# Extremal behavior of reduced type of one dimensional rings

Sarasij Maitra<EMAIL_ADDRESS>Department of Mathematics, Indian
Institute of Technology Delhi, Hauz Khas, India.  and  Vivek Mukundan
<EMAIL_ADDRESS>
###### Abstract.

Let $R$ be a domain that is a complete local $\mathbb{k}$ algebra in dimension
one. In an effort to address the Berger’s conjecture, a crucial invariant
reduced type $s(R)$ was introduced in [HMM21]. In this article, we study this
invariant and its max/min values separately and relate it to the valuation
semigroup of $R$. We justify the need to study $s(R)$ in the context of
numerical semigroup rings and consequently investigate the occurrence of the
extreme values of $s(R)$ for the Gorenstein, almost Gorenstein, and far-flung
Gorenstein complete numerical semigroup rings. Finally, we study the
finiteness of the category $\operatorname{CM}(R)$ of maximal Cohen Macaulay
modules and the category $\operatorname{Ref}(R)$ of reflexive modules for
rings which are of maximal/minimal reduced type and provide many
classifications.

###### Key words and phrases:

reduced type, one dimensional complete local domains, maximal reduced type,
minimal reduced type, finite CM type, finite ref type

###### 2010 Mathematics Subject Classification:

Primary: 13A15. Secondary: 13H05, 13Cxx, 13Gxx, 16W60

##  1\. Introduction

Let $(R,\mathfrak{m},\mathbb{k})$ be a non-regular one dimensional complete
local domain which is a $\mathbb{k}$-algebra. We assume that $\mathbb{k}$ is
algebraically closed of characteristic $0$. Assume
$\mathbb{k}\llbracket{\alpha_{1}t^{a_{1}},\dots\alpha_{n}t^{a_{n}}}\rrbracket\cong
R=\frac{\mathbb{k}\llbracket{X_{1},\dots,X_{n}}\rrbracket}{I}$,
$I\subseteq(X_{1},\dots,X_{n})^{2}$. Here $\alpha_{i}$ are units of
$\overline{R}$. Additionally, we also arrange the $x_{i}$ such that
$a_{1}<a_{2}<\dots<a_{n}$. The reduced type of $R$ is

$\displaystyle s(R):=\dim_{\mathbb{k}}\frac{\mathfrak{C}+x_{1}R}{x_{1}R},$

where $\mathfrak{C}$ is the conductor ideal of the ring $R$. Recall that the
conductor $\mathfrak{C}=R:_{\operatorname{Quot}(R)}\overline{R}$ is the
largest common ideal between $R$ and $\overline{R}$. This invariant was
defined in [HMM21] as a means of constructing an over ring $S$ of $R$. The
authors also demonstrated that constructing sufficient torsion elements in the
module of differentials $\Omega_{S}$ helped to solve the Berger’s conjecture
for $R$. This invariant was also used by the authors in [MM23] extensively to
build classes of rings which attests to the positivity of the conjecture. The
main objective of this article is to investigate the values attained by this
invariant $s(R)$ and its relationship to the Cohen-Macaulay type
$\operatorname{type}(R)$. As the name suggests, the reduced type $s(R)$ is
indeed related to $\operatorname{type}(R)$. For,
$\frac{\mathfrak{C}+x_{1}R}{x_{1}R}\subseteq\frac{x_{1}R:\mathfrak{m}}{x_{1}R}$
and the $\mathbb{k}$-vector space dimension of the latter object is precisely
$\operatorname{type}(R)$. Thus the maximum and minimum values of $s(R)$ are
captured in the inequality

$1\leqslant s(R)\leqslant\operatorname{type}(R).$

Following this observation, we say that $R$ is of minimal (resp. maximal)
reduced type when $s(R)=1$ (resp. $s(R)=\operatorname{type}(R)$).
Interestingly, when $R$ is Gorenstein (equivalently
$\operatorname{type}(R)=1$), $s(R)$ equals one, but not conversely (see for
example Example 2.6). In fact, Gorenstein rings are the only ones of both
maximal and minimal reduced type.

Notice that under the above setup, the integral closure
$\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$ is a $DVR$ (Discrete
Valuation Ring) and hence one can define an order valuation $v(\cdot)$ on the
ring $R$ induced from the one on $\overline{R}$, allowing us to define a
valuation semigroup $v(R)$ of $R$. There is a strong relationship between the
singularities of $R$ and the structural representations of the valuation
semigroup $v(R)$. For example, Kunz [Kun70] showed that $R$ is Gorenstein if
and only if $v(R)$ is symmetric. This has since then been generalized to
almost Gorenstein rings too. Barucci-Froberg [BF97] showed that $R$ is almost
Gorenstein (equivalently
$\ell(\overline{R}/R)=\ell(R/\mathfrak{C})+\operatorname{type}(R)-1$, where
$\ell(M)$ is the length of the module $M$) if and only if the valuation
semigroup $v(R)$ is almost symmetric and
$\operatorname{type}(R)=\operatorname{type}(v(R))$. The Cohen-Macaulay type
$\operatorname{type}(v(R))$ can be easily computed as the cardinality of the
pseudo-Frobenius set $\operatorname{PF}(v(R))$ (see Section 3 for a
definition). In fact there are some nice characterizations to check when the
valuation semigroup $v(R)$ is almost symmetric ([Nar13]). We show that the
reduced type $s(R)$ is also the cardinality of the set
$[c-a_{1},c-1]\backslash v(R)$ where $\mathfrak{C}=(t^{c})\overline{R}$, the
conductor of the ring $R$ (see Theorem 2.13). The integer $c$ is called the
conductor valuation of $R$. For the ring $R$ we can also define a complete
numerical semigroup ring
$\mathbb{k}\llbracket{v(R)}\rrbracket=\mathbb{k}\llbracket{t^{b_{1}},\dots,t^{b_{m}}}\rrbracket$
where $v(R)=\langle b_{1},\dots,b_{m}\rangle$. Along with [BDF97, Proposition
II.1.16] the following set of inequalities become apparent.

$s\left(\mathbb{k}\llbracket{v(R)}\rrbracket\right)=s(R)\leqslant\operatorname{type}(R)\leqslant|\operatorname{PF}(v(R))|=\operatorname{type}(\mathbb{k}\llbracket{v(R)}\rrbracket).$

Using this set of inequalities we conclude:

######  Theorem A (Proposition 3.4).

Let $R$ be as described at the beginning of the article.

  1. (1)

The ring $R$ is of minimal reduced type if and only if
$\mathbb{k}\llbracket{v(R)}\rrbracket$ is of minimal reduced type.

  2. (2)

The ring $R$ is of maximal reduced type if
$\mathbb{k}\llbracket{v(R)}\rrbracket$ is of maximal reduced type. The
converse holds if additionally
$\operatorname{type}(R)=\operatorname{type}(\mathbb{k}\llbracket{v(R)}\rrbracket)$.

In fact, when $R$ is of minimal multiplicity or when $R$ is almost Gorenstein,
then the statements in (2) are equivalent.

The above result indicates that the exact nature of the minimal/maximal
reduced type of $R$ can often be captured by studying the minimal/maximal
reduced type of the complete numerical semigroup
$\mathbb{k}\llbracket{v(R)}\rrbracket$. As such, we focus our attention to
complete numerical semi-group rings and provide explicit characterization
(Proposition 3.7 and Proposition 3.8) of maximal/minimal reduced type rings in
terms of the distribution of the pseudo-Frobenius numbers. For instance, the
above study yields the following criteria.

######  Theorem B (Corollaries 3.10 and 3.11).

Let $R=\mathbb{k}\llbracket{t^{a_{1}},t^{a_{2}},\ldots,t^{a_{n}}}\rrbracket$
be a complete numerical semigroup ring of minimal multiplicity. Then $R$ is of
maximal (resp. minimal) reduced type if and only if $a_{2}\geqslant
a_{n}-a_{1}+1$ (resp. $a_{n-1}+a_{1}-1<a_{n}$).

When $R$ is almost Gorenstein, the maximal and minimal reduced type property
bounds the conductor valuation $c$ of $R$ (Proposition 3.16). But when the
type is 2, these statements are actually equivalent. These types of rings are
called Kunz rings.

######  Theorem C (Corollary 3.19).

Let $R=\mathbb{k}\llbracket{t^{a_{1}},t^{a_{2}},\ldots,t^{a_{n}}}\rrbracket$
be a complete numerical semigroup ring which is almost Gorenstein of type 2
with conductor valuation $c$. Then $R$ is of maximal (resp. minimal) reduced
type if and only if $c\leqslant 2a_{1}-1$ (resp. $c>2a_{1}-1$).

Herzog, Kumashiro and Stamate in [HKS21] introduced the notion of far-flung
Gorenstein rings (see Section 3.2 for a definition). We show that the reduced
type of such rings are at least $2$ (Proposition 3.22) and hence they are
never of minimal reduced type. But the maximal reduced type can be understood
from constraints in $\operatorname{type}(R)$.

######  Theorem D (Theorems 3.23 and 3.25).

Let $R$ be a far-flung Gorenstein complete numerical semigroup ring of
$\operatorname{type}(R)\leqslant 3$ or $\operatorname{type}(R)=4$ of non-
minimal multiplicity. Then $R$ has maximal reduced type.

Our methods do not immediately generalize to higher types in such rings. The
lack of answers to [HKS21, Questions 5.5] is the primary obstruction. Bounds
on multiplicity of $R$ also have implications on the maximal reduced type of
such rings (Theorem 3.26 and Proposition 3.27).

In Section 3.3 we study the gluing on two semigroups as one of the interesting
means of constructing more minimal reduced type examples.

In Section 4, we study the finiteness of the categories of maximal Cohen-
Macaulay modules, $\operatorname{CM}(R)$, and the category of reflexive
modules, $\operatorname{Ref}(R)$, primarily in the context of numerical
semigroup rings of maximal/minimal reduced type (refer to Section 4 for the
relevant definitions). A category is said to be of finite type if there are
only finitely many indecomposable objects upto isomorphism. Notably, such
finiteness studies have been of interest in both algebra and representation
theory; the central idea of the latter is to analyze an algebraic object based
the way it acts on sets. The representation theory of maximal Cohen-Macaulay
modules has garnered high interest in the last 50 years, for instance these
modules play an important role in Hochster’s study of homological conjectures.
We refer the reader to various sources such as [Her78], [Knö86],[GK85],
[BGS87], [Sol89], [Yos90], [Wie98], [LW12], etc. which discuss such
classifications in great details. In particular, it is well-known that for
finiteness of $\operatorname{CM}(R)$, one only needs to focus on multiplicity
at most $3$. In Theorem 4.6, we first establish the explicit forms of
numerical semigroup rings of minimal multiplicity $3$ such that
$\operatorname{CM}(R)$ is of finite type and we use this as our tool to study
$\operatorname{Ref}(R)$ subsequently. The study of the category
$\operatorname{Ref}(R)$ has attracted quite a bit of attention recently, for
instance see [KK22], [Kob22], [DMS23], etc. The categories
$\operatorname{Ref}(R)$ and $\operatorname{CM}(R)$ coincide when $R$ is
Gorenstein due to the work of Vasconcelos [Vas68]. Outside the Gorenstein
situation, however, the finiteness problem still is open. We provide explicit
descriptions of the almost Gorenstein scenario in the context of
maximal/minimal reduced type.

######  Theorem E (Theorems 4.9, 4.12 and 4.13, Proposition 4.11).

Suppose $R$ is a complete numerical semigroup ring such that $R$ is almost
Gorenstein but not Gorenstein. Then the following statements are true.

  1. (1)

Suppose $R$ is of minimal reduced type. Then $\operatorname{Ref}(R)$ is of
finite type if and only if $R$ is either
$\mathbb{k}\llbracket{t^{3},t^{7},t^{11}}\rrbracket$ or
$\mathbb{k}\llbracket{t^{3},t^{8},t^{13}}\rrbracket$.

  2. (2)

Suppose $R$ is of maximal reduced type 2. If $R$ is of minimal multiplicity,
then $\operatorname{CM}(R)$ (and hence $\operatorname{Ref}(R)$) is of finite
type).

