * •

$\mathfrak{f}(R^{\prime\prime}/R^{\prime})\mathfrak{f}(R^{\prime}/R)\subset\mathfrak{f}(R^{\prime\prime}/R)\subset\mathfrak{f}(R^{\prime}/R)$
for numerical semigroup algebras $R^{\prime\prime}/R^{\prime}$ and
$R^{\prime}/R$.

######  Example 2.1.

Let
$R=\kappa[\\![\mathbf{u}^{4},\mathbf{u}^{6},\mathbf{u}^{7},\mathbf{u}^{9}]\\!]$,
$R^{\prime}=\kappa[\\![\mathbf{u}^{2},\mathbf{u}^{5}]\\!]$,
$R^{\prime\prime}=\kappa[\\![\mathbf{u}^{2},\mathbf{u}^{3}]\\!]$ and
$R^{\prime\prime\prime}=\kappa[\\![\mathbf{u}]\\!]$. Then

$\displaystyle\mathfrak{f}(R^{\prime\prime\prime}/R^{\prime})$
$\displaystyle=$ $\displaystyle
R^{\prime\prime\prime}\mathbf{u}^{4}=R^{\prime}\mathbf{u}^{4}+R^{\prime}\mathbf{u}^{5},$
$\displaystyle\mathfrak{f}(R^{\prime\prime}/R^{\prime})$ $\displaystyle=$
$\displaystyle
R^{\prime\prime}\mathbf{u}^{2}=R^{\prime}\mathbf{u}^{2}+R^{\prime}\mathbf{u}^{5},$
$\displaystyle\mathfrak{f}(R^{\prime}/R)$ $\displaystyle=$ $\displaystyle
R^{\prime}\mathbf{u}^{4}+R^{\prime}\mathbf{u}^{7}=R\mathbf{u}^{4}+R\mathbf{u}^{6}+R\mathbf{u}^{7}+R\mathbf{u}^{9}.$

Monomials in $R^{\prime}$ but not in $R$ are called gap monomials of
$R^{\prime}/R$. Monomials in $R$ but not in $\mathfrak{f}(R^{\prime}/R)$ are
called sporadic monomials of $R^{\prime}/R$. The set of gap monomials and the
set of sporadic monomials of $R^{\prime}/R$ are denoted by
$\operatorname{G}(R^{\prime}/R)$ and $\operatorname{N}(R^{\prime}/R)$,
respectively. For an equi-gcd numerical semigroup algebra
$\kappa[\\![\mathbf{u}]\\!]/R$, sporadic monomials and gap monomials are
exponential versions of sporadic elements and gaps of the numerical semigroup
$\log_{\mathbf{u}}R$. For a numerical semigroup $S\subset{\mathbb{N}}$, the
set of gaps is finite if and only if $\gcd(S)=1$. This well-known fact is a
special case of the following result for numerical semigroup algebras.

######  Proposition 2.2.

For a numerical semigroup algebra $R^{\prime}/R$ in the variable $\mathbf{u}$,
the following conditions are equivalent.

  * •

$R^{\prime}/R$ is equi-gcd.

  * •

$\mathfrak{f}(R^{\prime}/R)\neq 0$.

  * •

The set $\operatorname{G}(R^{\prime}/R)$ is finite.

  * •

The set $\operatorname{N}(R^{\prime}/R)$ is finite.

###### Proof.

We may assume that $\log_{\mathbf{u}}R^{\prime}\subset{\mathbb{N}}$ and
$\gcd(\log_{\mathbf{u}}R^{\prime})=1$. So
$\mathfrak{f}(\kappa[\\![\mathbf{u}]\\!]/R^{\prime})=\kappa[\\![\mathbf{u}]\\!]\mathbf{u}^{w^{\prime}+1}$,
where $w^{\prime}$ is the Frobenius number of $\log_{\mathbf{u}}R^{\prime}$.
Let $p=\gcd(\log_{\mathbf{u}}R)$.

Consider first the case that $R^{\prime}/R$ is equi-gcd, equivalently $p=1$.
Let $w$ be the Frobenius number of $\log_{\mathbf{u}}R$. Then
$\mathfrak{f}(\kappa[\\![\mathbf{u}]\\!]/R)=\kappa[\\![\mathbf{u}]\\!]\mathbf{u}^{w+1}$
and $\mathbf{u}^{w+w^{\prime}+1}\in\mathfrak{f}(R^{\prime}/R)$. As a subset of
the finite set $\operatorname{G}(\kappa[\\![\mathbf{u}]\\!]/R)$ (resp.
$\operatorname{N}(\kappa[\\![\mathbf{u}]\\!]/R)$), the set
$\operatorname{G}(R^{\prime}/R)$ (resp. $\operatorname{N}(R^{\prime}/R)$) is
finite.

Now we consider the case that $R^{\prime}/R$ is not equi-gcd, equivalently
$p>1$. There are infinitely many monomials $\mathbf{u}^{s}$ in $R^{\prime}$
such that $p$ does not divide $s$. Therefore $\operatorname{G}(R^{\prime}/R)$
is infinite. Choose such a monomial $\mathbf{u}^{s}$. Then
$\mathbf{u}^{t}\mathbf{u}^{s}\not\in R$ for any $\mathbf{u}^{t}\in R$. That
means $\mathbf{u}^{t}\not\in\mathfrak{f}(R^{\prime}/R)$. Therefore
$\mathfrak{f}(R^{\prime}/R)=0$. The set $\operatorname{N}(R^{\prime}/R)$
consists of all monomials in $R$ and hence is also infinite. ∎

If an equi-gcd numerical semigroup algebra $R^{\prime}/R$ is flat, then
$R^{\prime}=R$ by [4, Proposition 3.5]. Beyond flatness, there are properties
of non-trivial equi-gcd numerical semigroup algebras worth studying. As seen
in [4, 5], Apéry monomials form an essential tool to study flat numerical
semigroup algebras. For equi-gcd numerical semigroup algebras, Frobenius
monomials and Pseudo-Frobenius monomials play an important role. In a
numerical semigroup algebra $R^{\prime}/R$ in the variable $\mathbf{u}$, we
have two partial orders:

  * •

$\mathbf{u}^{s}\preceq_{R}\mathbf{u}^{t}$ if and only if $\mathbf{u}^{t-s}\in
R$.

  * •

$\mathbf{u}^{s}\preceq_{R^{\prime}}\mathbf{u}^{t}$ if and only if
$\mathbf{u}^{t-s}\in R^{\prime}$.

We consider maximal elements in $\operatorname{G}(R^{\prime}/R)$ with respect
to these two partial orders. Those maximal with respect to
$\preceq_{R^{\prime}}$ are called Frobenius monomials of $R^{\prime}/R$. Those
maximal with respect to $\preceq_{R}$ are called pseudo-Frobenius monomials of
$R^{\prime}/R$. Frobenius monomials are pseudo-Frobenius monomials. The sets
of Frobenius monomials and pseudo-Frobenius monomials are denoted by
$\operatorname{F}(R^{\prime}/R)$ and $\operatorname{PF}(R^{\prime}/R)$. The
cardinalities of these sets are called the Frobenius type (F-type for short)
and the Cohen-Macaulay type (CM-type for short) of $R^{\prime}/R$,
respectively. Cohen-Macaulay type was introduced for numerical semigroups [3].
This notion of type coincides with the Cohen-Macaulay type of a numerical
semigroup ring [16, Theorem 3.1]. The Frobenius type of an equi-gcd numerical
semigroup algebra $\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$
is always one. Beyond the classical case, the Frobenius type has an
application somewhat dual to the Cohen-Macaulay type in the comparison of the
number of gap monomials and the number of sporadic monomials. See Proposition
2.6.

Frobenius monomials and pseudo-Frobenius monomials generalize Frobenius
numbers and pseudo-Frobenius numbers in the non-trivial case. Assume that
$S\subset{\mathbb{N}}$ is a numerical semigroup with $\gcd(S)=1$. Recall that
a number $w\in{\mathbb{Z}}\setminus S$ is called pseudo-Frobenius in [13] if
$w+s\in S$ for any $0\neq s\in S$. For the trivial case $S={\mathbb{N}}$,
there is a unique Frobenius number and a unique pseudo-Frobenius number,
namely $-1$. However
$\operatorname{F}(\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}]\\!])=\operatorname{PF}(\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}]\\!])=\emptyset$
in our definition and if $S\neq{\mathbb{N}}$, the only Frobenius monomial of
$\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$ is
$\mathbf{u}^{w}$, where $w$ is the Frobenius number of $S$. For the non-
trivial case $S\neq{\mathbb{N}}$, the exponents of pseudo-Frobenius monomials
of $\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$ are exactly
pseudo-Frobenius numbers of $S$. We remark that all pseudo-Frobenius numbers
are positive in the nontrivial case. Note that $0\neq w+1\in S$ in such a
case. If there is a negative pseudo-Frobenius number $-s$, then $0\neq
w+1-s\in S$. By induction, $w+1-ns\in S$ for all $n\geq 1$. This is
impossible, since $S\subset{\mathbb{N}}$.

######  Example 2.3.

Let $R=\kappa[\\![\mathbf{u}^{4},\mathbf{u}^{6},\mathbf{u}^{9}]\\!]$ and
$R^{\prime}=R[\\![\mathbf{u}^{5}]\\!]$. Then
$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{5},\mathbf{u}^{11}\\}$,
$\operatorname{N}(R^{\prime}/R)=\\{1,\mathbf{u}^{6}\\}$ and
$\operatorname{PF}(R^{\prime}/R)=\operatorname{F}(R^{\prime}/R)=\\{\mathbf{u}^{11}\\}$.

######  Example 2.4.

Let $R=\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{7},\mathbf{u}^{9}]\\!]$ and
$R^{\prime}=R[\\![\mathbf{u}^{8},\mathbf{u}^{11}]\\!]$. Then
$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{8},\mathbf{u}^{11},\mathbf{u}^{13}\\}$,
$\operatorname{N}(R^{\prime}/R)=\\{1,\mathbf{u}^{5}\\}$ and
$\operatorname{PF}(R^{\prime}/R)=\operatorname{F}(R^{\prime}/R)=\\{\mathbf{u}^{11},\mathbf{u}^{13}\\}$.

######  Example 2.5.

Let
$R=\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{7},\mathbf{u}^{11},\mathbf{u}^{13}]\\!]$
and $R^{\prime}=R[\\![\mathbf{u}^{3}]\\!]$. Then
$\operatorname{N}(R^{\prime}/R)=\\{1,\mathbf{u}^{5}\\}$,
$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{3},\mathbf{u}^{6},\mathbf{u}^{8},\mathbf{u}^{9}\\}$,
$\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{6},\mathbf{u}^{8},\mathbf{u}^{9}\\}$
and $\operatorname{F}(R^{\prime}/R)=\\{\mathbf{u}^{8},\mathbf{u}^{9}\\}$.

In the classical case, the ratio of $|\operatorname{G}(R^{\prime}/R)|$ to
$|\operatorname{N}(R^{\prime}/R)|$ is bounded above by the CM-type of
$R^{\prime}/R$ [3, Theorem 20]. For equi-gcd numerical semigroup algebras,
this result still holds. Moreover, the inverse of the F-type provides a lower
bound for the ratio.