  3. (3)

Suppose $R$ is of maximal reduced type and
$e(\mathfrak{m}:\mathfrak{m})\leqslant 2$. Then $\operatorname{Ref}(R)$ is of
finite type if and only if $R$ is either
$\mathbb{k}\llbracket{t^{a_{1}},t^{a_{1}+1},\dots,t^{2a_{1}-1}}\rrbracket$ or
$\mathbb{k}\llbracket{t^{a_{1}},t^{a_{1}+2},t^{a_{1}+3},\dots,t^{2a_{1}+1}}\rrbracket$.

  4. (4)

Suppose $R$ is of maximal reduced type and $e(\mathfrak{m}:\mathfrak{m})=3$,
then $\operatorname{Ref}(R)$ is of finite type if and only if
$R=\mathbb{k}\llbracket{t^{a_{1}},t^{a_{1}+1},t^{a_{1}+3},\dots,t^{2a_{1}-1}}\rrbracket,a_{1}\geqslant
4$.

### Acknowledgements

Much of this article is dedicated to answering Craig Huneke’s questions about
reduced type. We owe him a great deal for the understanding of this invariant.
Additionally, we would like to thank Anurag Singh, Kei-ichi Watanabe, and
Hailong Dao for various discussions and for providing us with valuable
references.

##  2\. Maximal and Minimal Reduced type for one dimensional complete local
domains

Let $(R,\mathfrak{m},\mathbb{k})$ be a non-regular one dimensional complete
local domain which is a $k$-algebra. We assume that $\mathbb{k}$ is
algebraically closed of characteristic $0$. The integral closure of $R$ in its
field of fractions $K$ if denoted by $\overline{R}$. Notice that
$\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$. Using this and the fact that
$R=\frac{\mathbb{k}\llbracket{X_{1},\dots,X_{n}}\rrbracket}{I}$,
$I\subseteq(X_{1},\dots,X_{n})^{2}$, we can write each of the images of
$X_{i}$ modulo $I$, denoted $x_{i}$, as $x_{i}=\alpha_{i}t^{a_{i}}$. Here
$\alpha_{i}$ are units of $\overline{R}$. We also arrange the $x_{i}$ such
that $a_{1}<a_{2}<\dots<a_{n}$. Here $n$ is the embedding dimension of $R$,
which is defined as the minimal number of generators of the maximal ideal
$\mathfrak{m}$, denoted $\operatorname{edim}(R)$. By our assumption,
$n\geqslant 2$.

There is a valuation on $\overline{R}$ (a DVR) which in turn induces a
valuation on $R$ given by

$\displaystyle v\left(\sum_{i}c_{i}t^{i}\right)=\min\\{i~{}|~{}c_{i}\neq
0\\}\text{ where }c_{i}\in\mathbb{k}.$

Also, we set $v(\alpha)=0$ if $\alpha$ is a unit in $\overline{R}$. This
valuation extends to the valuation on $\operatorname{Frac}R$, the fraction
field of $R$, given by $v(\frac{\alpha}{\beta})=v(\alpha)-v(\beta)$ for
$\alpha,\beta\in R$. For any $R$ submodule $M$ of $\operatorname{Frac}R$, we
let $v(M)$ to be the collection $\\{v(\alpha)~{}|~{}\alpha\in M\\}$. The set
$v(R)$ forms a semigroup called the valuation semigroup of $R$.

Let $\mathfrak{C}=R:_{K}\overline{R}$ denote the conductor ideal of $R$ in
$\overline{R}$. Observe that $\mathfrak{C}=(t^{c+i})_{i\geqslant 0}$, where
$c$ is the positive integer such that $t^{c-1}\not\in R$ but $t^{c+i}\in R$
for all $i\geqslant 0$.

######  Remark 2.1.

Notice that $\mathfrak{m}\overline{R}=x_{1}\overline{R}$ and thus

$\mathfrak{m}=\mathfrak{m}\overline{R}\cap R=x_{1}\overline{R}\cap
R=\overline{x_{1}},$

i.e., $x_{1}$ is a minimal reduction of $\mathfrak{m}$. Since
$\mathfrak{m}\overline{R}=x_{1}\overline{R}$, we get that
$\mathfrak{m}\mathfrak{C}=x_{1}\mathfrak{C}$. Hence,
$\mu(\mathfrak{C})=\ell(\mathfrak{C}/x_{1}\mathfrak{C})=e(x_{1};\mathfrak{C})=e(x_{1};R)\operatorname{rank}_{R}(\mathfrak{C})=\ell(R/x_{1}R)\operatorname{rank}_{R}(\mathfrak{C})=e(R)$.
The last few equalities follow from well-known properties of the Hilbert-
Samuel multiplicity, and also the fact that a non-zero ideal is a maximal
Cohen-Macaulay module of rank one.

The following result which we will keep using in the sequel is due to Herzog
and Kunz. Many of the length computations reduce to counting arguments in the
valuation semigroup.

######  Proposition 2.2.

[HK71, Proposition 2.9] Suppose $M_{1}\subseteq M_{2}$ are two $R$ submodules
of $\operatorname{Frac}R$ with $M_{1}\neq 0$. Then $M_{2}/M_{1}$ has finite
length and its length $\ell(M_{2}/M_{1})$ is the number of elements in
$v(M_{2})$ but not in $v(M_{1})$.

######  Definition 2.3.

[HMM21] We define the reduced type of $R$, denoted $s(R)$ as follows:

$s(R):=\dim_{\mathbb{k}}\frac{\mathfrak{C}+x_{1}R}{x_{1}R}$

Since $\mathfrak{m}\mathfrak{C}=x_{1}\mathfrak{C}\subseteq x_{1}R$, the above
definition makes sense. Also notice by the same inclusions, we get that
$s(R)\leqslant\dim_{\mathbb{k}}\frac{x_{1}R:_{R}\mathfrak{m}}{x_{1}R}$. The
latter quantity is precisely the type of $R$ and this justifies the
nomenclature. Further, it is well-known that $\mathfrak{C}$ is not contained
in any proper principal ideal. Thus, we get that

$1\leqslant s(R)\leqslant\operatorname{type}(R).$

######  Definition 2.4.

We say $R$ has maximal reduced type if $s(R)=\operatorname{type}(R)$.
Analogously, we say that $R$ has minimal reduced type if $s(R)=1$.

######  Question 2.5.

Can we classify when $R$ has maximal or minimal reduced type?

Notice that if $R$ is Gorenstein, then $R$ has maximal (and minimal) reduced
type since $\operatorname{type}(R)=1$. However, $s(R)=1$, does not force $R$
to have type one.

######  Example 2.6.

Let $R=\mathbb{k}\llbracket{t^{5},t^{8},t^{9},t^{11}}\rrbracket$. The
conductor $\mathfrak{C}=(t^{13})\overline{R}$ and the valuation semigroup is
$\\{5,8,9,10,11,13\rightarrow\\}$. Thus the reduced type $s(R)=1$, but the
$\operatorname{type}(R)=2$ and hence not a Gorenstein ring.

######  Definition 2.7.

We say that $R$ is of minimal multiplicity if $\operatorname{edim}R=e(R)$.

In the rest of the article $\ell(M)$ denotes the length of a module $M$.

######  Proposition 2.8.

Let $(R,\mathfrak{m},\mathbb{k})$ be as above. Let $\mathfrak{C}$ denote the
conductor and $x_{1}$ denote the minimal reduction of $\mathfrak{m}$ as in the
definition of $s(R)$. Then $R$ has maximal reduced type if any of the
following holds.

  1. (1)

$\ell(R/\mathfrak{C})\leqslant 2$.

  2. (2)

$\ell(R/\mathfrak{C})=3$ and $R$ does not have minimal multiplicity.

###### Proof.

Notice that maximal reduced type implies that we have the equality

$\mathfrak{C}+x_{1}R=x_{1}R:_{R}\mathfrak{m}.$

Observe that we always have the following inclusions:

$\mathfrak{C}\subseteq\mathfrak{C}+x_{1}R\subseteq
x_{1}R:_{R}\mathfrak{m}\subseteq\mathfrak{m}.$

If $\mathfrak{C}+x_{1}R=\mathfrak{C}$, then $x_{1}\in\mathfrak{C}$.

If $\ell(R/\mathfrak{C})=1$, then $\mathfrak{C}=\mathfrak{m}$ and hence all
the inclusions are equalities and we are done. So assume
$\ell(R/\mathfrak{C})=2$. If $\mathfrak{C}+x_{1}R=\mathfrak{C}$, then
$x_{1}\in\mathfrak{C}$. Thus,
$\mathfrak{m}=\overline{x_{1}}\subseteq\overline{\mathfrak{C}}=\mathfrak{C}\subseteq\mathfrak{m}$.
This shows that $\mathfrak{m}=\mathfrak{C}$ but this contradicts the fact,
that $\ell(R/\mathfrak{C})=2$. Hence,
$\mathfrak{C}\subsetneq\mathfrak{C}+x_{1}R$. Since
$\ell(\mathfrak{m}/\mathfrak{C})=1$, this shows that
$\mathfrak{C}+x_{1}R=x_{1}:_{R}\mathfrak{m}=\mathfrak{m}$ and we are done with
the proof of $(a)$.

For $(b)$, first we recall the well-known fact that $R$ is of minimal
multiplicity if and only if $\mathfrak{m}^{2}=x_{1}\mathfrak{m}$. We include a
quick proof here. Since $\operatorname{edim}(R)=e(R)$, we have that
$\ell(\mathfrak{m}/\mathfrak{m}^{2})=\ell(R/x_{1}R)$. The LHS is
$\ell(R/\mathfrak{m}^{2})-1$ whereas the RHS is $\ell(R/x_{1}\mathfrak{m})-1$
(using the inclusions $\mathfrak{m}^{2}\subseteq\mathfrak{m}\subseteq R$ and
$x_{1}\mathfrak{m}\subseteq x_{1}R\subseteq R$ respectively). Thus,
$\ell(R/x_{1}\mathfrak{m})=\ell(R/\mathfrak{m}^{2})$ which shows that
$\mathfrak{m}^{2}=x_{1}\mathfrak{m}$. The converse is clear as well as the
arguments are reversible.

Since $x_{1}R$ is a minimal reduction of $\mathfrak{m}$, we get that
$x_{1}R\cap\mathfrak{m}^{2}=x_{1}\mathfrak{m}$. Hence,
$\mathfrak{m}^{2}=x_{1}\mathfrak{m}$ if and only if
$x_{1}R:_{R}\mathfrak{m}=\mathfrak{m}$. Since by assumption, $R$ is not of
minimal multiplicity, we get the following chain of ideals:

$\mathfrak{C}\subsetneq\mathfrak{C}+x_{1}R\subseteq
x_{1}R:_{R}\mathfrak{m}\subsetneq\mathfrak{m}.$

The conclusion now follows because $\ell(\mathfrak{m}/\mathfrak{C})=2$. ∎

######  Remark 2.9.

The arguments in the first part show that $\mathfrak{m}=\mathfrak{C}$ if and
only if $\mathfrak{C}=\mathfrak{C}+x_{1}R$. In fact, one can precisely
describe $R$ in the case $\ell(R/\mathfrak{C})=2$ (see [DMS23, Proposition 6.3
(2)]).

The following example shows that the minimal multiplicity condition cannot be
ignored. The computation of $\operatorname{type}(R)$ in the following examples
can be obtained by computing the cardinality of the pseudo-Frobenius set
$\operatorname{PF}(v(R))$ (we refer to the next section for a definition of
pseudo-Frobenius set).

######  Example 2.10.

Let $R=\mathbb{k}\llbracket{t^{4},t^{6},t^{9},t^{11}}\rrbracket$. Here
$\mathfrak{C}=(t^{8},t^{9},t^{10},t^{11})$. Thus, $\ell(R/\mathfrak{C})=3$
(using Proposition 2.2, it is enough to notice that the valuations missing
from the conductor are $0,4,6$). Also, $R$ has minimal multiplicity. Finally
note that $s(R)=2$ whereas $\operatorname{type}(R)=3$.

For higher colength, the equality can fail as the following examples show.

######  Example 2.11.

Let $R=\mathbb{k}\llbracket{t^{4},t^{6},t^{11},t^{13}}\rrbracket$. Here,
$\mathfrak{C}=(t^{10},t^{11},t^{12},t^{13})$ and thus
$\ell(R/\mathfrak{C})=4$. Also, $R$ has minimal multiplicity. But $s(R)=2$
whereas $\operatorname{type}(R)=3$.

######  Example 2.12.

Let $R=\mathbb{k}\llbracket{t^{5},t^{7},t^{8},t^{11}}\rrbracket$. The
conductor is $\mathfrak{C}=(t^{10})\overline{R}$. We get that
$\ell(R/\mathfrak{C})=4$, $s(R)=2,\operatorname{type}(R)=3$. Here $R$ is not
of minimal multiplicity.

######  Theorem 2.13.

The quantity $s(R)$ equals the number of integers in the interval
$[c-a_{1},c-1]$ that are missing from the value group of $R$.

###### Proof.

Notice that $s(R)=\ell(\mathfrak{C}/(\mathfrak{C}\cap x_{1}R))$. Hence, we
need to count the valuations that are in $\mathfrak{C}$ but missing from the
valuation set coming from $\mathfrak{C}\cap x_{1}R$. Since
$\mathfrak{m}\mathfrak{C}=x_{1}\mathfrak{C}\subseteq\mathfrak{C}\cap x_{1}R$,
the only relevant valuations must come from
$\mathfrak{C}\setminus\mathfrak{m}\mathfrak{C}$. Hence, we can focus on the
following generating set of $\mathfrak{C}$: $t^{c},\dots,t^{c+a_{1}-1}$.

Suppose for some $1\leqslant i\leqslant a_{1}$, $t^{c+a_{1}-i}=x_{1}\alpha$
for some $\alpha\in R$. Then we get that $c+a_{1}-i=a_{1}+k$ for some $k$
lying in the value group of $R$. Hence, $k=c-i\in[c-a_{1},c]$. Thus the
missing valuations from $[c-a_{1},c-1]$ precisely correspond to the elements
in the generating set which cannot be written as a multiple of $x_{1}$. ∎

######  Corollary 2.14.

We have $s(R)=s(\mathbb{k}\llbracket{v(R)}\rrbracket)$.

###### Proof.

The proof is clear from Theorem 2.13. ∎

##  3\. Numerical Semigroup Rings

For a semigroup $H$ minimally generated by $b_{1}<\cdots<b_{m}$, we define a
complete numerical semigroup
$\mathbb{k}\llbracket{H}\rrbracket=\mathbb{k}\llbracket{t^{b_{1}},\dots,t^{b_{m}}}\rrbracket$.
So for the numerical semigroup $v(R)$ appearing in the previous section, we
can generate a complete numerical semigroup ring
$\mathbb{k}\llbracket{v(R)}\rrbracket$.

It is a natural question whether our study can be transferred to the study of
this numerical semigroup ring. For that we need the interpretations of type
and reduced type in terms of valuations. One of the simplest relationships
between $R$ and $\mathbb{k}\llbracket{v(R)}\rrbracket$ is that their conductor
valuations remain the same.

It is well-known that the type of a complete numerical semigroup ring
$\mathbb{k}\llbracket{H}\rrbracket$ is given by the cardinality of the pseudo-
Frobenius set, denoted $\operatorname{PF}(H)$. It is defined as follows:

$\operatorname{PF}(H)=\\{x\in\mathbb{Z}\setminus H\mid x+h\in H\text{~{}for
all~{}}0\neq h\in H\\}.$

The largest element in this set is called the Frobenius number of $H$, denoted
$F(H)$. In our notations, $F(H)=c-1$.

In the previous section we saw that the reduced type is invariant under the
study of $R$ versus $\mathbb{k}\llbracket{v(R)}\rrbracket$. But the type of
$R$ is more sensitive through this change of rings. The next result Theorem
3.2 which appears in [BDF97, Proposition II.1.16], captures this behaviour. We
provide a proof for the convenience of the reader. Also, in the following
$|S|$ refers to the cardinality of a set $S$.

######  Remark 3.1.

We recall that for any two ideals of $R$, we can identify
$\operatorname{Hom}_{R}(I,J)$ with
$J:I=\\{\alpha\in\operatorname{Frac}(R)\mid\alpha I\subseteq J\\}$ via the
following: for any non-zero $x,y\in I$ and $f:I\to J$, we have
$\frac{f(x)}{x}=\frac{f(y)}{y}$ using $R$-linearity of $f$. Hence, for $0\neq
r\in I$, we get $f(r)=r\frac{f(rx)}{rx}=r\frac{f(x)}{x}$, a multiplication by
$\frac{f(x)}{x}\in\operatorname{Frac}(R)$.

######  Theorem 3.2.

We have the inequality

$s(R)\leqslant\operatorname{type}(R)\leqslant|\operatorname{PF}(v(R))|=\operatorname{type}(\mathbb{k}\llbracket{v(R)}\rrbracket).$

###### Proof.

The first inequality follows from our discussion of reduced type above. For
the second one, we make the following observations. Let $\omega_{R}$ be a
canonical ideal of $R$. Then
$\operatorname{type}(R)=\ell(\omega_{R}/\mathfrak{m}\omega_{R})=\ell(\operatorname{Hom}(\omega_{R}/\mathfrak{m}\omega_{R},E(\mathbb{k})))$,
where $E(\mathbb{k})$ is the injective hull of $\mathbb{k}$. The latter
equality is due to [BH98, Proposition 3.2.12(b)] while the former is due to
[BH98, Theorem 3.3.11(c)]. Applying $\operatorname{Hom}_{R}(\cdot,\omega_{R})$
to the exact sequence

$0\to\mathfrak{m}\omega_{R}\to\omega_{R}\to\omega_{R}/\mathfrak{m}\omega_{R}\to
0,$

we obtain

$0\to\operatorname{Hom}_{R}(\omega_{R},\omega_{R})\to\operatorname{Hom}_{R}(\mathfrak{m}\omega_{R},\omega_{R})\to\operatorname{Ext}^{1}_{R}(\omega_{R}/\mathfrak{m}\omega_{R},\omega_{R})\to
0$

where we used the facts that $\operatorname{Hom}_{R}(M,N)=0$ if $M$ is torsion
and $N$ is torsion-free, and also that
$\operatorname{Ext}^{1}_{R}(\omega_{R},\omega_{R})=0$ ([BH98, Theorem
3.3.10(c)(iii)]) Now using local duality [BH98, Theorem 3.5.8], we get that
$\operatorname{Ext}^{1}_{R}(\omega_{R}/\mathfrak{m}\omega_{R},\omega_{R})\cong\operatorname{Hom}_{R}(H^{0}_{\mathfrak{m}}(\omega_{R}/\mathfrak{m}\omega_{R}),E(\mathbb{k}))=\operatorname{Hom}(\omega_{R}/\mathfrak{m}\omega_{R},E(\mathbb{k}))$.
Thus

$\operatorname{type}(R)=\ell\Big{(}\frac{\operatorname{Hom}_{R}(\mathfrak{m}\omega_{R},\omega_{R})}{\operatorname{Hom}_{R}(\omega_{R},\omega_{R})}\Big{)}.$

Using the identification with colons as discussed before the theorem, we get
that

$\displaystyle\operatorname{type}(R)$
$\displaystyle=\ell\Big{(}\frac{\omega_{R}:\mathfrak{m}\omega_{R}}{\omega_{R}:\omega_{R}}\Big{)}$
$\displaystyle=\ell\Big{(}\frac{(\omega_{R}:\omega_{R}):\mathfrak{m}}{\omega_{R}:\omega_{R}}\Big{)}=\ell\Big{(}\frac{R:\mathfrak{m}}{R}\Big{)}.$

The first equality follows from Remark 3.1 while the last equality follows
from [BH98, Proposition 3.3.11(c)(ii)]. Then
$\ell(R:\mathfrak{m}/R)=|\left\\{v(R:\mathfrak{m})\setminus v(R)\right\\}|$.
Since $v(R:\mathfrak{m})\subseteq\\{x\in\mathbb{Z}\mid x+h\in v(R)\text{~{}for
all~{}}h\in v(R)\\}$, we have finished the proof. ∎

Thus the proof above shows that the type of $R$ and the type of
$\mathbb{k}\llbracket{v(R)}\rrbracket$ need not be the same. In fact, here is
a concrete example which reflects this.

######  Example 3.3.

[BDF97, Example II.1.19] Let
$R=\mathbb{k}\llbracket{t^{4},t^{6}+t^{7},t^{11}}\rrbracket$. Then
$v(R)=\\{0,4,6,8,10,11,12,13,14,....\\}$ (we used that
$t^{10}=t^{4}(t^{6}+t^{7})-t^{11}\in R$ and
$t^{13}=(t^{6}+t^{7})^{2}-(t^{4})^{3}-t^{10}t^{4}\in R$). Also,
$v(R:\mathfrak{m})=\\{0,4,6,7,8,9,...\\}$. Thus,
$\operatorname{type}(R)=|v(R:\mathfrak{m})\setminus v(R)|=2$. However,
$\operatorname{PF}(v(R))=\\{2,7,9\\}$.

######  Proposition 3.4.

The following statements hold.

  1. (1)

The ring $R$ is of minimal reduced type if and only if
$\mathbb{k}\llbracket{v(R)}\rrbracket$ is of minimal reduced type.

  2. (2)

The ring $R$ is of maximal reduced type if
$\mathbb{k}\llbracket{v(R)}\rrbracket$ is of maximal reduced type. The
converse holds if additionally
$\operatorname{type}(R)=\operatorname{type}(\mathbb{k}\llbracket{v(R)}\rrbracket)$.

###### Proof.

The case of minimal reduced type (i.e., when $s(R)=1$) follows from Corollary
2.14. For (2), we use Theorem 3.2 and Corollary 2.14. ∎

The converse of Statement (2) in Proposition 3.4 need not hold without the
added assumption. The following example illustrates this.

######  Example 3.5.

Let $R$ be as in Example 3.3. Then from the description of $v(R)$, we see that
$c=10$. Thus from Theorem 2.13, we get that $s(R)=2$ since
$\\{7,9\\}=[10-4,10-1]\setminus v(R)$. Thus $R$ is of maximal reduced type but
$\mathbb{k}\llbracket{v(R)}\rrbracket$ is not since its type is $3$ whereas
reduced type is $2$.

######  Corollary 3.6.

Assume $R$ is of minimal multiplicity. Then $R$ is of maximal (resp. minimal)
reduced type if and only if $\mathbb{k}\llbracket{v(R)}\rrbracket$ is of
maximal (resp. minimal) reduced type.

###### Proof.

It is well-known that for a Cohen-Macaulay local ring with $e(R)>1$,
$e(R)\geqslant\operatorname{type}(R)+1$ and $R$ is of minimal multiplicity if
and only if $e(R)=\operatorname{type}(R)+1$ [HKS21, Fact 2.6].

Since $e(R)=e(\mathbb{k}\llbracket{v(R)}\rrbracket)$, we get that
$e(\mathbb{k}\llbracket{v(R)}\rrbracket)=\operatorname{type}(R)+1\leqslant\operatorname{type}(\mathbb{k}\llbracket{v(R)}\rrbracket)+1$
using Theorem 3.2. But
$\operatorname{type}(\mathbb{k}\llbracket{v(R)}\rrbracket)+1\leqslant
e(\mathbb{k}\llbracket{v(R)}\rrbracket)$ as discussed before.

Thus,
$\operatorname{type}(v(R))=\operatorname{type}(\mathbb{k}\llbracket{v(R)}\rrbracket)$
and now Proposition 3.4 finishes the proof. ∎

The above discussions show that the study of maximal or minimal reduced type
to complete numerical semigroup rings gives us information on the maximal or
minimal reduced type for rings $R$.

About the notation for $R$: In the sequel, we will be studying the maximal and
minimal reduced type for complete numerical semigroup rings. Essentially, as
the above results suggest, studying the properties for
$\mathbb{k}\llbracket{v(R)}\rrbracket$ gives us information on the properties
for $R$. Now notice that $a_{1},\dots,a_{n}$ may not be a generating set for
$v(R)$ even though
$R=\mathbb{k}\llbracket{\alpha_{1}t^{a_{1}},\dots,\alpha_{n}t^{a_{n}}}\rrbracket$.
For example, for the ring
$R=\mathbb{k}\llbracket{t^{4},t^{6}+t^{7},t^{11}}\rrbracket$, the complete
numerical semigroup ring is
$\mathbb{k}\llbracket{t^{4},t^{6},t^{11},t^{13}}\rrbracket$ as opposed to
$\mathbb{k}\llbracket{t^{4},t^{6},t^{11}}\rrbracket$. Thus to avoid any
confusion, we will be using a more general notation
$\mathbb{k}\llbracket{H}\rrbracket$ for a complete numerical semigroup as
opposed to $\mathbb{k}\llbracket{v(R)}\rrbracket$.