######  Proposition 2.6.

Let $R^{\prime}/R$ be a non-trivial equi-gcd numerical semigroup algebra of
CM-type $n$ and F-type $n^{\prime}$. Then

$\frac{1}{n^{\prime}}\leq\frac{|\operatorname{G}(R^{\prime}/R)|}{|\operatorname{N}(R^{\prime}/R)|}\leq
n.$

###### Proof.

For any $\mathbf{u}^{s}\in\operatorname{G}(R^{\prime}/R)$, there exists a
pseudo-Frobenius monomial $\mathbf{u}^{w}$ such that $\mathbf{u}^{w-s}\in R$.
Clearly $\mathbf{u}^{w-s}\not\in\mathfrak{f}(R^{\prime}/R)$. The element
$(\mathbf{u}^{w-s},\mathbf{u}^{w})\in\operatorname{N}(R^{\prime}/R)\times\operatorname{PF}(R^{\prime}/R)$
determines $\mathbf{u}^{s}$. Therefore

$|\operatorname{G}(R^{\prime}/R)|\leq n|\operatorname{N}(R^{\prime}/R)|.$

Let $\mathbf{u}^{s}\in\operatorname{N}(R^{\prime}/R)$. Since
$\mathbf{u}^{s}\not\in\mathfrak{f}(R^{\prime}/R)$, there exists a monomial
$\mathbf{u}^{v}\in R^{\prime}$ such that $\mathbf{u}^{s+v}\notin R$, which
implies $\mathbf{u}^{v}\notin R$. Setting $w=s+v$, we have
$\mathbf{u}^{w}\in\operatorname{G}(R^{\prime}/R)$ and
$\mathbf{u}^{w-s}\in\operatorname{G}(R^{\prime}/R)$. Multiplying
$\mathbf{u}^{w}$ by a monomial of $R^{\prime}$, we may assume that
$\mathbf{u}^{w}$ is a Frobenius monomial. The element
$(\mathbf{u}^{w-s},\mathbf{u}^{w})\in\operatorname{G}(R^{\prime}/R)\times\operatorname{F}(R^{\prime}/R)$
determines $\mathbf{u}^{s}$. Therefore

$|\operatorname{N}(R^{\prime}/R)|\leq
n^{\prime}|\operatorname{G}(R^{\prime}/R)|.$

∎

In the situation where $\kappa[\\![\mathbf{u}]\\!]/R$ is equi-gcd, the
numerical semigroup ring $R$ is Gorenstein if and only if it has only one
pseudo-Frobenius monomial. Gorensteiness of $R$ has a criterion that
$|\operatorname{G}(\kappa[\\![\mathbf{u}]\\!]/R)|=|\operatorname{N}(\kappa[\\![\mathbf{u}]\\!]/R)|$.
We may generalize this classical criterion [6] to the relative situation as
follows.

######  Theorem 2.7.

Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra with a single
Frobenius monomial $\mathbf{u}^{w}$. The algebra has a single pseudo-Frobenius
monomial if and only if
$|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|$.

###### Proof.

Given $\mathbf{u}^{s}\in\operatorname{N}(R^{\prime}/R)$, there is
$\mathbf{u}^{t}\in\operatorname{G}(R^{\prime}/R)$ such that
$\mathbf{u}^{t+s}\in\operatorname{G}(R^{\prime}/R)$. Since $\mathbf{u}^{w}$ is
the unique Frobenius monomial, $\mathbf{u}^{w-t-s}\in R^{\prime}$ and hence
$\mathbf{u}^{w-s}=\mathbf{u}^{w-t-s}\mathbf{u}^{t}\in R^{\prime}$. But
$\mathbf{u}^{w-s}\not\in R$, since $\mathbf{u}^{s}\in R$ and
$\mathbf{u}^{w}=\mathbf{u}^{w-s}\mathbf{u}^{s}\not\in R$. So we have an
injective map
$\operatorname{N}(R^{\prime}/R)\to\operatorname{G}(R^{\prime}/R)$ sending
$\mathbf{u}^{s}$ to $\mathbf{u}^{w-s}$.

If $|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|$, the
above map is surjective. The preimage of
$\mathbf{u}^{t}\in\operatorname{G}(R^{\prime}/R)$ is $\mathbf{u}^{w-t}$. Since
$\mathbf{u}^{t}\preceq_{R}\mathbf{u}^{w}$, the Frobenius monomial is the only
pseudo-Frobenius monomial.

Now we assume that $\mathbf{u}^{w}$ is the only pseudo-Frobenius monomial of
$R^{\prime}/R$. Consider $\mathbf{u}^{s}\in\operatorname{G}(R^{\prime}/R)$.
Then $\mathbf{u}^{w-s}\in R$. But
$\mathbf{u}^{w}=\mathbf{u}^{w-s}\mathbf{u}^{s}\not\in R$. Hence
$\mathbf{u}^{w-s}\in\operatorname{N}(R^{\prime}/R)$. This shows that the map
$\operatorname{N}(R^{\prime}/R)\to\operatorname{G}(R^{\prime}/R)$ above is
also surjective. Therefore
$|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|$.

∎

######  Example 2.8.

Let
$R=\kappa[\\![\mathbf{u}^{8},\mathbf{u}^{12},\mathbf{u}^{19},\mathbf{u}^{21}]\\!]$
and $R^{\prime}=R[\\![\mathbf{u}^{22}]\\!]$. Then
$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{22},\mathbf{u}^{30},\mathbf{u}^{34}\\}$
and $\operatorname{N}(R^{\prime}/R)=\\{1,\mathbf{u}^{8},\mathbf{u}^{12}\\}$.
In this example,
$|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|$ even
though
$\operatorname{PF}(R^{\prime}/R)=\operatorname{F}(R^{\prime}/R)=\\{\mathbf{u}^{30},\mathbf{u}^{34}\\}$.

In examples 2.3 and 2.8, there is a monomial $\mathbf{u}^{s}$ such that
$\operatorname{G}(R^{\prime}/R)=\mathbf{u}^{s}\operatorname{N}(R^{\prime}/R)$.
This is not true in general for a numerical semigroup algebra $R^{\prime}/R$
such that $|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|$.

######  Example 2.9.

Let $R^{\prime}=\kappa[\\![\mathbf{u}]\\!]$ and
$R=\kappa[\\![\mathbf{u}^{3},\mathbf{u}^{5}]\\!]$. Then
$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u},\mathbf{u}^{2},\mathbf{u}^{4},\mathbf{u}^{7}\\}$
and
$\operatorname{N}(R^{\prime}/R)=\\{1,\mathbf{u}^{3},\mathbf{u}^{5},\mathbf{u}^{6}\\}$.
Note that
$\operatorname{G}(R^{\prime}/R)\neq\mathbf{u}^{s}\operatorname{N}(R^{\prime}/R)$
for any $\mathbf{u}^{s}\in R^{\prime}$. In this example,
$\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{7}\\}$.

######  Example 2.10.

Let
$R=\kappa[\\![\mathbf{u}^{11},\mathbf{u}^{15},\mathbf{u}^{19},\mathbf{u}^{21}]\\!]$
and $R^{\prime}=R[\\![\mathbf{u}^{31},\mathbf{u}^{39}]\\!]$. Then
$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{31},\mathbf{u}^{39},\mathbf{u}^{46},\mathbf{u}^{50}\\}$
and
$\operatorname{N}(R^{\prime}/R)=\\{1,\mathbf{u}^{11},\mathbf{u}^{15},\mathbf{u}^{19}\\}$.
Note that
$\operatorname{G}(R^{\prime}/R)\neq\mathbf{u}^{s}\operatorname{N}(R^{\prime}/R)$
for any $\mathbf{u}^{s}\in R^{\prime}$. In this example,
$\operatorname{PF}(R^{\prime}/R)=\operatorname{F}(R^{\prime}/R)=\\{\mathbf{u}^{46},\mathbf{u}^{50}\\}$.

##  3\. Irreducibility

To study an algebraic structure, one often searchs for simplest objects.
Irreducibility is such a notion for numerical semigroups and was first
introduced in [10]. In this section, we extend the notion to the relative
situation. Let $R^{\prime\prime}/R$ be an equi-gcd numerical semigroup
algebra. For any extension $R^{\prime}$ of $R$, the algebras
$R^{\prime\prime}/R^{\prime}$ and $R^{\prime}/R$ are also equi-gcd. The
coefficient ring $R$ is irreducible in $R^{\prime\prime}$ if it cannot be
expressed as the intersection of two non-trivial extensions. For the case
$R^{\prime\prime}=\kappa[\\![\mathbf{u}]\\!]$, the coefficient ring $R$ is
irreducible if and only if $\log_{\mathbf{u}}R$ is an irreducible numerical
semigroup. Recall that, in the classical case $\kappa[\\![\mathbf{u}]\\!]/R$,
the algebra has a single Frobenius monomial.

######  Example 3.1.

Let $R=\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{7},\mathbf{u}^{9}]\\!]$ and
$R^{\prime}=R[\\![\mathbf{u}^{8}]\\!]$. Then
$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{8},\mathbf{u}^{13}\\}$. Any
non-trivial extension of $R$ must contain $\mathbf{u}^{13}$. Therefore $R$ is
irreducible as a coefficient ring of $R^{\prime}$. Let
$R^{\prime\prime}=R^{\prime}[\\![\mathbf{u}^{11}]\\!]$. Then
$\operatorname{G}(R^{\prime\prime}/R)=\\{\mathbf{u}^{8},\mathbf{u}^{11},\mathbf{u}^{13}\\}$
and $R[\\![\mathbf{u}^{11}]\\!]\cap R[\\![\mathbf{u}^{13}]\\!]=R$. Therefore
$R$ is not irreducible as a coefficient ring of $R^{\prime\prime}$.

######  Example 3.2.

Let
$R=\kappa[\\![\mathbf{u}^{8},\mathbf{u}^{12},\mathbf{u}^{19},\mathbf{u}^{21}]\\!]$.
The algebra $R[\\![\mathbf{u}^{22}]\\!]/R$ given in Example 2.8 has gap
monomials $\mathbf{u}^{22},\mathbf{u}^{30},\mathbf{u}^{34}$. Since
$R=R[\\![\mathbf{u}^{30}]\\!]\cap R[\\![\mathbf{u}^{34}]\\!]$, it is not
irreducible.

######  Lemma 3.3.

Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra. If the
coefficient ring $R$ is irreducible, then $R^{\prime}/R$ has a single
Frobenius monomial.

###### Proof.

A Frobenius monomial $\mathbf{u}^{w}$ of $R^{\prime}/R$ is maximal in
$\operatorname{G}(R^{\prime}/R)$ with respect to $\preceq_{R^{\prime}}$, hence
$\mathbf{u}^{2w}\in R$ and
$\operatorname{G}(R[\\![\mathbf{u}^{w}]\\!]/R)=\\{\mathbf{u}^{w}\\}$. If
$R^{\prime}/R$ has two distinct Frobenius monomials $\mathbf{u}^{w_{1}}$ and
$\mathbf{u}^{w_{2}}$, then $R=R[\\![\mathbf{u}^{w_{1}}]\\!]\cap
R[\\![\mathbf{u}^{w_{2}}]\\!]$ is not irreducible. ∎

The following criterion for irreducibility extends the classical case [10].

######  Proposition 3.4.

Let $\mathbf{u}^{w}$ be a gap monomial of an equi-gcd numerical semigroup
algebra $R^{\prime}/R$. The following conditions are equivalent .

  * •

$R$ is irreducible with $\mathbf{u}^{w}$ as the single Frobenius monomial.

  * •

$R$ is maximal with respect to set inclusion among coefficient rings with
$\mathbf{u}^{w}$ as the single Frobenius monomial.

  * •

$R$ is maximal with respect to set inclusion among coefficient rings over
which $\mathbf{u}^{w}$ is a Frobenius monomial.

###### Proof.

For a Frobenius monomial $\mathbf{u}^{s}\in\operatorname{F}(R^{\prime}/R)$,
note that
$\operatorname{G}(R[\\![\mathbf{u}^{s}]\\!]/R)=\\{\mathbf{u}^{s}\\}$. In other
words, joining a Frobenius monomial to a coefficient ring creates only one
extra monomial. Therefore the set of coefficient rings with $\mathbf{u}^{w}$
as the single Frobenius monomial and the set of coefficient rings over which
$\mathbf{u}^{w}$ is a Frobenius monomial have the same maximal coefficient
rings with respect to set inclusion.

Let $R_{1}=R[\\![\mathbf{u}^{w}]\\!]$ and $R_{2}$ be an extension of $R$ such
that $\mathbf{u}^{w}\in\operatorname{F}(R^{\prime}/R_{2})$. If
$\operatorname{F}(R^{\prime}/R)=\\{\mathbf{u}^{w}\\}$, then
$\operatorname{G}(R_{1}/R)=\\{\mathbf{u}^{w}\\}$ and $R=R_{1}\cap R_{2}$. If
$R$ is furthermore irreducible, then $R_{2}=R$. So we obtain maximality. Let
$R_{3}$ and $R_{4}$ be proper extensions of $R$. If $R$ is maximal among
coefficient rings of $R^{\prime}$ with $\mathbf{u}^{w}$ as the single
Frobenius monomial, then $\mathbf{u}^{w}\in R_{3}\cap R_{4}$ and hence $R\neq
R_{3}\cap R_{4}$. Therefore maximality implies irreducibility. ∎

For an equi-gcd numerical semigroup algebra $R^{\prime}/R$ with a single
Frobenius monomial $\mathbf{u}^{w}$, we would like to find an irreducible
coefficient ring $R_{1}$ containing $R$ such that $\mathbf{u}^{w}$ is the
single Frobenius monomial of $R^{\prime}/R_{1}$. If $R$ is not irreducible, it
is not maximal and there is a monomial $\mathbf{u}^{s}$ contained in an
extension of $R$ such that $\mathbf{u}^{ns}\neq\mathbf{u}^{w}$ for any
$n\in\mathbb{N}$. In other words, $\mathbf{u}^{s}$ is not a radical of
$\mathbf{u}^{w}$. As a step to construct an irreducible coefficient ring, the
following result generalizes [9, Lemma 3.2].