######  Proposition 3.7.

For a numerical semigroup $H$, the ring $S=\mathbb{k}\llbracket{H}\rrbracket$
has maximal reduced type if and only if
$\min_{x\in\operatorname{PF}(H)}x\geqslant c-b_{1}$.

###### Proof.

It is clear that $[c-b_{1},c-1]\setminus v(S)\subseteq\operatorname{PF(H)}$ by
the definition of the latter. (This is another way of seeing in this case that
$s(S)\leqslant\operatorname{type}(S)$). Thus, $S$ is of maximal reduced type
if and only if $\operatorname{PF}(H)=[c-b_{1},c-1]\setminus v(S)$ if and only
if $\min_{x\in\operatorname{PF}(H)}x\geqslant c-b_{1}$. ∎

######  Proposition 3.8.

For a numerical semigroup $H$, the ring $S=k\llbracket{H}\rrbracket$ is of
minimal reduced type if and only if
$\displaystyle\max_{x\in\\{\operatorname{PF}(H)\setminus\\{c-1\\}}x<c-b_{1}$.

###### Proof.

It is clear that $s(S)=1$ if and only if $c-1$ is the only integer in the
interval $[c-b_{1},c-1]$ that should be missing from $v(S)$. Hence, all the
other elements in $\operatorname{PF}(H)$ must be strictly less than $c-b_{1}$.
∎

######  Remark 3.9.

In fact, it is clear that if $h\in H$, then
$c-1-h\not\in\operatorname{PF}(H)$. Hence, the above statement can also be
changed to minimal reduced type if and only if
$\displaystyle\max_{x\in\\{\operatorname{PF}(H)\setminus\\{c-1\\}}x\leqslant
c-b_{1}-2$.

######  Corollary 3.10.

For a numerical semigroup $H$, the ring, suppose
$S=\mathbb{k}\llbracket{H}\rrbracket$ has minimal multiplicity. Then $S$ is of
maximal reduced type if and only if $b_{2}\geqslant c$.

###### Proof.

Since $S$ has minimal multiplicity,

$\operatorname{PF}(H)=\\{b_{2}-b_{1},b_{3}-b_{1},\dots,b_{n}-b_{1}\\};$

this follows from [RGS09, Proposition 2.20, Proposition 3.31] Thus, $S$ has
maximal reduced type if and only if $b_{2}-b_{1}\geqslant c-b_{1}$ which
completes the proof. ∎

######  Corollary 3.11.

Suppose $S=\mathbb{k}\llbracket{H}\rrbracket$ has minimal multiplicity. Then
$S$ is of minimal reduced type if and only if $b_{n-1}+b_{1}-1<b_{n}$.

###### Proof.

Since $S$ has minimal multiplicity,

$\operatorname{PF}(H)=\\{b_{2}-b_{1},b_{3}-b_{1},\dots,b_{n}-b_{1}\\};$

this follows from [RGS09, Proposition 2.20, Proposition 3.31]. Thus, $S$ has
minimal reduced type if and only if $b_{n-1}-b_{1}<c-b_{1}$. But notice that
$c-1=b_{n}-b_{1}$ or $c=b_{n}-b_{1}+1$. Thus $S$ has minimal reduced type if
and only if

$\displaystyle b_{n-1}-b_{1}<c-b_{1}=b_{n}-b_{1}+1-b_{1}=b_{n}-2b_{1}+1.$

∎

In general, getting a complete classification of numerical semi-group rings of
maximal or minimal reduced type seems tricky. We however can describe the
condition abstractly using the notion of Apéry sets as well.

Recall that given $h\in H$, the Apéry set of $h$ in $H$ is defined to be

$\operatorname{Ap}(H,h):=\\{s\in H\mid s-h\not\in H\\}.$

One defines an order relation over the set of integers as follows:
$a\leqslant_{H}b$ if $b-a\in H$. Let
$M(h)=\operatorname{Maximal}_{\leqslant_{H}}\operatorname{Ap}(H,h)$.

######  Proposition 3.12.

Suppose $S=\mathbb{k}\llbracket{H}\rrbracket$ is a complete numerical semi-
group ring as above. Then $S$ is of maximal reduced type if and only if

$\min_{w\in M(e(R))}w\geqslant(\max\operatorname{Ap}(H,e(R))-e(R)+1.$

###### Proof.

By [RGS09, Proposition 2.20], we get that $\operatorname{PF}(H)=\\{w-e(R)\mid
w\in M(e(R))\\}$. Furthermore, by [RGS09, Proposition 2.12], we have that
$c=(\max\operatorname{Ap}(H,e(R))-e(R)+1$. The proof is now complete by
Proposition 3.7. ∎

###  3.1. Almost Gorenstein Rings

######  Definition 3.13.

[BF97, Definition-Proposition 20] The ring $R$ is called almost Gorenstein if

$\ell(\overline{R}/R)=\ell(R/\mathfrak{C})+\operatorname{type}(R)-1.$

This class of rings has recently attracted a lot of attention. See for
instance [GMP13], [GTT15], [HHS19], [DKT21], etc. among other sources.

The characterization of almost Gorenstein property for numerical semigroup
rings due to Nari is given below.

######  Theorem 3.14.

[Nar13, Theorem 2.4] Suppose $H$ is a numerical semigroup with pseudo-
Frobenius set $\operatorname{PF}(H)=\\{f_{1},\dots,f_{t}\\}$ and
$\mathbb{k}\llbracket{H}\rrbracket$ is a numerical semigroup ring. Then $H$ is
almost Gorenstein if and only if $f_{i}+f_{t-i}=c-1$ for $1\leqslant
i\leqslant c-1$.

######  Proposition 3.15.

Assume $R$ is almost Gorenstein. Then $R$ is of maximal (resp. minimal)
reduced type if and only if $k\llbracket{v(R)}\rrbracket$ is of maximal (resp.
minimal) reduced type.

###### Proof.

Since $R$ is almost Gorenstein, we have
$\operatorname{type}(R)=\operatorname{type}(\mathbb{k}\llbracket{v(R)}\rrbracket)$
by [BF97, Proposition 29]. The proof is now complete by Proposition 3.4. ∎

######  Proposition 3.16.

Suppose the numerical semi-group ring $S=k\llbracket{H}\rrbracket$ is almost
Gorenstein. Then the following statements hold.

  1. (1)

If $S$ is maximal reduced type then $c\leqslant 2b_{1}-1$,

  2. (2)

If $S$ is of minimal reduced type but not Gorenstein then $c\geqslant
2b_{1}+3$.

###### Proof.

Let $r=\operatorname{type}(R)$ and
$\operatorname{PF}(v(R))=\\{f_{1},\ldots,f_{r-1},c-1\\}$ with $f_{i}<f_{i+1}$.
From Theorem 3.14, we see that $R$ is almost Gorenstein if and only if
$f_{i}+f_{r-i}=c-1$ for $1\leqslant i\leqslant r-1$.

For $(a)$, assume that $R$ has maximal reduced type. Hence, by Proposition
3.7, $f_{i}\geqslant c-b_{1}$ for all $i$. Thus, $c-1=f_{i}+f_{r-i}\geqslant
2c-2b_{1}$ which upon simplification yields our desired conclusion.

For $(b)$, assume $R$ is of minimal reduced type. This implies that
$f_{r-1}\leqslant c-b_{1}-2$. Thus, $c-1=f_{1}+f_{r-1}\leqslant 2c-2b_{1}-4$.
Simplifying this, we get $c\geqslant 2b_{1}+3$. ∎

The following example shows that the converse of the above statements do not
hold in general.

######  Example 3.17.

Let
$R=\mathbb{k}\llbracket{t^{8},t^{11},t^{12},t^{13},t^{15},t^{17},t^{18}}\rrbracket$.
Here $\operatorname{PF}(H)=\\{4,5,7,9,10,14\\}$ and hence $R$ is almost
Gorenstein with $\operatorname{type}(R)=6$. Note however that $s(R)=4$. Thus,
$R$ is not of maximal reduced type even though $c\leqslant 2b_{1}-1$ (here,
$c=15,b_{1}=8$).

######  Example 3.18.

[Eto17, Example 3.9] Let
$R=\mathbb{k}\llbracket{t^{34},t^{51},t^{53},t^{70}}\rrbracket$. This is
almost Gorenstein of type $3$ since $\operatorname{PF}(H)=\\{17,814,831\\}$.
Here $c=832$ and hence satisfies $c\geqslant 2b_{1}+3$. However, notice that
$s(R)=2$.

However, in almost Gorenstein numerical semigroup rings of type 2, the
situation is very clear as the following corollary shows. Such rings are
called pseudo-symmetric numerical semi-group rings.

######  Corollary 3.19.

Suppose $S=\mathbb{k}\llbracket{H}\rrbracket$ is an almost Gorenstein
numerical semi-group ring of type $2$. Then the following statements hold.

  1. (1)

$S$ is maximal reduced type if and only if $c\leqslant 2b_{1}-1$,

  2. (2)

$S$ is of minimal reduced type if and only if $c>2b_{1}-1$.

###### Proof.

One of the directions for $(1)$ follows from Proposition 3.16. For the
converse, we first observe that since $\operatorname{type}(S)=2$, in the
notation of the proof of Proposition 3.16, we get that $2f_{1}=c-1$. Hence,
$\operatorname{PF}(H)=\\{\frac{c-1}{2},c-1\\}$. Since $c\leqslant 2b_{1}-1$,
we get that $\frac{c-1}{2}\geqslant c-b_{1}$ and hence $R$ is of maximal
reduced type by Proposition 3.7. This proves (1).

Since $\operatorname{type}(S)=2$, if the ring is not of maximal reduced type,
then it must be that $s(S)=1$. Thus, (2) follows directly from (1). ∎

The following examples illustrate both the cases.

######  Example 3.20.

Let $R=\mathbb{k}\llbracket{t^{4},t^{9},t^{11}}\rrbracket$. Then
$\operatorname{PF}(H)=\\{7,14\\}$ and hence $R$ is almost Gorenstein of type
2. Moreover, $s(R)=1$ by direct computation as well as by Corollary 3.19.

######  Example 3.21.

Let $R=\mathbb{k}\llbracket{t^{5},t^{6},t^{7},t^{9}}\rrbracket$. Here
$\operatorname{PF}(H)=\\{4,8\\}$ and hence $R$ is almost Gorenstein of type
$2$. Moreover, $s(R)=2$ by direct computation as well as by Corollary 3.19.

###  3.2. Far Flung Gorenstein Rings

Recently the notion of far-flung Gorenstein rings were introduced in [HKS21].
These are rings where the trace ideal of the canonical module equals
$\mathfrak{C}$. We do not go into the details of trace ideals as we will not
be using the notion in this article. We refer the reader to [Lin17], [GIK20]
and [HHS19] for further details. These rings are (evidently) non-Gorenstein,
and hence $\operatorname{type}(R)\geqslant 2$ (thus, $a_{1}=e(R)\geqslant 3$).

######  Proposition 3.22.

Suppose $S=\mathbb{k}\llbracket{H}\rrbracket$ is a far-flung Gorenstein
complete numerical semigroup ring. Then $s(S)\geqslant 2$.

###### Proof.

It is well-known that there is a canonical module
$\displaystyle\omega_{S}=\sum_{\alpha\in\operatorname{PF}(H)}St^{c-1-\alpha}$
where $c$ is the conductor valuation. Notice that $c-1\in\operatorname{PF}(H)$
always. Thus, $t^{0}=1$ is a generator of $\omega_{S}$. Moreover, from [HKS21,
Theorem 5.1], we get that $t^{1}$ is also a generator. Hence,
$\alpha=c-2\in\operatorname{PF}(H)$ as well. But notice that
$c-2\in[c-b_{1},c-1]\setminus v(R)$. Hence, $s(S)\geqslant 2$. ∎

######  Theorem 3.23.

Suppose $S=\mathbb{k}\llbracket{H}\rrbracket$ is a far-flung Gorenstein
complete numerical semi-group ring of $\operatorname{type}(S)\leq 3$, Then $S$
has maximal reduced type.

###### Proof.

From Proposition 3.22, the case when $\operatorname{type}(S)=2$ is clear.