######  Lemma 3.5.

Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra with a single
Frobenius monomial $\mathbf{u}^{w}$. Given a pseudo-Frobenius monomial
$\mathbf{u}^{s}$, if it is not a radical of $\mathbf{u}^{w}$, then
$\mathbf{u}^{w}$ is also the single Frobenius monomial of
$R^{\prime}/R[\\![\mathbf{u}^{s}]\\!]$.

###### Proof.

For any $1\neq\mathbf{u}^{a}\in R$, we have $1\neq\mathbf{u}^{s+a}\in R$ since
$\mathbf{u}^{s}\in\operatorname{PF}(R^{\prime}/R)$. By induction,
$\mathbf{u}^{ns+a}\in R$ for $n\geq 1$. Together with the assumption,
$\mathbf{u}^{w}\not\in R[\\![\mathbf{u}^{s}]\\!]$. A gap monomial
$\mathbf{u}^{t}$ of $R^{\prime}/R[\\![\mathbf{u}^{s}]\\!]$ is also a gap
monomial of $R^{\prime}/R$. Therefore
$\mathbf{u}^{t}\preceq_{R^{\prime}}\mathbf{u}^{w}$. In other words,
$\mathbf{u}^{w}$ is the single Frobenius monomial of
$R^{\prime}/R[\\![\mathbf{u}^{s}]\\!]$. ∎

######  Theorem 3.6.

Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra in the variable
$\mathbf{u}$. The coefficient ring $R$ is irreducible if and only if
$\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{w}\\}$ or
$\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{w},\mathbf{u}^{w/2}\\}$ for
some $\mathbf{u}^{w}\in G(R^{\prime}/R)$. For CM-type one algebras over an
irreducible coefficient ring,
$|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|$. For CM-
type two algebras over an irreducible coefficient ring,
$|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|+1$.

###### Proof.

Consider the case that the coefficient ring $R$ is irreducible. By Lemma 3.3,
$R^{\prime}/R$ has a single Frobenius monomial $\mathbf{u}^{w}$. It is shown
in Proposition 3.4 that $R$ is maximal among numerical semigroup subrings of
$R^{\prime}$ in the variable $\mathbf{u}$ with $\mathbf{u}^{w}$ as the single
Frobenius monomial. Assume that there is a pseudo-Frobenius monomial
$\mathbf{u}^{s}$ other than $\mathbf{u}^{w}$. By Lemma 3.5, $w=ns$ for some
$n\in\mathbb{N}$. Since $\mathbf{u}^{w}$ is the unique Frobenius monomial,
$\mathbf{u}^{ns-s}\in R^{\prime}$. Since $\mathbf{u}^{ns}\not\in R$ and
$\mathbf{u}^{s}$ is pseudo-Frobenius, the monomial $\mathbf{u}^{ns-s}$ outside
$R$ is also pseudo-Frobenius. By Lemma 3.5 again, $n-1$ divides $n$. This
implies that $n=2$.

Consider the case that $\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{w}\\}$.
It is shown in Theorem 2.7 that
$|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|$. Given
$\mathbf{u}^{s}\in\operatorname{G}(R^{\prime}/R)$, we obtain
$\mathbf{u}^{w-s}\in R$. Therefore any subring of $R^{\prime}$ properly
containing $R$ has to contain $\mathbf{u}^{w}$. By maximality, the coefficient
ring $R$ is irreducible.

Consider the case that
$\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{w},\mathbf{u}^{w/2}\\}$. Given
$\mathbf{u}^{s}\in\operatorname{G}(R^{\prime}/R)$ not equal
$\mathbf{u}^{w/2}$, either $\mathbf{u}^{s}\preceq_{R}\mathbf{u}^{w}$ or
$\mathbf{u}^{s}\preceq_{R}\mathbf{u}^{w/2}$. If
$\mathbf{u}^{s}\preceq_{R}\mathbf{u}^{w}$, then $\mathbf{u}^{w-s}\in R$. If
$\mathbf{u}^{s}\preceq_{R}\mathbf{u}^{w/2}$, then
$1\neq\mathbf{u}^{(w/2)-s}\in R$ and hence
$\mathbf{u}^{w-s}=\mathbf{u}^{(w/2)-s}\mathbf{u}^{w/2}\in R$. In either cases,
$\mathbf{u}^{w-s}\in R$. But
$\mathbf{u}^{w-s}\not\in\mathfrak{f}(R^{\prime}/R)$, since
$\mathbf{u}^{w}=\mathbf{u}^{s}\mathbf{u}^{w-s}\not\in R$. So we have an one-
to-one map
$\operatorname{G}(R^{\prime}/R)\setminus\\{\mathbf{u}^{w/2}\\}\to\operatorname{N}(R^{\prime}/R)$
given by $\mathbf{u}^{s}\mapsto\mathbf{u}^{w-s}$. As seen in the proof of
Theorem 2.7, $\mathbf{u}^{w-t}\in\operatorname{G}(R^{\prime}/R)$ for any
$\mathbf{u}^{t}\in N(R^{\prime}/R)$. Therefore the one-to-one map is also
onto, whence the identity
$|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|+1$. Given
$\mathbf{u}^{s}\in\operatorname{G}(R^{\prime}/R)$ not equal
$\mathbf{u}^{w/2}$, we obtain $\mathbf{u}^{w-s}\in R$. As the CM-type one
case, any subring of $R^{\prime}$ properly containing $R$ has to contain
$\mathbf{u}^{w}$. By maximality, the coefficient ring $R$ is irreducible. ∎

######  Example 3.7.

Let
$R=\kappa[\\![\mathbf{u}^{n+1},\mathbf{u}^{n+2},\ldots,\mathbf{u}^{2n-1},\mathbf{u}^{2n+1}]\\!]$
and $R^{\prime}=R[\\![\mathbf{u}^{n}]\\!]$, where $n>1$. If $n=2$, then
$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{2},\mathbf{u}^{4},\mathbf{u}^{7}\\}$
and $\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{7}\\}$. If $n>2$, then
$\operatorname{PF}(R^{\prime}/R)=\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{n},\mathbf{u}^{2n}\\}$.
The coefficient ring $R$ is irreducible.

##  4\. Symmetries

Symmetry for a set can be often regarded as a phenomenon observed from one-to-
one correspondence of elements in certain subsets determined by a simple rule.
Besides symmetric numerical semigroups that have been observed and well-
studied, there are also pseudo-symmetry and almost symmetry properties
standing out as variations. These notions of symmetry have a relative nature.
Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra in the variable
$\mathbf{u}$. In view of Theorem 2.7, symmetry of the coefficient ring $R$ is
about correspondence of elements in $\operatorname{G}(R^{\prime}/R)$ and
$\operatorname{N}(R^{\prime}/R)$. Pseudo-symmetry is about correspondence of
elements in $\operatorname{G}(R^{\prime}/R)\setminus\\{\mathbf{u}^{w/2}\\}$
and $\operatorname{N}(R^{\prime}/R)$, where $\mathbf{u}^{w}$ occurs as the
only Frobenius monomial of $R^{\prime}/R$. To be precise, we call the
coefficient ring $R$ symmetric, if
$\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{w}\\}$. We call $R$ pseudo-
symmetric, if
$\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{w},\mathbf{u}^{w/2}\\}$. As a
generalization of [10], Theorem 3.6 asserts that a coefficient ring is
irreducible if and only if it is symmetric or pseudo-symmetric. In this
section, we consider almost symmetry for equi-gcd numerical semigroup
algebras.

As a generalization of Apéry numbers to the relative case, Apéry monomials of
a numerical semigroup algebra $R^{\prime}/R$ can be described using the order
$\preceq_{R}$ as minimal elements among monomials in $R^{\prime}$. Among Apéry
monomials, maximal elements with respect to the order $\preceq_{R^{\prime}}$
are called maximal Apéry monomials. It is known that a flat numerical
semigroup algebra $R^{\prime}/R$ in the variable $\mathbf{u}$ is Gorenstein if
and only if it has a unique maximal Apéry monomial [5, Theorem 3.1]. Let
$\kappa[\\![\mathbf{u}^{S}]\\!]$ be a numerical semigroup ring and $s\in S$.
It is also known that $\kappa[\\![\mathbf{u}^{S}]\\!]$ is a Gorenstein ring if
and only if $\kappa[\\![\mathbf{u}^{S}]\\!]/\kappa[\\![\mathbf{u}^{s}]\\!]$ is
a Gorenstein algebra.

Assume that $S\subset{\mathbb{N}}$ and
$\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$ is equi-gcd.
Kunz’s symmetry criterion for the Gorenstein property of the ring
$\kappa[\\![\mathbf{u}^{S}]\\!]$ is in fact a statement of the algebra
$\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$. Let
$\mathbf{u}^{w}$ be the Frobenius monomial. Kunz shows that
$\kappa[\\![\mathbf{u}^{S}]\\!]$ is Gorenstein if and only if the map
$S\to{\mathbb{Z}}\setminus S$ given by $s\mapsto w-s$ is bijective [6].
Equivalently, $\kappa[\\![\mathbf{u}^{S}]\\!]$ is Gorenstein if and only if
the map
$\operatorname{N}(\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!])\to\operatorname{G}(\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!])$
given by $\mathbf{u}^{s}\mapsto\mathbf{u}^{w-s}$ is bijective. On the other
hand, the algebra $\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$
can describe maximal Apéry monomials of
$\kappa[\\![\mathbf{u}^{S}]\\!]/\kappa[\\![\mathbf{u}^{s}]\\!]$.

######  Lemma 4.1.

Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra in the variable
$\mathbf{u}$. Given $1\neq\mathbf{u}^{s}\in R$,

$\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{t-s}:\mathbf{u}^{t}\in\max_{\preceq_{R}}\left(\operatorname{Apr}(R/\kappa[\\![\mathbf{u}^{s}]\\!])\setminus\operatorname{Apr}(R^{\prime}/\kappa[\\![\mathbf{u}^{s}]\\!])\right)\\}.$

###### Proof.

The set
$\mathbf{u}^{s}\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{s+v}:\mathbf{u}^{v}\in\operatorname{PF}(R^{\prime}/R)\\}$
consists of exactly those elements in
$\mathbf{u}^{s}\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{s+v}:\mathbf{u}^{v}\in\operatorname{G}(R^{\prime}/R)\\}$
maximal with respect to the order $\preceq_{R}$. Note that
$\mathbf{u}^{s}\operatorname{PF}(R^{\prime}/R)\subset R$. Hence the set
$R\cap\mathbf{u}^{s}\operatorname{G}(R^{\prime}/R)$ has same maximal elements
with $\mathbf{u}^{s}\operatorname{G}(R^{\prime}/R)$. The lemma follows from
the fact that
$R\cap\mathbf{u}^{s}\operatorname{G}(R^{\prime}/R)=\operatorname{Apr}(R/\kappa[\\![\mathbf{u}^{s}]\\!])\setminus\operatorname{Apr}(R^{\prime}/\kappa[\\![\mathbf{u}^{s}]\\!])$.
∎

For the case that $R^{\prime}=\kappa[\\![\mathbf{u}]\\!]$ in Lemma 4.1, the
set
$\operatorname{Apr}(R/\kappa[\\![\mathbf{u}^{s}]\\!])\setminus\operatorname{Apr}(R^{\prime}/\kappa[\\![\mathbf{u}^{s}]\\!])$
consists of all elements
$\mathbf{u}^{t}\in\operatorname{Apr}(R/\kappa[\\![\mathbf{u}^{s}]\\!])$ such
that $t>s$. Let $\mathbf{u}^{w}$ be the Frobenius monomial of $R^{\prime}/R$.
Then
$\mathbf{u}^{w+s}\in\operatorname{Apr}(R/\kappa[\\![\mathbf{u}^{s}]\\!])$.
Given $\mathbf{u}^{t}\in\operatorname{Apr}(R/\kappa[\\![\mathbf{u}^{s}]\\!])$,
if $t<s$, then $\mathbf{u}^{t}\preceq_{R}\mathbf{u}^{w+s}$. So we recover [13,
Proposition 2.20] that

$\operatorname{PF}(\kappa[\\![\mathbf{u}]\\!]/R)=\\{\mathbf{u}^{t-s}:\mathbf{u}^{t}\in\max_{\preceq_{R}}\left(\operatorname{Apr}(R/\kappa[\\![\mathbf{u}^{s}]\\!])\right)\\}.$

In particular, $R/\kappa[\\![\mathbf{u}^{s}]\\!]$ has a unique maximal Apéry
monomial if and only if $\kappa[\\![\mathbf{u}]\\!]/R$ has a unique pseudo-
Frobenius monomial.