Suppose $\operatorname{type}(S)=3$ and $S$ has minimal multiplicity. Then we
know that $b_{1}=e(S)=4=\operatorname{edim}(S)$ ([Sal80, Proposition 3.1]). As
in the proof of Corollary 3.10, we get that

$\operatorname{PF}(H)=\\{b_{2}-4,b_{3}-4,b_{4}-4\\}$

and hence $c-1=b_{4}-4$ or $b_{4}=c+3$. Using the proof of Proposition 3.22,
we see that $c-2\in\operatorname{PF}(H)$ and hence, $b_{3}=c+2$. By [HKS21,
Corollary 6.2], we know that

$\\{2b_{4}-3,\cdots,2b_{4}\\}\subseteq\\{b_{i}+b_{j},2\leqslant i,j\leqslant
4\\}.$

Notice that $2b_{4}=2c+6,2b_{4}-1=2c+5=b_{3}+b_{4},2b_{4}-2=2c+4=2b_{3}$. If
$b_{2}<c$, $2b_{4}-3$ cannot be generated by $b_{i}+b_{j}$. Hence
$b_{2}\geqslant c$ and so we are done by Corollary 3.10.

Finally, suppose $\operatorname{type}(S)=3$ and $S$ is not of minimal
multiplicity. First, we notice that $b_{1}=e(S)=5$ by [HKS21, Corollary 5.3].
Choosing
$\omega_{S}=\sum\limits_{\alpha\in\operatorname{PF}(H)}St^{c-1-\alpha}$ as in
the proof of Proposition 3.22, we know that $t^{0},t^{1}$ are generators of
$\omega_{S}$, i.e., $c-1,c-2\in\operatorname{PF}(H)$. Since
$\operatorname{type}(S)=3$, [HKS21, Proposition 6.1 (ii)] gives us that
$\omega_{S}=\langle 1,t,t^{2}\rangle$ or $\omega_{S}=\langle
1,t,t^{3}\rangle$.

Assume that $\omega_{S}=\langle 1,t,t^{2}\rangle$. Then we know that

$\operatorname{PF}(H)=\\{c-3,c-2,c-1\\}.$

Since $c-3>c-5$, we are done by Proposition 3.7. Next assume that
$\omega_{R}=\langle 1,t,t^{3}\rangle$. Then

$\operatorname{PF}(H)=\\{c-4,c-2,c-1\\}$

and again Proposition 3.7 finishes the proof. ∎

However, even if $S$ is far-flung Gorenstein numerical semigroup of minimal
multiplicity with $\operatorname{type}(S)\geqslant 4$, it is not guaranteed
that the ring will be of maximal reduced type. The following example
illustrates this.

######  Example 3.24.

Let $R=\mathbb{k}\llbracket{t^{5},t^{11},t^{17},t^{18},t^{19}}\rrbracket$.
This ring is of minimal multiplicity with
$e(R)=5,\operatorname{type}(R)=4,n=5$. Also notice that $R$ is far-flung
Gorenstein by [HKS21, Corollary 6.2]. However, $a_{2}=11<c=15$ and hence $R$
is not of maximal reduced type by Corollary 3.10.

If the ring is not of minimal multiplicity, then far flung Gorenstein rings of
type $4$ indeed have maximal reduced type.

######  Theorem 3.25.

Suppose $S$ is a far flung Gorenstein complete numerical semi-group ring of
$\operatorname{type}(S)=4$ which has non minimal multiplicity. Then $S$ has
maximal reduced type.

###### Proof.

As in the previous proof, choose
$\omega_{S}=\sum\limits_{\alpha\in\operatorname{PF}(H)}t^{c-1-\alpha}$. Since
$\operatorname{type}(S)=4$, we have $\omega_{S}=\langle
1,t,t^{a},t^{b}\rangle$ where $a<b$. Also, it follows that $6\leqslant
e(S)\leqslant 9$ ([HKS21, Corollary 5.3]). Since $S$ is far flung Gorenstein,
by [HKS21, Proposition 6.1],

(3.1)
$\displaystyle\\{0,1,2,3,\cdots,e(R)-1\\}\subseteq\\{(c-1-\alpha)+(c-1-\beta)\mid\alpha,\beta\in\operatorname{PF}(H)\\}.$

Thus we have two cases for the description of $\omega_{R}$:

  1. (1)

$\omega_{S}=\langle 1,t,t^{2},t^{b}\rangle$

  2. (2)

$\omega_{S}=\langle 1,t,t^{3},t^{b}\rangle$

Notice that $c-1-b$ is a minimal element in $\operatorname{PF}(H)$, the pseudo
Frobenius set. Thus $R$ is of maximal reduced type, if and only if
$c-1-b\geqslant c-b_{1}$ or $e(S)=b_{1}\geqslant b+1$.

Suppose $e(S)\leqslant b$. Now if case (1) holds, then
$B\cap[5,e(S)-1]=\emptyset$ which contradicts (3.2). On the other hand, if
case(2) holds, then $B\cap\\{5\\}=\emptyset$, again contradicsts (3.2). Thus
we have $e(S)\geqslant b+1$ and hence $S$ is of maximal reduced type. ∎

When the type is more than $4$, we need more tools to show that it is of
maximum reduced type. In [HKS21, Corollary 5.3], the authors connected the the
maximum bound for the multiplicity with the Rohrbach number $\overline{n}(r)$
where $r=\operatorname{type}(R)$. The Rohrbach number is defined as follows:
Suppose $A$ is a set of non negative integers with cardinality $r$. Let $n(A)$
is the integer such that $A+A=\\{a+b~{}|~{}a,b\in A\\}$ contains consecutive
integers $0,1,\dots n(A)-1$ but not $n(A)$. The Rohrbach number is the integer

$\displaystyle\overline{n}(r)=\max\\{n(A)~{}|~{}|A|=r\\}.$

######  Theorem 3.26.

Suppose $S=\mathbb{k}\llbracket{H}\rrbracket$ is a far-flung Gorenstein
numerical semigroup ring of type $r$. If $e(S)>\overline{n}(r-1)+1$, then $S$
has maximal reduced type.

###### Proof.

Since $S$ is a far-flung Gorenstein numerical semigroup ring, by [HKS21,
Theorem 5.1], there exists a canonical module $C=\langle
f_{1},\dots,f_{r}\rangle$ where $n_{i}=v(f_{i}),n_{1}<n_{2}<\cdots<n_{r}$ such
that $\\{0,1,\dots,e-1\\}\subseteq A+A$ where $A=\\{n_{1},\dots,n_{r}\\}$.
Here $e=e(S)$.

Suppose $n_{r}>e-1$. Then $\\{0,1,\dots,e-1\\}\subseteq A^{\prime}+A^{\prime}$
where $A^{\prime}=\\{n_{1},\dots,n_{r-1}\\}$. Thus the consecutive set of
numbers generated by $A^{\prime}+A^{\prime}$ is $0,1,\dots,n(A^{\prime})-1$.
But by hypothesis, $\overline{n}(r-1)<e-1$ and thus, by definition we have
$n(A^{\prime})\leqslant\overline{n}(r-1)<e-1$. This contradiction forces
$n_{r}\leqslant e-1$. Thus the pseudo Frobenius set
$PF(H)=\\{c-1-n_{1},\dots,c-1-n_{r}\\}$. Since $n_{r}\leqslant e-1$, we get
$c-1-n_{r}\geqslant c-b_{1}$. Thus by Proposition 3.7, we have $R$ is of
maximum reduced type. ∎

The bound $\overline{n}(r-1)<e-1$ may not be sharp. In the following result,
$r=5,\overline{n}(r)-r+1=9$.

######  Proposition 3.27.

Suppose $S$ is far flung Gorenstein complete numerical semigroup ring of
$\operatorname{type}(S)=5$. If $e(S)\geqslant 9$, then $S$ is of maximal
reduced type.

###### Proof.

As in the previous proof, choose
$\omega_{S}=\sum\limits_{\alpha\in\operatorname{PF}(H)}t^{c-1-\alpha}$. Since
$\operatorname{type}(S)=4$, we have $\omega_{S}=\langle
1,t,t^{a},t^{b},t^{d}\rangle$ where $a<b<d$. Also, it follows that $7\leqslant
e(S)\leqslant 13$ ([HKS21, Corollary 5.3]). Since $S$ is far flung Gorenstein,
by [HKS21, Proposition 6.1],

(3.2)
$\displaystyle\\{0,1,2,3,\cdots,e(S)-1\\}\subseteq\\{(c-1-\alpha)+(c-1-\beta)\mid\alpha,\beta\in\operatorname{PF}(H)\\}.$

Thus we have two cases for the description of $\omega_{S}$:

  1. (1)

$\omega_{S}=\langle 1,t,t^{2},t^{b},t^{d}\rangle$

  2. (2)

$\omega_{S}=\langle 1,t,t^{3},t^{b},t^{d}\rangle$

Notice that $c-1-d$ is a minimal element in $\operatorname{PF}(H)$, the pseudo
Frobenius set. Thus $S$ is of maximal reduced type, if and only if
$c-1-d\geqslant c-b_{1}$ or $e(S)=b_{1}\geqslant d+1$.

Suppose $e=e(S)\leqslant d$. Now suppose case (1) holds. Let
$A=\\{0,1,2,b,d\\}$. Since $A+A$ should generate consecutive integers
$1,\dots,e-1$ and $e\leqslant d$, we see that $A^{\prime}+A^{\prime}$ where
$A^{\prime}=\\{0,1,2,b\\}$ should generate $\\{0,\dots,e-1\\}$. Thus our
choice should satisfy $b\leqslant 5$. Our choices for $b$ are $3,4,5$. But
with these choices, one can easily see that either $7$ or $8$ is missing from
the set $A^{\prime}+A^{\prime}$. This is a contradiction.

Now suppose (2) holds. Let $A=\\{0,1,3,b,d\\}$. Since $A+A$ should generate
consecutive integers $1,\dots,e-1$ and $e\leqslant d$, we see that
$A^{\prime}+A^{\prime}$ where $A^{\prime}=\\{0,1,2,b\\}$ should generate
$\\{0,\dots,e-1\\}$. Thus our choice should satisfy $b\leqslant 5$. Our
choices for $b$ are $2,4,5$. But with these choices, one can easily check that
either $6$ or $7$ is missing from the set $A^{\prime}+A^{\prime}$. This is a
contradiction.

Thus these contradictions force $e=b_{1}\geqslant d+1$. ∎

The hypothesis of Theorem 3.26 is not sharp even when
$\operatorname{type}(S)=6$ as the following example shows:

######  Example 3.28.

Let
$R=\mathbb{k}\llbracket{t^{12},t^{13},t^{14},t^{15},t^{16},t^{19}}\rrbracket$.
Then conductor $\mathfrak{C}_{R}=(t^{24})$. The pseudo Frobenius set can be
computed to be $\operatorname{PF}(H)=\\{17,18,20,21,22,23\\}$. Here the
reduced type $s=6$ and also equals the type $r=6$. Notice that the
multiplicity $e=b_{1}=12<n(5)=13$.

Pushing the proof along the same lines, one can hope to ask if such tight
bounds for the multiplicity exists for the ring $S$ to have maximal reduced
type.

######  Question 3.29.

If $S$ is far flung Gorenstein complete numerical semigroup with
$e(S)\geqslant\overline{n}(r)-r+1$ where $r=\operatorname{type}(S)$, then $S$
is of maximal reduced type.

So far, all the examples of far-flung Gorenstein rings with non-minimal
multiplicity we have tried satisfy $s(R)=\operatorname{type}(R)$. So it is
natural to ask the following question.

######  Question 3.30.

If $R$ is far-flung Gorenstein with $e(R)>\operatorname{edim}(R)$, does $R$
have maximal reduced type?

Beyond $r=\operatorname{type}(R)=5$, one would need a lot more calculations
and also knowledge about the valuation semigroup of the far flung Gorenstein
ring. In fact even though examples are given to show that there exists far
flung Gorenstein rings with multiplicity $e=\overline{n}(r)$, one may need to
answer [HKS21, Question 5.5] in its entirety to improve Theorem 3.26 and
answer the above question.

There are rings which are not far-flung Gorenstein but still of maximal
reduced type.