Note that maximal Apéry monomials of $R^{\prime}/R$ can be characterized as
those Apéry monomial $\mathbf{u}^{w}$ such that
$\mathbf{u}^{w}\mathbf{u}^{w^{\prime}}$ is not Apéry for any non-trivial Apéry
monomial $\mathbf{u}^{w^{\prime}}$. As an analogue, a pseudo-Frobenius
monomial $\mathbf{u}^{w}$ is called maximal, if
$\mathbf{u}^{w}\mathbf{u}^{w^{\prime}}$ is not pseudo-Frobenius for any
pseudo-Frobenius monomial $\mathbf{u}^{w^{\prime}}$. Generalizing symmetry,
the coefficient ring $R$ is called almost symmetric in $R^{\prime}$, if the
algebra has a unique maximal pseudo-Frobenius monomial.

When $R^{\prime}=\kappa[\\![\mathbf{u}]\\!]$, the coefficient ring $R$ is
almost symmetric precisely when $\log_{\mathbf{u}}R$ is almost symmetric as it
is defined in [1]. Generalizing [7, Theorem 2.4], the next proposition
justifies our terminology. Note that, over an almost symmetric coefficient
ring, a numerical semigroup algebra has a unique Frobenius monomial. By
Theorem 3.6, an irreducible coefficient ring of an equi-gcd numerical
semigroup algebra is almost symmetric. Therefore our notions extend the
classical hierarchy that almost symmetric numerical semigroups include
symmetric numerical semigroups and pseudo-symmetric numerical semigroups.

######  Proposition 4.2.

Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra with a single
Frobenius monomial $\mathbf{u}^{w}$. Write
$\operatorname{PF}(R^{\prime}/R)\setminus\\{\mathbf{u}^{w}\\}=\\{\mathbf{u}^{t_{1}},\dots,\mathbf{u}^{t_{n-1}}\\}$,
where $t_{1}<\cdots<t_{n-1}$. The following statements are equivalent.

  * •

$R$ is almost symmetric in $R^{\prime}$.

  * •

$\mathbf{u}^{t_{i}}\mathbf{u}^{t_{n-i}}=\mathbf{u}^{w}$ for $i=1,\ldots,n-1$.

  * •

$|\operatorname{G}(R^{\prime}/R)|=|\operatorname{N}(R^{\prime}/R)|+n-1$.

###### Proof.

If $\mathbf{u}^{t_{i}}\mathbf{u}^{t_{n-i}}=\mathbf{u}^{w}$ for all $i$, then
all $\mathbf{u}^{t_{i}}$ are not maximal. With the only maximal pseudo-
Frobenius monomial $\mathbf{u}^{w}$, the coefficient ring $R$ is almost
symmetric.

Conversely, if $R$ is almost symmetric, the only maximal pseudo-Frobenius
monomial has to be $\mathbf{u}^{w}$. For each $\mathbf{u}^{t_{i}}$, there is
an $\mathbf{u}^{t_{j_{1}}}$ such that
$\mathbf{u}^{t_{i}}\mathbf{u}^{t_{j_{1}}}$ is pseudo-Frobenius. If the product
is not yet maximal, there is an $\mathbf{u}^{t_{j_{2}}}$ such that
$\mathbf{u}^{t_{i}}\mathbf{u}^{t_{j_{1}}}\mathbf{u}^{t_{j_{2}}}$ is pseudo-
Frobenius. Since $\mathbf{u}^{t_{j_{1}}}\mathbf{u}^{t_{j_{2}}}\not\in R$ and
$\mathbf{u}^{s}\mathbf{u}^{t_{j_{1}}}\mathbf{u}^{t_{j_{2}}}\in R$ for any
$1\neq\mathbf{u}^{s}\in R$, we obtain a pseudo-Frobenius monomial
$\mathbf{u}^{t_{j_{1}}+t_{j_{2}}}$. Keep multiplying pseudo-Frobenius
monomials, we reach some $\mathbf{u}^{t_{j}}$ such that
$\mathbf{u}^{t_{i}}\mathbf{u}^{t_{j}}=\mathbf{u}^{w}$. Sorting the numbers
$w-t_{i}$, we see $j=n-i$.

There is a map

$\varphi:\operatorname{N}(R^{\prime}/R)\cup(\operatorname{PF}(R^{\prime}/R)\setminus\\{\mathbf{u}^{w}\\})\longrightarrow\operatorname{G}(R^{\prime}/R)$

sending $\mathbf{u}^{s}$ to $\mathbf{u}^{w-s}$. Since the map is injective,

(2) $|\operatorname{G}(R^{\prime}/R)|\geq|\operatorname{N}(R^{\prime}/R)|+n-1$

and the equality holds if and only if $\varphi$ is surjective. The image of
$\operatorname{N}(R^{\prime}/R)\setminus\\{1\\}$ is contained in
$\operatorname{G}(R^{\prime}/R)\setminus\operatorname{PF}(R^{\prime}/R)$. If
$\varphi$ is surjective, every $\mathbf{u}^{t_{i}}$ is the image of some
$\mathbf{u}^{t_{j}}$. Taking the increasing sequence into account, this
implies that $t_{i}+t_{n-i}=w$. Conversely, assume that $t_{i}+t_{n-i}=w$.
Then $\operatorname{PF}(R^{\prime}/R)$ is in the image of $\varphi$. To show
that $\varphi$ is surjective, we take an element
$\mathbf{u}^{v}\in\operatorname{G}(R^{\prime}/R)\setminus\operatorname{PF}(R^{\prime}/R)$.
Then $\mathbf{u}^{s}\mathbf{u}^{v}\in\operatorname{PF}(R^{\prime}/R)$ for some
$1\neq\mathbf{u}^{s}\in R$. From the assumption, either
$\mathbf{u}^{s+v}=\mathbf{u}^{w}$ or
$\mathbf{u}^{w-s-v}\in\operatorname{PF}(R^{\prime}/R)$. Since
$\mathbf{u}^{w-v}=\mathbf{u}^{s}\mathbf{u}^{w-s-v}\in R$ and
$\mathbf{u}^{w-v}\mathbf{u}^{v}=\mathbf{u}^{w}\not\in R$, we obtain an element
$\mathbf{u}^{w-v}\in\operatorname{N}(R^{\prime}/R)$ such that
$\varphi(\mathbf{u}^{w-v})=\mathbf{u}^{v}$. ∎

For a numerical semigroup algebra $R^{\prime}/R$ with F-type one,
$|\operatorname{PF}(R^{\prime}/R)\setminus\operatorname{F}(R^{\prime}/R)|$
provides a lower bound for
$|\operatorname{G}(R^{\prime}/R)|-|\operatorname{N}(R^{\prime}/R)|$. The
inequality does not hold in general. The number
$|\operatorname{G}(R^{\prime}/R)|-|\operatorname{N}(R^{\prime}/R)|$ is not
even necessarily positive, when the F-type is greater than one.

######  Example 4.3.

Let
$R^{\prime}=\kappa[\\![\mathbf{u}^{10},\mathbf{u}^{11},\mathbf{u}^{12},\mathbf{u}^{13},\mathbf{u}^{14},\mathbf{u}^{15}]\\!]$
and $R$ be its coefficient ring such that
$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{12},\mathbf{u}^{13},\mathbf{u}^{23},\mathbf{u}^{27}\\}$.
Then
$\operatorname{N}(R^{\prime}/R)=\\{1,\mathbf{u}^{10},\mathbf{u}^{11},\mathbf{u}^{14},\mathbf{u}^{15}\\}$
has more elements than $\operatorname{G}(R^{\prime}/R)$. Here the F-type is
two.

######  Example 4.4.

For $n\in\mathbb{N}$, let
$R=\kappa[\\![\mathbf{u}^{2n+1},\mathbf{u}^{2n+2},\ldots,\mathbf{u}^{4n+1}]\\!]$
and $R^{\prime}=R[\\![\mathbf{u}^{2}]\\!]$. Then
$\operatorname{PF}(R^{\prime}/R)=\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{2},\mathbf{u}^{4},\ldots,\mathbf{u}^{2n}\\}$
and $\operatorname{F}(R^{\prime}/R)=\\{\mathbf{u}^{2n}\\}$. The coefficient
ring $R$ is almost symmetric. Note that $R$ is irreducible if and only if
$n\leq 2$.

##  5\. Radicals

Let $R^{\prime\prime}/R$ be a numerical semigroup algebra in the variable
$\mathbf{u}$. To illustrate the relative nature of an extension $R^{\prime}$
of $R$, we often write
$R^{\prime}=R[\\![\mathbf{u}^{w_{1}},\ldots,\mathbf{u}^{w_{n}}]\\!]$ by
joining
$\mathbf{u}^{w_{1}},\ldots,\mathbf{u}^{w_{n}}\in\operatorname{G}(R^{\prime\prime}/R)$
to $R$. Clearly, $\operatorname{G}(R^{\prime\prime}/R)$ is the disjoint union
of $\operatorname{G}(R^{\prime\prime}/R^{\prime})$ and
$\operatorname{G}(R^{\prime}/R)$. If
$\mathbf{u}^{w_{1}},\ldots,\mathbf{u}^{w_{n}}\in\operatorname{F}(R^{\prime\prime}/R)$,
then
$\operatorname{G}(R^{\prime}/R)=\operatorname{F}(R^{\prime}/R)=\\{\mathbf{u}^{w_{1}},\ldots,\mathbf{u}^{w_{n}}\\}$.
Recall that a monomial $\mathbf{u}^{s}\in R^{\prime\prime}$ is the $n$th
radical of a monomial in $R$ if $\mathbf{u}^{ns}\in R$. Besides trivial
extensions $R$ and $R^{\prime\prime}$, radicals of the coefficient ring $R$
provide more interesting examples of extensions. Given $n\in{\mathbb{N}}$, the
extension of $R$ in $R^{\prime\prime}$ generated by all $n$th radicals of
monomials of $R$ is called the $n$th radical of $R$ in $R^{\prime\prime}$. If
$R^{\prime\prime}$ is understood from the context, we denote the $n$th radical
by $\sqrt[n]{R}$.

For the case $R^{\prime\prime}=\kappa[\\![\mathbf{u}]\\!]$, radicals are
investigated in the logarithmic form as quotients of the numerical semigroup
$\log_{\mathbf{u}}R$. It is known that a numerical semigroup is one half of a
symmetric numerical semigroup [11]. In fact, there are infinitely many such
symmetric numerical semigroups [12]. Even more, for each $n\geq 2$, a
numerical semigroup can be expressed as quotients of infinitely many symmetric
numerical semigroups by $n$ [17, Theorem 5]. For each $n\geq 3$, a numerical
semigroup can be also expressed as quotients of infinitely many pseudo-
symmetric numerical semigroups by $n$ [17, Theorem 6]. In this section, we
show that the classical results in [17] have a relative nature.

######  Example 5.1.

Let
$R=\kappa[\\![\mathbf{u}^{6},\mathbf{u}^{10},\mathbf{u}^{14},\mathbf{u}^{2n+1}]\\!]$,
where $n\in{\mathbb{N}}$. Then
$\sqrt[2]{R}=\kappa[\\![\mathbf{u}^{3},\mathbf{u}^{5},\mathbf{u}^{7}]\\!]$ in
$\kappa[\\![\mathbf{u}^{3},\mathbf{u}^{4},\mathbf{u}^{5}]\\!]$.

Numerical duplications are a source of radicals of coefficient rings. They are
first introduced in the logarithmic form [2]. In our notation, a numerical
duplication of a numerical semigroup ring $R$ is the numerical semigroup ring
$R^{\prime}=R[\\![\mathbf{u}^{s_{1}}\mathbf{u}^{s_{0}/2},\ldots,\mathbf{u}^{s_{n}}\mathbf{u}^{s_{0}/2}]\\!]$,
where $\mathbf{u}^{s_{0}},\ldots,\mathbf{u}^{s_{n}}\in R$ but
$\mathbf{u}^{s_{0}/2}\not\in R$. In $R^{\prime}$, we have
$\sqrt[2]{R}=R^{\prime}$. Flatness of numerical semigroup algebras is a main
theme for the study of singularities. However, as seen in the following
proposition, there are no extensions of the coefficient ring other than
radicals in a flat algebra.

######  Proposition 5.2.

Let $R^{\prime\prime}/R$ be a numerical semigroup algebra and let $R^{\prime}$
be an extension of the coefficient ring $R$. If $R^{\prime\prime}/R^{\prime}$
and $R^{\prime\prime}/R$ are both flat, then $R^{\prime}$ is a radical of $R$.

###### Proof.

We may assume that the numerical semigroup rings are in the variable
$\mathbf{u}$ and $\log_{\mathbf{u}}R^{\prime\prime}\subset{\mathbb{N}}$. Let
$d^{\prime\prime}=\gcd(\log_{\mathbf{u}}R^{\prime\prime})$,
$d^{\prime}=\gcd(\log_{\mathbf{u}}R^{\prime})$ and
$d=\gcd(\log_{\mathbf{u}}R)$. An element of $R^{\prime}$ can be written as
$\mathbf{u}^{sd^{\prime}}$ for some $s\in{\mathbb{N}}$. Let $q=d/d^{\prime}$.
We claim $R^{\prime}=\sqrt[q]{R}$. On the one hand,
$\mathbf{u}^{sd^{\prime}q}=\mathbf{u}^{sd}\in\kappa[\\![\mathbf{u}^{d/d^{\prime\prime}}]\\!]\cap
R^{\prime\prime}=R$, where the last equality holds by flatness of
$R^{\prime\prime}/R$ [4, Proposition 3.5]. Therefore
$R^{\prime}\subset\sqrt[q]{R}$. On the other hand, consider
$\mathbf{u}^{sd^{\prime}}\in\sqrt[q]{R}$. Then $\mathbf{u}^{sd}\in R$ and
hence $s\in{\mathbb{N}}$. The inclusion $\sqrt[q]{R}\subset R^{\prime}$
follows from
$\mathbf{u}^{sd^{\prime}}\in\kappa[\\![\mathbf{u}^{d^{\prime}/d^{\prime\prime}}]\\!]\cap
R^{\prime\prime}=R^{\prime}$, where the last equality holds by flatness of
$R^{\prime\prime}/R^{\prime}$ [ibid.]. ∎

Note that an equi-gcd algebra
$\kappa[\\![\mathbf{u}]\\!]/\kappa[\\![\mathbf{u}^{S}]\\!]$ is never flat,
unless $S={\mathbb{N}}$. We can say more about radicals for equi-gcd numerical
semigroup algebras in general. Let $n\geq 2$ and $R^{\prime\prime}/R^{\prime}$
be an equi-gcd numerical semigroup algebra in the variable $\mathbf{u}$. Using
a similar idea as [17], we will show that there are infinitely many equi-gcd
numerical semigroup algebras $R^{\prime\prime}/R$ such that
$R^{\prime}=\sqrt[n]{R}$ and $R$ is symmetric in $R^{\prime\prime}$. We may
assume that $\log_{\mathbf{u}}R^{\prime\prime}\subset{\mathbb{N}}$ and
$\gcd(\log_{\mathbf{u}}R^{\prime\prime})=1$. In its logarithmic form, the
condition $R^{\prime}=\sqrt[n]{R}$ can be stated as
$nS^{\prime}=\log_{\mathbf{u}}R\cap n\log_{\mathbf{u}}R^{\prime\prime}$, where
$S^{\prime}=\log_{\mathbf{u}}R^{\prime}$. Since $R^{\prime\prime}/R$ is
required to be equi-gcd, there should be a monomial $\mathbf{u}^{h}\in
R^{\prime\prime}$ such that $\mathbf{u}^{s}\in R$ for $s>h$. To construct a
symmetric coefficient ring $R$, we will choose an $h$ and join to

$R_{0}:=\kappa[\\![\mathbf{u}^{nS^{\prime}},\mathbf{u}^{h+1},\mathbf{u}^{h+2},\dots,\mathbf{u}^{2h+1}]\\!]$

certain monomials $\mathbf{u}^{s}\in R^{\prime\prime}$ satisfying $s\not\in
n\log_{\mathbf{u}}R^{\prime\prime}$. Note that $R^{\prime\prime}/R_{0}$ is
equi-gcd. Denote
$\\{\mathbf{u}^{w}\\}=\operatorname{F}(\kappa[\\![\mathbf{u}]\\!]/R^{\prime})$.
In other words, $w$ is the Frobenius number of $\log_{\mathbf{u}}R^{\prime}$.
If $h>w$, then $R_{0}\subset R^{\prime}$. If $h>nw$, then
$nS^{\prime}=\log_{\mathbf{u}}R_{0}\cap n\log_{\mathbf{u}}R^{\prime\prime}$.
The algebra $\kappa[\\![\mathbf{u}]\\!]/R^{\prime}$ has finitely many gap
monomials $\mathbf{u}^{w_{1}},\mathbf{u}^{w_{2}},\ldots$. If $h>nw+w$, then

$R_{1}:=R_{0}[\\![\mathbf{u}^{h-nw_{1}},\mathbf{u}^{h-nw_{2}},\ldots]\\!]\subset
R^{\prime}.$

######  Lemma 5.3.

Assume that $h>2nw$. Then
$\operatorname{G}(R_{1}/R_{0})=\\{\mathbf{u}^{h-nw_{1}},\mathbf{u}^{h-nw_{2}},\ldots\\}$.

###### Proof.

The condition $h>2nw$ implies $2h-nw_{i}-nw_{j}>h$. Hence
$\mathbf{u}^{h-nw_{i}}\mathbf{u}^{h-nw_{j}}\in R_{0}$. Monomials of $R_{0}$
are of the form $\mathbf{u}^{t}$ or $\mathbf{u}^{ns}$, where $t>h$ and
$\mathbf{u}^{s}\in R^{\prime}$. Clearly,
$\mathbf{u}^{t}\mathbf{u}^{h-nw_{i}}\in R_{0}$. If $s>w_{i}$, then
$ns+h-nw_{i}>h$ and hence $\mathbf{u}^{ns}\mathbf{u}^{h-nw_{i}}\in R_{0}$. If
$s<w_{i}$, then
$\mathbf{u}^{w_{i}-s}\in\operatorname{G}(\kappa[\\![\mathbf{u}]\\!]/R^{\prime})$
and $\mathbf{u}^{ns}\mathbf{u}^{h-nw_{i}}=\mathbf{u}^{h-nw_{j}}$ for some $j$.
We conclude that joining $\mathbf{u}^{h-nw_{1}},\ldots,\mathbf{u}^{h-nw_{m}}$
to $R_{0}$ does not create other gap monomials. ∎

Now we assume furthermore $n\nmid h$ besides the condition $h>2nw$. Then
$\mathbf{u}^{h}\in\operatorname{G}(R^{\prime}/R_{1})$ and
$nS^{\prime}=\log_{\mathbf{u}}R_{1}\cap n\log_{\mathbf{u}}R^{\prime\prime}$.
Before joining to $R_{1}$ more monomials, we consider a monomial
$\mathbf{u}^{s}\in\operatorname{G}(R^{\prime\prime}/R^{\prime})\subset\operatorname{G}(\kappa[\\![\mathbf{u}]\\!]/R^{\prime})$.
By the assumption $h>2nw$, we have $h-s\geq h-w>w$ and hence
$\mathbf{u}^{s}\preceq_{R^{\prime\prime}}\mathbf{u}^{h}$. This implies
$\operatorname{F}(R^{\prime\prime}/R_{1})\subset R^{\prime}$. Let
$\mathbf{u}^{t_{1}},\mathbf{u}^{t_{2}},\ldots$ be those monomials in
$\operatorname{G}(R^{\prime}/R_{1})$ satisfying
$\mathbf{u}^{t_{i}}\npreceq_{R^{\prime\prime}}\mathbf{u}^{h}$. Define

$R_{2}:=R_{1}[\\![\mathbf{u}^{t_{1}},\mathbf{u}^{t_{2}},\ldots]\\!]\subset
R^{\prime}.$

Then
$\operatorname{G}(R_{2}/R_{1})=\\{\mathbf{u}^{t_{1}},\mathbf{u}^{t_{2}},\ldots\\}$.
By the choice of $t_{i}$, the algebra $R^{\prime\prime}/R_{2}$ has a single
Frobenius monomial $\mathbf{u}^{h}$.

######  Theorem 5.4.

Let $R^{\prime\prime}/R^{\prime}$ be an equi-gcd numerical semigroup algebra.
Given $n\geq 2$, the coefficient ring $R^{\prime}$ is the $n$th radical of
infinitely many coefficient ring $R$, which can be taken to be symmetric in
$R^{\prime\prime}$. If $n\geq 3$, $R$ can be taken to be pseudo-symmetric.

###### Proof.

For $h\in{\mathbb{N}}$ satisfying $n\nmid h$ and $h>2nw$, we construct $R_{1}$
and $R_{2}$ as above so that
$\operatorname{F}(R^{\prime\prime}/R_{2})=\\{\mathbf{u}^{h}\\}$. By
Proposition 3.4, there is an irreducible coefficient ring $R$ of
$R^{\prime\prime}$ extending $R_{2}$ such that
$\operatorname{F}(R^{\prime\prime}/R)=\\{\mathbf{u}^{h}\\}$. Clearly,
$\sqrt[n]{R}\supset\sqrt[n]{R_{1}}=R^{\prime}$. On the other hand, if
$\mathbf{u}^{ns}\in R$ for some $\mathbf{u}^{s}\in R^{\prime\prime}$, then
$\mathbf{u}^{s}\in R^{\prime}$; otherwise $\mathbf{u}^{h-ns}\in R_{1}\subset
R_{2}\subset R$ contradicting to $\mathbf{u}^{h}\not\in R$. Therefore
$\sqrt[n]{R}=R^{\prime}$. In particular, $R\subset R^{\prime}$. Since $R$ is
irreducible,
$\operatorname{PF}(R^{\prime}/R)\subset\\{\mathbf{u}^{h},\mathbf{u}^{h/2}\\}$
by Theorem 3.6. If we choose $h$ such that $\mathbf{u}^{h/2}\notin
R^{\prime\prime}$, then
$\operatorname{PF}(R^{\prime\prime}/R)=\\{\mathbf{u}^{h}\\}$, which means $R$
is symmetric in $R^{\prime\prime}$. There are infinitely many such $h$. For
instance, we may choose $h$ to be an odd integer. If $\mathbf{u}^{h/2}\in
R^{\prime\prime}$, then
$\mathbf{u}^{h/2}\in\operatorname{G}(R^{\prime\prime}/R)$ and
$\mathbf{u}^{h/2}\npreceq_{R}\mathbf{u}^{h}$. Therefore,
$\operatorname{PF}(R^{\prime}/R)=\\{\mathbf{u}^{h},\mathbf{u}^{h/2}\\}$, i.e.
$R$ is pseudo-symmetric in $R^{\prime\prime}$. If $n\geq 3$, there are
infinitely many such $h$. For instance, we may choose $h$ to be of the form
$ns+2$, where $s\in{\mathbb{N}}$. ∎

######  Example 5.5.

Consider
$R^{\prime}=\kappa[\\![\mathbf{u}^{4},\mathbf{u}^{6},\mathbf{u}^{7},\mathbf{u}^{9}]\\!]$
and
$R^{\prime\prime}=\kappa[\\![\mathbf{u}^{4},\mathbf{u}^{5},\mathbf{u}^{6},\mathbf{u}^{7}]\\!]$.
Then
$\operatorname{F}(\kappa[\\![\mathbf{u}]\\!]/R^{\prime})=\\{\mathbf{u}^{5}\\}$.
In the above construction for $n=2$ and $h=21$, we have

$\displaystyle R_{0}$ $\displaystyle=$
$\displaystyle\kappa[\\![\mathbf{u}^{8},\mathbf{u}^{12},\mathbf{u}^{14},\mathbf{u}^{18},\mathbf{u}^{23},\mathbf{u}^{25},\mathbf{u}^{27},\mathbf{u}^{29}]\\!],$
$\displaystyle R_{1}$ $\displaystyle=$ $\displaystyle
R_{0}[\\![\mathbf{u}^{11},\mathbf{u}^{15},\mathbf{u}^{17},\mathbf{u}^{19}]\\!]=\kappa[\\![\mathbf{u}^{8},\mathbf{u}^{11},\mathbf{u}^{12},\mathbf{u}^{14},\mathbf{u}^{15},\mathbf{u}^{17},\mathbf{u}^{18}]\\!]$

and $R_{2}=R_{1}$. Then $R=R_{2}$ is symmetric in $R^{\prime\prime}$ and
$\sqrt[2]{R}=R^{\prime}$. For $n=3$ and $h=31$ we have