######  Example 3.31.

Let $R=\mathbb{k}\llbracket{t^{4},t^{5},t^{11}}\rrbracket$. Here
$\mathfrak{C}=(t^{8})\overline{R}$. Also, observe that
$\operatorname{PF}(H)=\\{6,7\\}=[8-4,7]\setminus v(R)$. Thus
$s(R)=2=\operatorname{type}(R)$. However, $R$ is not far-flung Gorenstein. To
see this, observe that $2(c-1)-e(R)+1=11$ cannot be written as a sum of two
(not necessarily distinct) terms from $\operatorname{PF}(H)$. Thus, $R$ is not
far-flung Gorenstein using [HKS21, Theorem 6.1 iii].

###  3.3. Gluing of Numerical Semigroups

######  Definition 3.32.

Let $H_{1}=\langle a_{1},a_{2},\ldots,a_{n}\rangle$ and $H_{2}=\langle
b_{1},b_{2},\ldots,b_{m}\rangle$ be two numerical semigroups. Take $y\in
H_{1}\setminus\\{a_{1},a_{2},\ldots,a_{n}\\}$ and $x\in
H_{2}\setminus\\{b_{1},b_{2},\ldots,b_{m}\\}$ such that $(x,y)=1$. We say that

$H=\langle xH_{1},yH_{2}\rangle=\langle
xa_{1},xa_{2},\ldots,xa_{n},yb_{1},yb_{2},\ldots,yb_{m}\rangle$

is a gluing of $H_{1}$ and $H_{2}$.

The following results are well-known.

######  Theorem 3.33.

[Nar13, Proposition 6.6] Let $x,y,H_{1},H_{2}$ be as in Definition 3.32 and
let $H=\langle xH_{1},yH_{2}\rangle$ be the gluing. Let
$S=\mathbb{k}\llbracket{H}\rrbracket$ and
$S_{i}=\mathbb{k}\llbracket{H_{i}}\rrbracket$ be the corresponding semigroup
rings with $c_{S},c_{S_{i}}$ denoting the respective conductor valuations.
Then the following statements hold.

  1. (1)

$\displaystyle\operatorname{PF}(H)=\\{xf+yg+xy\mid
f\in\operatorname{PF}(H_{1}),g\in\operatorname{PF}(H_{2})\\}.$

  2. (2)

$\displaystyle c_{S}-1=x(c_{S_{1}}-1)+y(c_{S_{2}}-1)+xy.$

  3. (3)

$\displaystyle\operatorname{type}(S)=\operatorname{type}(S_{1})\operatorname{type}(S_{2})$.

It is natural to expect a similar statement for the reduced type of the glued
numerical semigroup ring $S$. This is precisely the content of the next
theorem. We first observe that the multiplicity of the glued numerical
semigroup ring is given by

$\min\\{x{a_{1}},y{b_{1}}\\}.$

######  Theorem 3.34.

Let $S,S_{i}$ be as in Theorem 3.33. Then

$s(S)\leqslant s(S_{1})s(S_{2}).$

###### Proof.

Let
$\operatorname{PF}(H_{1})=\\{f_{1}<f_{2}<\cdots<f_{\operatorname{type}(S_{1})}\\}$
and
$\operatorname{PF}(H_{2})=\\{g_{1}<g_{2}<\cdots<g_{\operatorname{type}(S_{2})}\\}$.
Observe that
$f_{\operatorname{type}(S_{1})}=c_{S_{1}}-1,g_{\operatorname{type}(S_{2})}=c_{S_{2}}-1$.
Also, notice that the multiplicity of $S$ is $\min\\{x{a_{1}},y{b_{1}}\\}$.

Let $z\in[c_{S}-\min\\{x{a_{1}},y{b_{1}}\\},c_{S}-1]\setminus v(S)$. By
Theorem 2.13, we need to count all such $z$ to determine $s(S)$. Since
$z\in\operatorname{PF}(H)$, by Theorem 3.33(1), we get that
$z=xf_{i}+yg_{j}+xy$ for some
$f_{i}\in\operatorname{PF}(H_{1}),g_{j}\in\operatorname{PF}(H_{2})$.

Suppose $g_{j}\not\in[c_{S_{2}}-b_{1},c_{S_{2}}-1]$. Then we can write
$g_{j}=c_{S_{2}}-b_{1}-k$ with $k\geqslant 2$ (since
$c_{S_{2}}-b_{1}-1\not\in\operatorname{PF}(H_{2})$. Thus, we get

$z=xf_{i}+yg_{j}+xy=xf_{i}+y(c_{S_{2}}-b_{1}-k)+xy.$

Since $z\geqslant c_{S}-\min\\{xa_{1},yb_{1}\\}$, using Theorem 3.33(2), we
get that

$\displaystyle xf_{i}+y(c_{S_{2}}-b_{1}-k)+xy$ $\displaystyle\geqslant
x(c_{S_{1}}-1)+y(c_{S_{2}}-1)+xy+1-\min\\{xa_{1},yb_{1}\\}$

which upon simplification gives

$\min\\{xa_{1},yb_{1}\\}-1\geqslant x(c_{S_{1}}-1-f_{i})+y(b_{1}+k-1)\geqslant
y(b_{1}+k-1)\geqslant y(b_{1}+1).$

This gives

$yb_{1}-1\geqslant\min\\{xa_{1},yb_{1}\\}-1\geqslant y(b_{1}+1)$

but this is a contradiction since $y>a_{1}\geqslant 1$. Hence,
$g_{j}\in[c_{S_{2}}-b_{1},c_{S_{2}}-1]$.

A similar argument with $f_{i}$ shows that
$f_{i}\in[c_{S_{1}}-a_{1},c_{S_{1}}-1]$. Since this holds for any such $z$, we
conclude that $s(S)\leqslant s(S_{1})s(S_{2}).$ ∎

######  Corollary 3.35.

Let $S,S_{i}$ be as in Theorem 3.33. If $S_{1}$ and $S_{2}$ are of minimal
reduced type, then so is $S$.

###### Proof.

The proof is immediate from Theorem 3.34. ∎

However, the converse of the above statement need not hold, i.e., if $s(S)=1$,
then $S_{1},S_{2}$ simultaneously need not have minimal reduced type. The
following example illustrates this.

######  Example 3.36.

Consider $S_{1}$ to be the ring in Example 3.20 and $S_{2}$ as in Example
3.21. Let $x=10,y=13.$ Then $S=\mathbb{k}\llbracket{\langle
xH_{1},yH_{2}\rangle}\rrbracket=\mathbb{k}\llbracket{t^{40},t^{65},t^{78},t^{90},t^{91},t^{110},t^{117}}\rrbracket$.
Here $s(S_{1})s(S_{2})=1\times 2=2$. However, $s(S)=1$: the conductor
valuation $c_{S}=10(14)+13(8)+10\times 13+1=375$ using Theorem 3.33(2). The
multiplicity of $S$ is $40$. From Theorem 3.33(1), Examples 3.20 and 3.21, we
can compute the pseudo-Frobenius set:

$\operatorname{PF}(H)=\\{252,304,322,374\\}.$

Since $322<375-40=335$, we get that $s(R)=1$.

We do have the following statement in the converse direction which is
reflected in Example 3.36.

######  Theorem 3.37.

Let $S,S_{i}$ be as in Theorem 3.33. If $s(S)=1$, then either $s(S_{1})=1$ or
$s(S_{2})=1$.

###### Proof.

We will prove the statement by contradiction. Assume that $s(S_{i})\geqslant
2$ for $i=1,2$. Let $f_{i},g_{j}$ be chosen as in the proof of Theorem 3.34.

Consider
$z_{1}=x(c_{S_{1}}-1)+yg_{\operatorname{type}(S_{2})-1}+xy\in\operatorname{PF}(H)$
(the last part is due to Theorem 3.33(1)). Since $s(S_{2})\geqslant 2$, we can
write $g_{\operatorname{type}(S_{2})-1}=c_{S_{2}}-b_{1}+k_{1}$ with
$b_{1}-1>k_{1}\geqslant 0$. Hence,
$z_{1}=x(c_{S_{1}}-1)+y(c_{S_{2}}-1)+xy+y(1-b_{1}+k_{1})=c_{S}-1+y(1-b_{1}+k_{1})$,
where the last equality follows from Theorem 3.33(2). Since $s(S)=1$, we have
that $\displaystyle c_{S}-1+y(1-b_{1}+k_{1})\leqslant
c_{S}-\min\\{xa_{1},yb_{1}\\}-2$ which upon simplification, gives
$\displaystyle\min\\{xa_{1},yb_{1}\\}+1\leqslant y(b_{1}-1-k_{1})\leqslant
y(b_{1}-1)$. Thus, we must have $xa_{1}<yb_{1}$.

Since $s(S_{1})\geqslant 2$,
$f_{\operatorname{type}(S_{1})-1}=c_{S_{1}}-a_{1}+k_{2}$ with $0\leqslant
k_{2}<a_{1}-1$. Consider
$z_{2}=xf_{\operatorname{type}(S_{1})-1}+y(c_{S_{2}}-1)+xy=x(c_{S_{1}}-1)+y(c_{S_{2}}-1)+xy+x(1-a_{1}+k_{2})=c_{S}-1+x(1-a_{1}+k_{2})\in\operatorname{PF}(H)$.
Since $s(S)=1$, we get that $c_{S}-1+x(1-a_{1}+k_{2})\leqslant
c_{S}-\min\\{xa_{1},yb_{1}\\}-2$ which simplifies to
$\min\\{xa_{1},yb_{1}\\}+1\leqslant x(a_{1}-1-k_{2})\leqslant x(a_{1}-1)$.
This shows that $yb_{1}<xa_{1}$, a contradiction.

Thus either $s(S_{1})=1$ or $s(S_{2})=1$. ∎

Even if exactly one of $S_{1}$ or $S_{2}$ is of minimal reduced type, $s(S)$
need not be one.

######  Example 3.38.

Let $S_{1}$ be the ring in Example 3.20 and let $S_{2}$ be the ring in Example
3.28. Let $x=24,y=13$. Then $S=\mathbb{k}\llbracket{\langle
xH_{1},yH_{2}\rangle}\rrbracket=\mathbb{k}\llbracket{t^{96},t^{156},t^{169},t^{182},t^{195},t^{208},t^{216},t^{264},t^{247}}\rrbracket$.
Here, $c_{S}=24(14)+13(23)+13\times 24+1=948$ whereas the multiplicity is
$96$. Moreover,
$\operatorname{PF}(H)=\\{701,714,740,753,766,779,869,882,908,921,934,947\\}$.
Since there are $6$ numbers greater than $948-96=852$, we get that $s(S)=6$.
However, notice that $s(S_{1})=1$ as computed in Example 3.20.

Unfortunately, an analogue of Corollary 3.35 for maximal reduced may not be
feasible as the following example shows.

######  Example 3.39.

Let $S_{1}=\mathbb{k}\llbracket{t^{5},t^{6},t^{7},t^{9}}\rrbracket$ as in
Example 3.21 and let
$S_{2}=\mathbb{k}\llbracket{t^{3},t^{7},t^{8}}\rrbracket$. Notice that
$s(S_{2})=2=\operatorname{type}(S_{2})$ since
$\operatorname{PF}(H_{2})=\\{4,5\\}$. Thus both $S_{1}$ and $S_{2}$ are of
maximal reduced type. Choose $x=6,y=11$ to get that
$S=\mathbb{k}\llbracket{\langle
xH_{1},yH_{2}\rangle}\rrbracket=\mathbb{k}\llbracket{t^{30},t^{33},t^{36},t^{42},t^{54},t^{77},t^{88}}\rrbracket$.
Here, $c_{S}=170$ and $\operatorname{PF}(H)=\\{134,145,158,169\\}$. Since
$134<170-30$, we get that $s(S)=3$ whereas $\operatorname{type}(S)=4$. Hence
$S$ is not of maximal reduced type.

However, if the glued ring has maximal reduced type, then both the starting
rings must have the same as the following result shows.

######  Theorem 3.40.

Suppose $S,S_{i}$ be as in Theorem 3.33. If $S$ is of maximal reduced type,
then both $S_{1},S_{2}$ must have maximal reduced type.