$\displaystyle R_{0}$ $\displaystyle=$
$\displaystyle\kappa[\\![\mathbf{u}^{12},\mathbf{u}^{18},\mathbf{u}^{21},\mathbf{u}^{27},\mathbf{u}^{32},\mathbf{u}^{34},\mathbf{u}^{35},\mathbf{u}^{37},\mathbf{u}^{38},\mathbf{u}^{40},\mathbf{u}^{41},\mathbf{u}^{43}]\\!],$
$\displaystyle R_{1}$ $\displaystyle=$ $\displaystyle
R_{0}[\\![\mathbf{u}^{16},\mathbf{u}^{22},\mathbf{u}^{25},\mathbf{u}^{28}]\\!]=\kappa[\\![\mathbf{u}^{12},\mathbf{u}^{16},\mathbf{u}^{18},\mathbf{u}^{21},\mathbf{u}^{22},\mathbf{u}^{25},\mathbf{u}^{27},\mathbf{u}^{35}]\\!],$
$\displaystyle R_{2}$ $\displaystyle=$ $\displaystyle
R_{1}[\\![\mathbf{u}^{29}]\\!]=\kappa[\\![\mathbf{u}^{12},\mathbf{u}^{16},\mathbf{u}^{18},\mathbf{u}^{21},\mathbf{u}^{22},\mathbf{u}^{25},\mathbf{u}^{27},\mathbf{u}^{29},\mathbf{u}^{35}]\\!].$

Note that $R_{2}$ is not symmetric in $R^{\prime\prime}$. Indeed,
$\operatorname{PF}(R^{\prime\prime}/R_{2})=\\{\mathbf{u}^{17},\mathbf{u}^{20},\mathbf{u}^{23},\mathbf{u}^{26},\mathbf{u}^{31}\\}$.
However,
$R=R_{2}[\\![\mathbf{u}^{17},\mathbf{u}^{20},\mathbf{u}^{23},\mathbf{u}^{26}]\\!]$
is symmetric in $R^{\prime\prime}$ and $\sqrt[3]{R}=R^{\prime}$.

##  6\. Fundamental Gap Monomials

Let $R^{\prime\prime}/R$ be a numerical semigroup algebra in the variable
$\mathbf{u}$. Intersections of radicals of $R$ have been studied in [8] for
the case $R^{\prime\prime}=\kappa[\\![\mathbf{u}]\\!]$ under the name
“arithmetic extension” from the logarithmic viewpoint. In this section, we
exhibit the relative nature of the connection between arithmetic extensions
and fundamental gaps.

######  Example 6.1.

Let
$R=\kappa[\\![\mathbf{u}^{7},\mathbf{u}^{8},\mathbf{u}^{9},\mathbf{u}^{10},\mathbf{u}^{11},\mathbf{u}^{13}]\\!]$
and $R^{\prime\prime}=R[\\![\mathbf{u}^{3},\mathbf{u}^{4}]\\!]$. Then
$\operatorname{G}(R^{\prime\prime}/R)=\\{\mathbf{u}^{3},\mathbf{u}^{4},\mathbf{u}^{6},\mathbf{u}^{12}\\}$.
The extension $R[\\![\mathbf{u}^{12}]\\!]=R[\\![\mathbf{u}^{4}]\\!]\cap
R[\\![\mathbf{u}^{3}]\\!]$ of $R$ is the intersection
$\sqrt[2]{R}\cap\sqrt[3]{R}$ of radicals.

######  Example 6.2.

Let $R=\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6},\mathbf{u}^{13}]\\!]$ and
$R^{\prime\prime}=R[\\![\mathbf{u}^{7},\mathbf{u}^{8}]\\!]$. Then
$\operatorname{G}(R^{\prime\prime}/R)=\\{\mathbf{u}^{7},\mathbf{u}^{8},\mathbf{u}^{14}\\}$.
The coefficient ring $R$ has three non-trivial extensions in
$R^{\prime\prime}$. The extensions $R[\\![\mathbf{u}^{7}]\\!]$ and
$R[\\![\mathbf{u}^{14}]\\!]$ are not intersection of radicals, while
$R[\\![\mathbf{u}^{8}]\\!]=\sqrt[2]{R}$.

######  Example 6.3.

The coefficient ring $\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6}]\\!]$ of
$\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6},\mathbf{u}^{19}]\\!]$ has only
trivial extensions. The extension
$\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6},\mathbf{u}^{19}]\\!]$ of
$\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6}]\\!]$ in
$\kappa[\\![\mathbf{u}]\\!]$ is not an intersection of radicals.

######  Example 6.4.

The coefficient ring
$\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6},\mathbf{u}^{14}]\\!]$ of
$\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6},\mathbf{u}^{7},\mathbf{u}^{8}]\\!]$
has two non-trivial extensions
$\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6},\mathbf{u}^{7}]\\!]$ and
$\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6},\mathbf{u}^{8}]\\!]$. Both are not
intersection of radicals.

Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra. The elements in

$\operatorname{FG}(R^{\prime}/R):=\operatorname{G}(R^{\prime}/R)\cap\sqrt[2]{R}\cap\sqrt[3]{R}$

are called the fundamental gap monomials of $R^{\prime}/R$. The gap monomial
with the largest exponent is a fundamental gap monomial. For the case
$R^{\prime}=\kappa[\\![\mathbf{u}]\\!]$, the exponents of monomials in
$\operatorname{FG}(R^{\prime}/R)$ are the fundamental gaps of the numerical
semigroup $\log_{\mathbf{u}}R$ first introduced in [14]. Note that
$\operatorname{FG}(R^{\prime}/R)\subset\sqrt[n]{R}$ for $n\geq 2$. Therefore
any intersection of radicals of $R$ contains
$\operatorname{FG}(R^{\prime}/R)$.

######  Example 6.5.

Let $R=\kappa[\\![\mathbf{u}^{5},\mathbf{u}^{6}]\\!]$ and
$R^{\prime}=R[\\![\mathbf{u}^{7},\mathbf{u}^{8}]\\!]$. Then

$\operatorname{G}(R^{\prime}/R)=\\{\mathbf{u}^{7},\mathbf{u}^{8},\mathbf{u}^{13},\mathbf{u}^{14},\mathbf{u}^{19}\\}.$

Radicals of $R$ are
$\sqrt[2]{R}=R[\\![\mathbf{u}^{8},\mathbf{u}^{13},\mathbf{u}^{14},\mathbf{u}^{19}]\\!]=R[\\![\mathbf{u}^{8}]\\!]$
and $\sqrt[n]{R}=R^{\prime}$ for $n>2$. Hence
$\operatorname{FG}(R^{\prime}/R)=\\{\mathbf{u}^{8},\mathbf{u}^{13},\mathbf{u}^{14},\mathbf{u}^{19}\\}$.

As observed in the classical case [8, Lemma 10], equi-gcd numerical semigroup
algebras with a single fundamental gap monomial have interesting properties.

######  Lemma 6.6.

Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra in the variable
$\mathbf{u}$. A monomial $\mathbf{u}^{s}\in\operatorname{G}(R^{\prime}/R)$ is
the only fundamental gap monomial if and only if any gap monomial is a radical
of $\mathbf{u}^{s}$.

###### Proof.

Assume first that $\operatorname{FG}(R^{\prime}/R)=\\{\mathbf{u}^{s}\\}$.
Given
$\mathbf{u}^{t}\in\operatorname{G}(R^{\prime}/R)\setminus\\{\mathbf{u}^{s}\\}$,
we have $\mathbf{u}^{t}\not\in\sqrt[n_{1}]{R}$ for some $n_{1}\geq 2$. Hence
$\mathbf{u}^{tn_{1}}\in\operatorname{G}(R^{\prime}/R)$. If
$\mathbf{u}^{tn_{1}}\neq\mathbf{u}^{s}$, there exists $n_{2}\geq 2$ such that
$\mathbf{u}^{tn_{1}n_{2}}\in\operatorname{G}(R^{\prime}/R)$. Repeating this
procedure, eventually we find $n\geq 2$ such that
$\mathbf{u}^{tn}=\mathbf{u}^{s}$.

On the other hand, we assume that any gap monomial is a radical of
$\mathbf{u}^{s}$. Given
$\mathbf{u}^{t}\in\operatorname{G}(R^{\prime}/R)\setminus\\{\mathbf{u}^{s}\\}$,
there exists $n\geq 2$ such that $\mathbf{u}^{tn}=\mathbf{u}^{s}$. Hence
$\mathbf{u}^{t}\not\in\sqrt[n]{R}$ and
$\mathbf{u}^{t}\not\in\operatorname{FG}(R^{\prime}/R)$. ∎

######  Proposition 6.7.

Any numerical semigroup ring has finitely many equi-gcd coefficient rings,
over which there is a single fundamental gap monomial.

###### Proof.

We may work on a numerical semigroup ring $R^{\prime}$ in the variable
$\mathbf{u}$ such that $\log_{\mathbf{u}}R^{\prime}\subset{\mathbb{N}}$ and
$\gcd(\log_{\mathbf{u}}R^{\prime})=1$. Choose an integer $n>4$ such that
$\mathbf{u}^{s}\in R^{\prime}$ for any $s\geq n$. If the proposition was not
true, there would exist an equi-gcd coefficient ring $R$, over which there is
a single fundamental gap monomial $\mathbf{u}^{w}$ for some $w>2n$. Let

$m=\begin{cases}\frac{w+1}{2},&\text{ if }w\text{ is odd;}\\\
\frac{w+2}{2},&\text{ if }w\text{ is even}.\end{cases}$

Then $\mathbf{u}^{m},\mathbf{u}^{w-m}\in R^{\prime}$ are not radicals of
$\mathbf{u}^{w}$. Hence $\mathbf{u}^{m},\mathbf{u}^{w-m}\in R$ by Lemma 6.6.
However, $\mathbf{u}^{m}\mathbf{u}^{w-m}\in R$ contradicts to
$\mathbf{u}^{w}\in\operatorname{G}(R^{\prime}/R)$. ∎

######  Example 6.8.

It is shown in the logarithmic form [8, Lemma 11] that the numerical semigroup
rings $\kappa[\\![\mathbf{u}[\\!]$,
$\kappa[\\![\mathbf{u}^{2},\mathbf{u}^{3}[\\!]$,
$\kappa[\\![\mathbf{u}^{3},\mathbf{u}^{4},\mathbf{u}^{5}]\\!]$,
$\kappa[\\![\mathbf{u}^{2},\mathbf{u}^{5}]\\!]$,
$\kappa[\\![\mathbf{u}^{3},\mathbf{u}^{5},\mathbf{u}^{7}]\\!]$ and
$\kappa[\\![\mathbf{u}^{4},\mathbf{u}^{5},\mathbf{u}^{7}]\\!]$ are all equi-
gcd coefficient rings of $\kappa[\\![\mathbf{u}]\\!]$, over which there is a
single fundamental gap monomial.

######  Example 6.9.

Let $R$ be a non-trivial equi-gcd coefficient ring of
$\kappa[\\![\mathbf{u}^{2},\mathbf{u}^{3}]\\!]$. Then
$\operatorname{FG}(\kappa[\\![\mathbf{u}]\\!]/R)=\operatorname{FG}(\kappa[\\![\mathbf{u}^{2},\mathbf{u}^{3}]\\!]/R)$.
Non-trivial coefficient rings of
$\kappa[\\![\mathbf{u}^{2},\mathbf{u}^{3}]\\!]$, over which, there is a single
fundamental gap monomials are
$\kappa[\\![\mathbf{u}^{3},\mathbf{u}^{4},\mathbf{u}^{5}]\\!]$,
$\kappa[\\![\mathbf{u}^{2},\mathbf{u}^{5}]\\!]$,
$\kappa[\\![\mathbf{u}^{3},\mathbf{u}^{5},\mathbf{u}^{7}]\\!]$ and
$\kappa[\\![\mathbf{u}^{4},\mathbf{u}^{5},\mathbf{u}^{7}]\\!]$.

######  Corollary 6.10.

Any numerical semigroup ring has finitely many equi-gcd coefficient rings,
whose extensions are all intersections of radicals.

###### Proof.

Let $R^{\prime}$ be a numerical semigroup ring in the variable $\mathbf{u}$.
Given an equi-gcd coefficient ring $R$ whose extensions are all intersections
of radicals, it suffices to prove that $\operatorname{FG}(R^{\prime}/R)$ is a
singleton. Assume the contrary that $\operatorname{FG}(R^{\prime}/R)$ contains
monomials $\mathbf{u}^{t}$ and $\mathbf{u}^{s}$ such that $t<s$. Then the
extension $R[\\![\mathbf{u}^{s}]\\!]$ is not an intersection of radicals,
because it does not contain $\mathbf{u}^{t}$. ∎

######  Proposition 6.11.

Let $R^{\prime}/R$ be an equi-gcd numerical semigroup algebra in the variable
$\mathbf{u}$. Then $\operatorname{FG}(R^{\prime}/R)$ is a singleton if and
only if all extensions of $R$ in $R^{\prime}$ are intersections of radicals.

###### Proof.

Assume first that
$\mathbf{u}^{s},\mathbf{u}^{t}\in\operatorname{FG}(R^{\prime}/R)$ and $t<s$.
Then the extension $R[\\![\mathbf{u}^{s}]\\!]$ is not an intersection of
radicals, because it does not contain $\mathbf{u}^{t}$. On the other hand, we
assume that $\operatorname{FG}(R^{\prime}/R)=\\{\mathbf{u}^{w}\\}$. Consider
an extension $R[\\![\mathbf{u}^{w_{1}},\ldots,\mathbf{u}^{w_{n}}[\\!]$, where
$\mathbf{u}^{w_{1}},\ldots,\mathbf{u}^{w_{n}}\in\operatorname{G}(R^{\prime}/R)$.
For each $\mathbf{u}^{w^{\prime}}\in\operatorname{G}(R^{\prime}/R)\setminus
R[\\![\mathbf{u}^{w_{1}},\ldots,\mathbf{u}^{w_{n}}[\\!]$, there exits
$d^{\prime}\in{\mathbb{N}}$ such that $d^{\prime}w^{\prime}=w$. Hence
$\mathbf{u}^{w^{\prime}}\not\in\sqrt[d^{\prime}]{R}$. We claim that
$\mathbf{u}^{w_{i}}\in\sqrt[d^{\prime}]{R}$ for all $i$. It then follows that
$R[\\![\mathbf{u}^{w_{1}},\ldots,\mathbf{u}^{w_{n}}[\\!]$ is the intersection
of all such radicals of $R$. If
$\mathbf{u}^{w_{i}}\not\in\sqrt[d^{\prime}]{R}$, there exists
$d_{i}\in{\mathbb{N}}$ such that $d_{i}d^{\prime}w_{i}=w$. Together with
$d^{\prime}w^{\prime}=w$, we obtain $d_{i}w_{i}=w^{\prime}$ contradicting to
$\mathbf{u}^{w^{\prime}}\not\in
R[\\![\mathbf{u}^{w_{1}},\ldots,\mathbf{u}^{w_{n}}[\\!]$.

∎

## Acknowledgments

This work has been initiated during the visit of Raheleh Jafari to the
Institute of Mathematics, Academia Sinica in 2018. The authors would like to
thank the institute for the great hospitality and support.

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#  Symmetries in 3D photoelectron momentum spectroscopy as precursory methods
for dichroic and enantiosensitive measurements

Michael Davino  Department of Physics, University of Connecticut, Storrs, CT
06268, USA  Edward McManus  Department of Physics, University of Connecticut,
Storrs, CT 06268, USA  Tobias Saule  Department of Physics, University of
Connecticut, Storrs, CT 06268, USA  Phi-Hung Tran  Department of Physics,
University of Connecticut, Storrs, CT 06268, USA  Andrés F. Ordóñez
Department of Physics, Imperial College London, SW7 2AZ London, UK  Department
of Chemistry, Queen Mary University of London, E1 4NS London, UK  George
Gibson  Department of Physics, University of Connecticut, Storrs, CT 06268,
USA  Anh-Thu Le  Department of Physics, University of Connecticut, Storrs, CT
06268, USA  Carlos A. Trallero-Herrero  Department of Physics, University of
Connecticut, Storrs, CT 06268, USA<EMAIL_ADDRESS>
###### Abstract

3D photoelectron angular distributions (PADs) are measured from an atomic
target ionized by ultrafast, elliptical fields of opposite handedness.
Comparing these PADs to one another and to numeric simulations, a difficult to
avoid systematic error in their orientation is identified and subsequently
corrected by imposing the dichroic symmetry by which they are necessarily
related. We show that this correction can be directly applied to molecular
targets in the same fields. This paves the way for measurement of
enantiosensitive information which has yet to be accessed experimentally.

††preprint: APS/123-QED

Chirality has been, and continues to be, of significant interest to both
fundamental science and technological applications. While biological
homochirality is among the most profound unsolved scientific questions
Meierhenrich (2013); Sallembien et al. (2022), chiral molecules are already
fundamental for drug production Lin et al. (2011) and serve as disease
biomarkers for cancer, Alzheimer’s, diabetes and more Liu et al. (2023).
Despite the need for reliable methods, chiral recognition remains challenging
Brandt et al. (2017); Liu et al. (2023). Photoelectron momentum spectrosopy
techniques present a promising alternative at both the fundamental level Wanie
et al. (2024) and the applied level Comby et al. (2023, 2018). Specifically,
photoelectron angular distributions (PADs) obtained from ionization of chiral
molecules by circular or elliptically polarized strong fields not only contain
information pertaining to ionization dynamics Eckle et al. (2008); Trabert et
al. (2021); Hockett et al. (2015), but also for the identification of chiral
enantiomers Beaulieu et al. (2016a); Comby et al. (2018).

The interplay between field handedness and molecular chirality in ionization
manifests as asymmetries in the measured PAD. Such measurements are highly
relevant as they can be both dichroic (opposite for opposite field handedness)
and enantiosensitive (opposite for opposite enantiomers); a particularly
powerful example of this is photoelectron circular dichroism (PECD) Ritchie
(1976); Beaulieu et al. (2016a). PECD manifests as a forward-backward
asymmetry in the PAD along the axis of laser propagation for randomly oriented
chiral molecules when ionized by circular or elliptical fields and can be
tracked on ultrafast time scales Beaulieu et al. (2017); Ayuso et al. (2022);
Beaulieu et al. (2016b); Comby et al. (2016). The forward-backward asymmetry
of PECD does not require measurement of the full 3D momentum distribution to
resolve; the 2D projection as measured by conventional velocity map imaging
(VMI) techniques is generally sufficient. However, information regarding
molecular chirality is not exclusively contained within this asymmetry.
Theoretical work has shown that the azimuthal (in the plane transverse to
field propagation) dependence of the PAD, inaccessible to 2D VMI, contains
information about molecular chirality Ordonez and Smirnova (2022); Demekhin
(2019) which has not been accessed experimentally. Further, although standard
reconstruction methods have typically allowed for 3D information to be
recovered from 2D measurements in cylindrical symmetric systems, recent work
suggests that symmetry breaking is expected to occur in any case where linear
and circular fields are used together Sparling et al. (2023). Such cases are
particularly relevant for studying chiral molecules.

In this letter we provide analytic methods foundational for identifying
dichroism and enantiosensitivity in experimentally obtained 3D PADs. Using a
novel 3D VMI setup Davino et al. (2023) we measure strong field ionization of
atomic xenon (Xe) by elliptically polarized light of different handedness. The
measured 3D PADs are dichroic and non cylindrically symmetric. They are found
to obey the expected dichroic symmetry rules, but only after correction for
systematic errors in the azimuthal orientation. Further, we show that this
correction may be directly applied to a molecular species (oxygen, O2) ionized
by the same fields. Thus our methodology may be broadly applied and is
particularly relevant for molecular targets where the numeric simulations
required to calibrate the correction may be infeasible. We argue this is a
necessary precursor to studies involving chiral targets where azimuthal
orientation is critical to the extraction of maximum enantiosensitive
information.

The natural representation of 3D PADs is in the spherical harmonics,

$S(k,\theta,\varphi)=\sum_{L,M}\beta_{L,M}(k)Y_{L,M}(\theta,\varphi),$ (1)

where $k\propto|\vec{p}|$, $\theta$ is the angle from the optical propagation
axis, $\hat{z}$, and the azimuthal angle, $\varphi$, is in the plane of
polarization. The coefficients, $\beta_{L,M}(k)$, are the anisotropy
parameters defined in 3D. The angle $\varphi$ is measured relative to the
field polarization, namely, the ellipse major axis which is aligned to
$\hat{x}$. Common VMI methods utilize reconstruction techniques which require
a cylindrically symmetric system and therefore only $M=0$ coefficients are
accessible. By performing direct 3D measurements, no assumption of the PAD
symmetry is required, and therefore complementary enansiosensitive information
will be contained within the $M\neq 0$ terms Ordonez and Smirnova (2022). The
information lost to conventional VMI measurements becomes increasingly
relevant in the case of elliptically polarized light and in the multi-photon
regime where many $M$ terms are allowed.

Figure 1: Experimental and simulated results for Xe+. (a) Slices of the 3D
photoelectron momentum distribution in the plane of polarization for various
$p_{z}$; experiment and simulation are represented in the top and bottom rows
respectively. Experimental distributions are integrated over a range $\pm
0.05$ amu from the quoted $p_{z}$. (b) Spherical PADs of the first ATI ring
for experiment (left) and simulation (right). Each is generated from the
maximum of their first ATI peak, the experimental peak at $p=0.245$ amu
(integrated over $\pm 0.007$ amu) and simulation at $p=0.225$. Dashed black
lines in (a) and (b) represent same line, placed at the peak of the first ATI
shell for $\theta=\pi/2$. All distributions are separately normalized for easy
visualization of structure.

The information encoded in PADs is neatly encapsulated in $\beta_{L,M}$ which
are subject to selection rules dictated by the target symmetry Ordonez and
Smirnova (2022); Demekhin (2019). Several important instances are highlighted
here for the case of elliptically polarized light with ellipticity
$\varepsilon=|E_{min}|/|E_{max}|$. The field handedness will be denoted as
right ($r$) or left ($\ell$) handed, and molecular handedness will be denoted
as positive ($+$) or negative ($-$). Also, because any measured distribution
is necessarily real, it is advantageous to use the real spherical harmonics,
thus the anisotropy parameters are also real.

  1. i.

$\beta_{L,M}=0$ for odd $M$

This is a direct consequence of the two-fold symmetry of the system in the
plane of polarization.

  1. ii.

Enansiosensitive coefficients have odd $L$

This condition, familiar from the forward-backward asymmetry in PECD, stems
from considering a mirror reflection in the polarization plane. Such
reflection exchanges enantiomers without modifying the rotation direction of
the field, which leads to the relation

$S^{+}(k,\theta,\varphi)=S^{-}(k,\pi-\theta,\varphi),$ (2)

which holds for both field handedness.

  1. iii.

Dichroic coefficients have either odd $L$ and $M\geq 0$ or even $L$ and $M<0$

This condition stems from considering a 180 deg rotation of the system about
the $x-$axis. Such reflection reverses the rotation direction of the field
without modifying the enantiomers, which leads to the relation

$S^{(r)}(k,\theta,\varphi)=S^{(\ell)}(k,\pi-\theta,-\varphi),$ (3)

which holds for both enantiomers. For an achiral sample, mirror symmetry about
the polarization plane modifies Eq. 3 to be

$S^{(r)}(k,\theta,\varphi)=S^{(\ell)}(k,\theta,-\varphi).$ (4)

This is the relevant dichroic relation for the work presented in this letter
and has the direct consequence that $\beta_{L,M}=0$ for odd $L$.

Finally, for dichroic PADs satisfying either Eq. 3 or 4, the
$\varphi$-inversion manifests in the anisotropy parameters as

$\beta_{L,M}^{(r)}=\begin{cases}\beta_{L,M}^{(\ell)},&M\geq 0\\\
-\beta_{L,M}^{(\ell)},&M<0.\end{cases}$ (5)

Figure 2: (a) Photoelectron yield as a function of $\varphi$ for the peak of
the first ATI ring at $p_{z}=0$, or equivalently, $\theta=\pi$ (see dashed
black lines in Fig. 1). This is shown for left (experiment and simulation) and
right (experimental only) handed polarization with $\varepsilon=0.3$. The
right handed results are $\varphi$-inverted for direct comparison. (b)
$\chi^{2}$ as a function of an introduced variable angular offset,
$\varphi_{o}$. This is shown for the $S^{(\ell)}$ minimized to the simulated
PAD (black), and the $S^{(r)}$ minimized to the corrected $S^{(\ell)}$ (red).
The curves indicate fits to a quadratic function, and the minima are shown
with errorbars.