###### Proof.

Using Theorem 3.33(3) and the assumption that $s(S)=\operatorname{type}(S)$,
we get that

$\operatorname{type}(S_{1})\operatorname{type}(S_{2})=\operatorname{type}(S)=s(S)\leqslant
s(S_{1})s(S_{2})$

where the last inequality follows from Theorem 3.34. Since $1\leqslant
s(S_{i})\leqslant\operatorname{type}(S_{i})$ and all are integers, we get that
$s(S_{i})=\operatorname{type}(S_{i})$ must hold for each $i$, thereby
finishing the proof. ∎

###  3.4. Dual of numerical semigroup

In this section, we study the behaviour of

$\operatorname{End}_{S}(\mathfrak{n}):=\operatorname{Hom}_{S}(\mathfrak{n},\mathfrak{n})$

where $S=\mathbb{k}\llbracket{H}\rrbracket$ is a complete numerical semigroup
ring and $\mathfrak{n}$ is the maximal ideal of $S$. We also make the
following notation. We have a surjection

$\displaystyle\mathbb{k}\llbracket{Y_{1},\dots,Y_{m}}\rrbracket\twoheadrightarrow\mathbb{k}\llbracket{H}\rrbracket$

which gives us that
$S=\mathbb{k}\llbracket{H}\rrbracket\cong\frac{\mathbb{k}\llbracket{Y_{1},\dots,Y_{m}}\rrbracket}{J}$
where $J\subseteq(Y_{1},\dots,Y_{m})^{2}$. Also, $y_{i}$ denote the image of
$Y_{i}$ in $S$. Thus we denote $\mathfrak{n}=(y_{1},\dots,y_{m})$. Here notice
that $\overline{S}=\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$.

By identifying with colons, we see that the endomorphism ring is the same as
$\displaystyle\mathfrak{n}:\mathfrak{n}=S:\mathfrak{n}$. Clearly,
$H\cup\operatorname{PF}(H)\subseteq v(\mathfrak{n}:\mathfrak{n})$. As in the
proof of Theorem 3.2, notice that

$\displaystyle\ell\left(\frac{\mathfrak{n}:\mathfrak{n}}{S}\right)=\operatorname{type}(S)=|\operatorname{PF}(H)|$

Since $\ell\left(\frac{\mathfrak{n}:\mathfrak{n}}{S}\right)$ computes the
number of valuations $v(\mathfrak{n}:\mathfrak{n})$, but not in $H$. Now the
cardinality of $H\cup\operatorname{PF}(H)$ and $v(\mathfrak{n}:\mathfrak{n})$
are the same. Thus we have the equality
$H\cup\operatorname{PF}(H)=v(\mathfrak{n}:\mathfrak{n})$.

Let $B=\mathfrak{n}:\mathfrak{n}$. Notice that the conductor $C_{B}$ of $B$
has the form $\frac{\mathfrak{C}}{y_{1}}$ where $\mathfrak{C}$ is the
conductor of $S$. For,

$\displaystyle\mathfrak{C}_{B}=B:\overline{S}=(S:\mathfrak{n}):\overline{S}=S:\mathfrak{n}\overline{S}=S:(y_{1})\overline{S}=\frac{1}{y_{1}}(S:\overline{S})=\frac{1}{y_{1}}(\mathfrak{C}).$

where $\mathfrak{C}=(t^{c})\overline{S}$. Thus the conductor
$\mathfrak{C}_{B}=(t^{c-b_{1}})\overline{S}$ (see for example, [BDF97] or
[Nar13, Proposition 3.4]). The above discussion is precisely the motivation of
the discussion that follows.

######  Definition 3.41.

Let $H=\langle b_{1},\dots,b_{m}\rangle$ be a numerical semigroup. Let $M$
denote the maximal ideal of $H$. Then the dual $H^{*}$ of $H$ is the numerical
semigroup $M-M=\\{x\in\mathbb{Z}\setminus\\{0\\}~{}|~{}x+m\in H\text{ for all
}m\in M\\}$

######  Remark 3.42.

Notice that $H\subseteq H^{*},\operatorname{PF}(H)\subseteq H^{*}$. Now let
$\alpha\in H^{*}\backslash H$, then $\alpha+m\in H$ for all $m\in M$. By
definition $\alpha\in\operatorname{PF}(H)$. Thus
$H^{*}=H\cup\operatorname{PF}(H)$. Thus for a numerical semigroup ring
$\mathbb{k}\llbracket{H}\rrbracket$, we have
$H^{*}=v(\mathfrak{n}:\mathfrak{n})=v(B)$. Thus we can set
$B=\operatorname{End}_{S}(\mathfrak{n})=\mathbb{k}\llbracket{H^{*}}\rrbracket$.

Thus it is clear that the multiplicity can drop as we move from $H$ to
$H^{*}$. However, the following proposition gives two conditions which
guarantee that the multiplicities of $H$ and $H^{*}$ stay the same.

######  Proposition 3.43.

The following statements hold for a numerical semigroup ring
$S=\mathbb{k}\llbracket{H}\rrbracket$ with $H=\langle
b_{1},\cdots,b_{m}\rangle$ and conductor valuation $c$.

  1. (1)

Suppose $S$ is of minimal reduced type but not Gorenstein. Then the
multiplicity of $H$ and $H^{*}$ are equal. Moreover, the conductor valuation
$c\geqslant 2b_{1}+3$.

  2. (2)

Suppose $S$ is Gorenstein. Then multiplicity of $H^{*}$ is strictly less than
$b_{1}$ if and only if $S=\mathbb{k}\llbracket{t^{2},t^{3}}\rrbracket$.

###### Proof.

For (1), using [Nar13, Lemma 6.4], we observe that $c-1-f_{t-1}\leqslant
f_{1}$ where $\operatorname{PF}(H)=\\{f_{1}<\cdots<f_{t}\\}$ with
$t=\operatorname{type}(S)$. Since $S$ is of minimal reduced type,
$f_{t-1}\leqslant c-b_{1}-2$ ($c-b_{1}-1\not\in\operatorname{PF}(H)$). Such an
$f_{t-1}$ exists as $t\geqslant 2$ by assumption. Thus $c-1-f_{t-1}\geqslant
c-1+2+b_{1}-c\geqslant b_{1}+1$. Thus we have

$\displaystyle b_{1}+1\leqslant c-1-f_{t-1}\leqslant f_{1}$

which proves that the multiplicity of $H$ and $H^{*}$ are equal. The last
statement now follows by using the above established inequalities:

$c\geqslant b_{1}+1+f_{t-1}+1=b_{1}+2+f_{1}\geqslant 2b_{1}+3.$

For $(2)$, first notice that if
$R=\mathbb{k}\llbracket{t^{2},t^{3}}\rrbracket$, then
$\operatorname{PF}(H)=\\{1\\}$ and hence $H^{*}=\langle 1\rangle$, in
particular the multiplicity dropped strictly. Conversely, assume that the
multiplicity of $H^{*}$ is strictly less than $b_{1}$. Since $R$ is
Gorenstein, $\operatorname{PF}(H)=\\{c-1\\}$. Hence, by Remark 3.42, we must
have $c-1\leqslant b_{1}-1$, i.e., $c\leqslant b_{1}$. But $c\in H$ and hence
$c=b_{1}$. Thus, $\mathfrak{C}=\mathfrak{m}$ which shows that $S$ has minimal
multiplicity by Remark 2.1. Moreover, $b_{2}=b_{1}+1$ since
$\mathfrak{C}=\mathfrak{m}$. Since $R$ is Gorenstein, [HKS21, Fact 2.6]
immediately yields that $b_{1}=2$. So, $b_{2}=3$ and
$\operatorname{edim}(S)=b_{1}=2$. This finishes the proof. ∎

######  Corollary 3.44.

Let $S=\mathbb{k}\llbracket{H}\rrbracket$ with $c\geqslant b_{1}+1$ and and is
of minimal reduced type. Then
$e(\mathbb{k}\llbracket{H^{*}}\rrbracket)=e(S)=b_{1}$.

###### Proof.

If $S$ is not Gorenstein, then the result follows from Proposition 3.43(1). So
assume $S$ is Gorenstein. Since $c\geqslant b_{1}+1$, $S$ cannot be
$\mathbb{k}\llbracket{t^{2},t^{3}}\rrbracket$ and hence the result follows
from Proposition 3.43(2). ∎

Now a natural question is to study the behaviour of $s(S)$ under the
construction of the dual semigroup. The following proposition sheds light upon
this study.

######  Theorem 3.45.

Suppose $S=\mathbb{k}\llbracket{H}\rrbracket$. If
$\operatorname{End}_{S}(\mathfrak{n})$ is of minimal reduced type and
$e(\operatorname{End}_{S}(\mathfrak{n}))=e(S)$, then $S$ is of minimal reduced
type.

###### Proof.

Using Remark 3.42, we see that
$H^{*}=v(\operatorname{End}_{S}(\mathfrak{n}))$. Suppose $H^{*}$ is of minimal
reduced type and the multiplicity of $H$ and $H^{*}$ are equal. From the
discussion at the beginning of this subsection,
$\mathfrak{C}_{B}=(t^{c-b_{1}})\overline{S}$. Thus
$c-b_{1}-1\in\operatorname{PF}(H^{*})$. Since $s(B)=1$, every element of
$[c-2b_{1},c-b_{1}-2]$ is in $H^{*}=H\cup\operatorname{PF}(H)$.

Let $c-b_{1}+k\in[c-b_{1},c-2]$. Then
$c-2b_{1}+k\in[c-2b_{1},c-b_{1}-2]\subseteq H^{*}$. Thus either $c-2b_{1}+k\in
H$ or $c-2b_{1}+k\in\operatorname{PF}(H)$. If $c-2b_{1}+k\in H$, then
$c-2b_{1}+k+b_{1}=c-b_{1}+k\in H$. If $c-2b_{1}+k\in\operatorname{PF}(H)$,
then $c-2b_{1}+k+b_{1}=c-b_{1}+k\in H$. Thus every member of $[c-b_{1},c-2]$
is a member of $H$ and hence $S$ is of minimal reduced type by Proposition
3.8. ∎

In general, the minimality of reduced type need not stay preserved as the
following examples illustrate.

######  Example 3.46.

Let $R=\mathbb{k}\llbracket{t^{4},t^{5},t^{6}}\rrbracket$. Then
$\operatorname{PF}(H)=\\{7\\}$ which shows that $R$ is Gorenstein (hence
$s(R)=1$). However,
$B=\mathbb{k}\llbracket{H^{*}}\rrbracket=\mathbb{k}\llbracket{t^{4},t^{5},t^{6},t^{7}}\rrbracket$
which is of minimal multiplicity (hence $\operatorname{type}(B)=3$) with
conductor valuation $4$. Thus $s(B)=3$ by Corollary 3.10.

######  Example 3.47.

Let
$R=\mathbb{k}\llbracket{t^{40},t^{65},t^{78},t^{90},t^{91},t^{110},t^{117}}\rrbracket$.
From Example 3.36, we know that $s(R)=1$ and
$\operatorname{PF}(H)=\\{252,304,322,374\\}$ (so $R$ is not Gorenstein). We
construct the dual semigroup ring

$B=\mathbb{k}\llbracket{t^{40},t^{65},t^{78},t^{90},t^{91},t^{110},t^{117},t^{252},t^{304},t^{322},t^{374}}\rrbracket$

which has conductor valuation $c_{B}=335$. Moreover,

$\operatorname{PF}(H^{*})=\\{187,212,213,226,232,239,257,264,282,283,284,296,309,334\\}.$

Thus $s(B)=3$.

Unfortunately, an analogue of the previous theorem for maximal reduced type
does not hold. The following example establishes this.

######  Example 3.48.

Let $R=\mathbb{k}\llbracket{t^{5},t^{9},t^{11},t^{12}}\rrbracket$. Notice that
$c=14$ and $\operatorname{PF}(H)=\\{6,7,13\\}$. Here $s(R)=1$ and
$\operatorname{type}(R)=3$, thus $R$ is not of maximal reduced type. However,
$B=\mathbb{k}\llbracket{t^{5},t^{6},t^{7},t^{9}}\rrbracket$ which is of
maximal reduced type by Example 3.21.

The following examples show that the multiplicity can drop even when we start
with a ring of maximal reduced type. Moreover, the maximality of reduced type
need not stay preserved.

######  Example 3.49.

[DMS23, Example 7.14] Let
$R=\mathbb{k}\llbracket{t^{4},t^{6},t^{7},t^{9}}\rrbracket$ which has
$\mathfrak{C}=(t^{6},t^{7},t^{8},t^{9})$ and
$\operatorname{PF}(H)=\\{2,3,5\\}$. Thus,
$B=\mathbb{k}\llbracket{H^{*}}\rrbracket=\mathbb{k}\llbracket{t^{2},t^{3}}\rrbracket$.
Thus multiplicity has dropped. Notice however that both rings have maximal
reduced type.