The rules and relations presented in points i-iii and Eqs. 2-5 represent
strict conditions for the anisotropy parameters under various configurations
of the field and target handedness. From an experimental perspective, a
failure to observe these relations when they are relevant is most easily
attributed to imperfect field characterization with respect to the laboratory
frame. This is to say that the handedness, ellipticity, and ellipse
orientation of the field must be well known in the interaction region to
transform the PAD into the interaction frame which is defined by the ellipse
axes. It is in this interaction frame that, for achiral targets, Eqs. 4 and 5
must hold. We find that despite careful characterization of the fields outside
the interaction region, comparison of the measured PADs to Eq. 4 show clear
deviations from the strict selection rules. This error can attributed to
birefringence of optical windows and is difficult to avoid. We show that with
the aid of high-quality simulations, an angular offset in $\varphi$ may be
introduced to properly align the experimental ellipse axes. The resulting PADs
exhibit the expected dichroic relations Eqs. 4 and 5 to high precision.

The VMI apparatus used here is a double-sided electron VMI and ion time-of-
flight spectrometer capable of measuring 3D PADs in coincidence with ions
Davino et al. (2023). A Titanium:Sapphire laser system was used to generate
elliptically polarized laser pulses ($80$ fs, $47$ µJ). These pulses were
focused into the VMI chamber containing the target gas (Xe or O2), reaching an
ionizing intensity of $\sim 2\times 10^{13}$ W/cm2. The ellipticity of the
field and orientation of the ellipse axes were determined just prior to the
VMI entrance window by the relative throughput power of the pulse following a
wire grid polarizer for various angles of the polarizer. Generally, this
measurement has a precision of $\pm 0.01$ and $\pm 0.5$ deg for the
ellipticity and axes orientation respectively. The results shown in this
letter are for $\varepsilon=0.3$, but all findings were similarly shown for
$\varepsilon=0.6$. For futher details on the experimental setup, ellipticity
characterization, and results for $\varepsilon=0.6$ please see Supplemental
Materials.

Coincidence analysis is performed to isolate photoelectrons from Xe+ or
O${}_{2}^{+}$, and from these electron events 3D PADs are generated.
$\beta_{L,M}$ are extracted individually for specific $k$ and thus, from the
full 3D PAD, angular histograms $h_{ij}=h(\theta_{i},\varphi_{j})$ are
generated. These histograms are effectively 3D PADs of spherical ’shells’ for
a given radius $k$, and they will be addressed as such. The histograms are
normalized such that $\sum_{L,M}|\beta_{L,M}|^{2}$ approaches $1$ with the
number of terms summed. In all experiments we compute terms up to and
including $L=10$ yielding sums $>0.990$. The uncertainty of each anisotropy
parameter ($\sigma_{L,M}$) was determined by a data partitioning scheme of the
gated electron events, and is taken to be the standard deviation of the set of
$\beta_{L,M}$ as attained from the (10) different partitioned subsets. Here we
focus on spherical shells with $k$ corresponding to the first above threshold
ionization (ATI) peak as they contain the most structure and largest number of
electron events.

To simulate the experimental results, we solved the time-dependent Schrödinger
equation (TDSE) numerically for the target atom in an intense laser pulse. Our
method has been described previously (see, e.g., Hoang et al. (2017); Chen et
al. (2009)). Briefly, the target atom is treated within the single-active-
electron (SAE) approximation with a model potential $V(r)$ of five independent
parameters. More details can be found in Supplementary Materials. From the
theory point of view, one of the most important findings is that measured 3D
PADs provide very strict constraints on the model parameters.

Figure 3: $\beta_{L,M}$ decomposition up to $L=10$ from the first ATI shell of
Xe+ for ionization by elliptically polarized fields ($\varepsilon=0.3$) of
both right and left handedness, as well as simulation results for the left
handed case. (a) Log plot of the overall contribution of each $L$; this is
shown for the corrected ($\chi^{2}$ minimized) experimental PADs, although
they are effectively unchanged upon correction. Error bars extending beyond
the limit of the $y$-axis indicate they reach into negative values. (b) Plots
of individual $\beta_{L,M}$ for $L=2,4,6,8$. Experimental results are shown
before (top row) and after (bottom row) angular correction. $\beta_{L,M}$ are
discrete parameters, connecting line segments are merely a visual tool.

Fig. 1 sample PADs for both experiment and simulation corresponding to
ionization by left-handed fields. Shown in (a) are slices of the 3D
distribution in the plane of polarization for different momenta along the
laser propagation axis, $p_{z}$, and (b) presents sample spherical PADs
corresponding to the first ATI peak. Qualitatively, both types of PAD exhibit
exceptionally good agreement between experiment and simulation. Even the slice
distributions at $p_{z}=0.25$ amu, where the ATI structure is complicated and
experimental statistics are worse, there remain striking similarities. Also
note that differences in experiment versus simulation may be partially
attributed to the finite integration of the 3D PAD in the experimental case.

Fig. 2 (a) presents lineouts of the photoelectron yield as a function of
$\varphi$ for the first ATI ring at $p_{z}=0$ (equivalently, the first ATI
shell at $\theta=\pi$, see dashed black lines in Fig. 1). The right handed
results have been $\varphi$-inverted so that, by Eq. 4, it is expected the
three lineouts are identical. While the general structure of the angular
photoelectron yield is very comparable between each dataset, there is a clear
offset in $\varphi$ between each of the three. In particular, the two
experimental lineouts are offset in $\varphi$ by $>5$ deg. This is despite
measurement of the ellipse axes (and thus the PAD axes) to an estimated
accuracy of $\pm 0.5$ deg. However, we reiterate that this measurement is
performed prior to the VMI entrance window and we atribute this difference to
birefringence in the window under vacuum stress. Regardless of the reason for
the error in characterization, a poorly calibrated ellipticity at the
interaction region translates as a poor transformation from the lab frame to
the interaction frame.

To properly transform each experimental PAD ($S^{(\ell)}$ and $S^{(r)}$) into
the interaction frame, a two step procedure is introduced. First, an angular
offset $\varphi\rightarrow\varphi+\varphi_{o}$ is introduced to $S^{(\ell)}$,
to optimize the agreement between the measurement and simulation. In doing so
$S^{(\ell)}$ is aligned to the field ellipse axes. Then, $S^{(r)}$ can be
aligned in one of two equivalent ways. One approach is to simply repeat the
process of comparing to simulation. Alternatively, and the method used here,
Eq. 4 may be employed as an optimization condition where the newly corrected
$S^{(\ell)}$ serves as a reference to which $S^{(r)}$ is aligned by its own
angular offset. Note that in this approach, if the experimental PAD used as
reference is not itself properly aligned, Eq. 4 will still hold, but neither
of the two PADs will be in the interaction frame nor will their field axes
coincide. Done properly, the net result is two experimentally obtained PADs,
from opposite field handedness, in the interaction frame.

Optimization of the agreement between two PADs, whether it be experiment to
simulation or experiment to experiment, may be performed by conventional
$\chi^{2}$ minimization with respect to $\varphi_{o}$. The details can be
found in Supplemental Materials. Fig. 2 (b) presents $\chi^{2}$ minimization
for the first ATI shell utilizing left handed simulation as a reference to
orient the left (red) and right (black) handed experimental results. Each
$\chi^{2}$ is minimized within $\pm 1$ deg; this represents a significant
improvement in ellipse alignment considering the right handed results require
a correction of $\sim 10$ deg.

The consequences of this correction procedure on the anisotropy parameters in
Fig. 3 (a) shows the relative contribution of each $L$ to the full PAD
normalized as stated above. Fig. 3 (b) presents both the raw and corrected
$\beta_{L,M}$ for the first four even $L\neq 0$.

Figure 4: (a) Slice of the 3D PAD for O2 ionized by field ellipticity
$\varepsilon=0.3$. The slice is centered at $p_{z}=0$ and integrated over a
width of $\pm 0.05$ amu. (b) $\chi^{2}$ minimization for Xe and O2 ionized by
the same fields. Dashed lines indicate fits to a quadratic function, and the
offset angles for which they are minimized fall within the uncertainty of one
another.

Inspection of Fig. 3 reveals that, only after $\chi^{2}$ correction, each of
the expected selection rules and relations for an achiral target are well
represented. Most obvious is that $\beta_{L,M}=0$ lies within the error of
each parameter with odd $M$ and odd $L$. Further, it can be seen qualitatively
that the relation between coefficients from opposite handedness (Eq. 5) holds
very well up to $L=10$. To assess the agreement quantitatively we introduce a
weighted mean square deviation from Eq. 5,

$\eta^{2}=\frac{1}{N}\sum_{L,M}\frac{|\beta^{(\ell)}_{L,M}-\operatorname{sgn}\left[M\right]\beta^{(r)}_{L,M}|^{2}}{|\sigma^{(\ell)}_{L,M}|^{2}+|\sigma^{(r)}_{L,M}|^{2}},$
(6)

where $N$ is the number of summation terms, i.e. the number of $L,M$
combinations included. The factor of $\operatorname{sgn}\left[M\right]$ is
necessary to account for Eq. 5, that is, the right handed coefficients are
expected to be equal but opposite their left handed counterparts for $M<0$.
Thus the numerator is minimized for every $L$ and $M$ combination when Eq. 5
is satisfied. $\eta^{2}<1$ indicates that, on average, the anisotropy
parameters satisfy Eq. 5 to within their uncertainties. The anisotropy
parameters shown in Fig. 3 result in $\eta^{2}=4.09$ for the uncorrected PADs,
and $\eta^{2}=0.31$ for the corrected PADs. These values represent a
significant shift in the interpretation of results upon angular correction:
the original PADs would be said to not exhibit a dichroic relation whereas the
corrected PADs do. Furthermore, we need to emphasize that for more complex
PADs the weight of different $L$ components can not be determined a priori and
thus a cutoff in the expansion is very target dependent.

Next, we extend the same analysis to molecular targets to demonstrate the
generality of the azimuthal calibration. 3D PADs were measured for both Xe and
O2 for both field handedness. A sample of the O2 PAD at $p_{z}=0$ is presented
in Fig. 4 (a). For both species, an angular offset was introduced to $S^{(r)}$
and $\chi^{2}$ minimized to optimize Eq. 4; the results are presented in Fig.
4 (b). Note that the same field parameters ($80$ fs, $47$ µJ,
$\varepsilon=0.3$) are used here as have been used throughout this letter, but
$\chi^{2}$ minimization was performed without first orienting to simulation;
thus the discrepancy in the Xe angular offset between Figs. 2 and 4. The
critical result is that both Xe and O2 have a minimum $\chi^{2}$ for the same
offset angle. In other words, the correction angle maximizing the dichroic
relation (Eq. 5) for Xe, also does so for O2. This means that when simulations
are utilized to find offset angles which properly orient Xe in the interaction
frame, the same offset angles may be applied directly to other molecules,
including chiral molecules, for which such simulations cannot be done.

To conclude, we have shown the experimental capability to measure 3D PADs that
exhibit dichroic behavior. Since the field can not be characterized at the
interaction region, we find systematic errors that affect the lab-to-
interaction frame transformation. This transform is critical in extracting
dichroic and enantiosensitive information contained within the anisotropy
parameters that represent the PADs. As a correction, numeric simulations are
utilized in conjunction with the symmetry relations between PADs. In
particular, the dichroic relation is shown to be well satisfied with the
correction up to the $L=10$ term in the expansion. We extend the
characterization to O2 and demonstrate it shares the same azimuthal correction
angle as Xe, implying that the interaction frame transformation may be
calibrated in an atomic species and then applied to a molecular species where
numeric simulations are infeasible. This work paves the way for high-
resolution measurement of the enantiosensitive information contained within
the anisotropy parameters by providing a contrast of several orders of
magnitude between dichroic only (even L) and enantiosensitive (even and odd L)
values in $\beta_{L,M}$. Further, 3D PADs present theory with highly sensitive
experimental information for model validation without the assumption of any
symmetry.

Experimental work was done under Air Force Office of Scientific Research grant
FA9550-21-1-0387. Theoretical simulations were done by PT and A-TL under DOE
BES, Chemical Sciences, Geosciences, & Biosciences Division grant DE-
SC0023192. Data analysis carried out by EM supported under DOE BES, Chemical
Sciences, Geosciences, & Biosciences Division grant DE-SC0024508.

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