######  Example 3.50.

Let $R=\mathbb{k}\llbracket{t^{5},t^{7},t^{9}}\rrbracket$. Here $s(R)=2$ with
$\operatorname{PF}(H)=\\{11,13\\}$ and $R$ is of maximal reduced type. Note
that
$B=\mathbb{k}\llbracket{H^{*}}\rrbracket=\mathbb{k}\llbracket{t^{5},t^{7},t^{9},t^{11},t^{13}}\rrbracket$
is of minimal multiplicity with conductor valuation $9$. Thus, $B$ is not of
maximal reduced type by Corollary 3.10.

##  4\. The categories $\operatorname{Ref}(R)$ and $\operatorname{CM}(R)$

An $R$-module $M$ is said to be reflexive if the natural map
$M\to\operatorname{Hom}_{R}(\operatorname{Hom}_{R}(M,R),R)$ is an isomorphism
whereas $M$ is Maximal Cohen-Macaulay (MCM) if $0pt(M)=\dim(R)$. Let
$\operatorname{CM}(R)$ denote the category of maximal Cohen-Macaulay
$R$-modules and let $\operatorname{Ref}(R)$ denote the category of reflexive
$R$-modules. We say a category is of finite type if it has only finitely many
indecomposable objects up to isomorphism.

Since we restrict our attention to one dimensional non-regular complete local
domains containing an algebraically closed field $\mathbb{k}$ of
characteristic $0$, we know that $\displaystyle\\{\text{free $R$-
modules}\\}\subseteq\operatorname{Ref}(R)\subseteq\operatorname{CM}(R)$;
moreover, $R$ is Gorenstein if and only if
$\displaystyle\operatorname{Ref}(R)=\operatorname{CM}(R)$ ([DMS23, Remark
3.1]). Thus it follows that if $\operatorname{CM}(R)$ is of finite type, then
$\operatorname{Ref}(R)$ is also of finite type.

The study of finiteness of $\operatorname{CM}(R)$ and the corresponding
classification question of $R$ has been of interest for quite some time. For
more details on these results along with discussions on the notions of ADE-
singularities, we refer the reader to [LW12, Chapter 4 §3, Chapters 9, 10].
For the convenience of the reader we also list the possible ADE hypersurface
singularities as in [LW12, §3, Page 70,71]

  1. (1)

$(A_{n}):x^{2}+y^{n+1},n\geqslant 1$

  2. (2)

$(D_{n}):x^{2}y+y^{n-1},n\geqslant 4$

  3. (3)

$(E_{6}):x^{3}+y^{4}$

  4. (4)

$(E_{7}):x^{3}+xy^{3}$

  5. (5)

$(E_{8}):x^{3}+y^{5}$

We first discuss Gorenstein rings which are precisely the rings of both
maximal and minimal reduced types. The following result is well-known in the
literature and provides a complete picture in the Gorenstein setup for all
dimensions.

######  Theorem 4.1.

[LW12, Corollary 10.7] Let $(R,\mathfrak{m},k)$ be an excellent, Gorenstein
ring containing a field of characteristic different from $2,3,5$, and let $K$
be an algebraic closure of $k$. Assume $d=\dim R\geqslant 1$ and that $k$ is
perfect. Then $\operatorname{CM}(R)$ is of finite type if and only if there is
a non-zero non-unit $f\in k\llbracket{x_{0},\cdots,x_{d}}\rrbracket$ such that
$\widehat{R}\cong k\llbracket{x_{0},\cdots,x_{d}}\rrbracket/(f)$ and
$K\llbracket{x_{0},\cdots,x_{d}}\rrbracket/(f)$ is a complete $ADE$
hypersurface singularity.

Note that in our notations, we have $d=1$, $\mathbb{k}=k=K$ and
$\widehat{R}=R$, where $\widehat{R}$ denotes the $\mathfrak{m}$-adic
completion of $R$. Thus, due to Theorem 4.1, it is enough to focus on non-
Gorenstein rings.

Here, in order to study $\operatorname{CM}(R)$, we need the following results.

######  Theorem 4.2.

[LW12, Theorem 4.10] Let $(R,\mathfrak{m},k)$ be an analytically unramified
local ring of dimension one. Then $\operatorname{CM}(R)$ is of finite type if
and only if $e(R)\leqslant 3$ and
$\mu(\frac{\mathfrak{m}\overline{R}+R}{R})\leqslant 1$.

######  Remark 4.3.

So, $\operatorname{CM}(R)$ is of finite type for $R$ only if $e(R)\leqslant 3$
which in turn shows that either $R$ is of minimal reduced type or of maximal
reduced type.

######  Theorem 4.4.

[DMS23, Propositions 6.12, 7.7] Let $(R,\mathfrak{m})$ be a one-dimensional
Cohen-Macaulay local ring. Let $B=\operatorname{End}_{R}(\mathfrak{m})$. If
$\operatorname{CM}(B)$ is of finite type, then $\operatorname{Ref}(R)$ is of
finite type. Moreover, if $R$ is almost Gorenstein, then the converse also
holds.

######  Proposition 4.5.

Let $R=\mathbb{k}\llbracket{X_{1},\cdots,X_{n}}\rrbracket/I$ be a one
dimensional complete local domain with $k$ an algebraically closed field of
characteristic $0$ with $\operatorname{edim}(R)\geqslant 2$. Let the integral
closure $\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$. If
$\operatorname{CM}(R)$ is of finite type, then
$\operatorname{CM}(\mathbb{k}\llbracket{H}\rrbracket)$ is of finite type for
any numerical semigroup $H$ with
$R\subseteq\mathbb{k}\llbracket{H}\rrbracket\subseteq\overline{R}$.

###### Proof.

Notice that
$R\subseteq\mathbb{k}\llbracket{H}\rrbracket\subseteq\operatorname{Frac}(R)$
for any such given numerical semigroup $H$. Now the result is immediate from
[LW12, 4.14(iv)]. ∎

Notice that $H$ need not be $v(R)$ in the above proof. For example if
$R=\mathbb{k}\llbracket{t^{3}+t^{4},t^{5},t^{7}}\rrbracket$, then
$v(R)=\langle 3,5,7\rangle$ and hence
$\mathbb{k}\llbracket{v(R)}\rrbracket=\mathbb{k}\llbracket{t^{3},t^{5},t^{7}}\rrbracket$,
but $R\not\subseteq\mathbb{k}\llbracket{v(R)}\rrbracket$. But it is however
true that $R\subseteq\mathbb{k}\llbracket{t^{3},t^{4},t^{5}}\rrbracket$. Thus,
studying $\operatorname{CM}(R)$ for a numerical semigroup ring of maximal or
minimal reduced type can help to some extent in understanding
$\operatorname{CM}(R)$. The link between the category of reflexive modules
however is not so clear unless we have $R$ Gorenstein to begin with.
Nevertheless, we restrict our attention to
$R=\mathbb{k}\llbracket{H}\rrbracket$ for the rest of this section.

######  Theorem 4.6.

Let $S=\mathbb{k}\llbracket{H}\rrbracket$ be a numerical semigroup ring which
is not Gorenstein. Then $\operatorname{CM}(S)$ is of finite type if and only
if one of the following holds:

  1. (1)

$H=\langle 3,4,5\rangle$

  2. (2)

$H=\langle 3,5,7\rangle$.

###### Proof.

We use the same notation as in the beginning of Section 3.4. Assume
$\operatorname{CM}(S)$ is of finite type. Then $e(S)\leqslant 3$ by Theorem
4.2 and since $S$ is not Gorenstein, we get that $\operatorname{type}(S)=2$,
i.e., $S$ is of minimal multiplicity. Hence $H=\langle 3,b_{2},b_{3}\rangle$.
Now again by Theorem 4.2, we must have
$\mu_{S}(\mathfrak{n}\overline{S}+S/S)\leqslant 1$. From Remark 2.1, we get
that
$L:=\frac{\mathfrak{n}\overline{S}+S}{S}=\frac{\mathfrak{n}\overline{S}}{(\mathfrak{n}\overline{S}\cap
S)}=\frac{y_{1}\overline{S}}{\mathfrak{n}}$. Hence

$1\geqslant\mu(L)=\ell\Big{(}\frac{y_{1}\overline{S}}{\mathfrak{n}y_{1}\overline{S}+\mathfrak{n}}\Big{)}=\ell\Big{(}\frac{y_{1}\overline{S}}{y_{1}^{2}\overline{S}+\mathfrak{n}}\Big{)}.$

Now $v(y_{1}\overline{S})=\\{3,4,5,\rightarrow\\}$ whereas
$v(y_{1}^{2}\overline{S}+\mathfrak{n})=\\{3,b_{2},b_{3}\\}\cup\\{6,7,8,\rightarrow\\}$.
Thus the only possibilities of $b_{2}$ are $4,5$. Suppose $b_{2}=4$. Notice
that the ring $\mathbb{k}\llbracket{t^{3},t^{4}}\rrbracket$ has the conductor
$\mathfrak{C}=(t^{6})\overline{R}$. Thus any element $t^{i},i\geqslant 6$ can
be generated by $t^{3},t^{4}$. Now since $\operatorname{edim}S=3$, we must
have $b_{3}=5$. Similarly, if $b_{2}=5$, then $b_{3}=7$ (the conductor of
$\mathbb{k}\llbracket{t^{3},t^{5}}\rrbracket$ is $(t^{8})\overline{S}$). This
finishes one direction of the proof.

Conversely, if $H$ is of the two forms as stated, then the same argument in
the above paragraph shows that $L$ is generated by at most one element in both
cases. Since $e(S)=3$, we can now apply Theorem 4.2. ∎

######  Theorem 4.7.

If $S=\mathbb{k}\llbracket{H}\rrbracket$ is of minimal reduced type, then
$\operatorname{CM}(S)$ is of finite type if and only if $S$ is a complete
$ADE$ hypersurface singularity of multiplicity $2$ or $3$.

###### Proof.

If $S$ is non-Gorenstein with $\operatorname{CM}(S)$ of finite type, then $S$
is of maximal reduced type using Corollary 3.10 on the two forms of $H$ we
obtained in Theorem 4.6. So, if $S$ is of minimal reduced type with
$\operatorname{CM}(S)$ of finite type, then $S$ must be Gorenstein. The
conclusion for the last part of the statement now follows from Theorem 4.1. ∎

######  Theorem 4.8.

Suppose $S=\mathbb{k}\llbracket{H}\rrbracket$ be a numerical semigroup ring
which is not almost Gorenstein. Let $B=\operatorname{End}_{R}(\mathfrak{m})$.
Assume that $e(S)=e(B)$. Then $\operatorname{CM}(B)$ is of finite type if and
only if one of the following holds:

  1. (1)

$H=\langle 3,7,8\rangle$

  2. (2)

$H=\langle 3,8,10\rangle$.

In particular, if $S$ is non-almost Gorenstein of minimal reduced type, then
$\operatorname{CM}(B)$ is not of finite type.

###### Proof.

Since $S$ is not almost Gorenstein, $B$ is not Gorenstein by [GMP13, Theorem
5.1]. So, by Theorem 4.6, we get that $B$ is constructed either by
$H^{*}=\langle 3,4,5\rangle$ or by $H^{*}=\langle 3,5,7\rangle$. Since
$e(S)=3$ and $S$ is not Gorenstein, it follows that $H=\langle
3,b_{2},b_{3}\rangle$. Note that $\operatorname{PF}(H)=\\{b_{2}-3,b_{3}-3\\}$.
Since $e(B)=e(S)$, the construction in Remark 3.42 shows that
$b_{2}-3\geqslant 4$. Hence $b_{2}\geqslant 7$. If $H^{*}=\langle
3,4,5\rangle$, then $b_{2}=7$ and $b_{3}=8$. If $H^{*}=\langle 3,5,7\rangle$,
then $b_{2}=8$ and $b_{3}=10$. This finishes one direction of the proof.

Conversely, given the forms of $H$ as in the statement, it is clear that $B$
is given by either $\langle 3,4,5\rangle$ or $\langle 3,5,7\rangle$. The proof
is complete by Theorem 4.2.

For the last part, we use Proposition 3.43(1) along with the observation that
both the forms of $H$ given in the statement of the above corollary are of
maximal reduced type by Corollary 3.10. ∎

######  Theorem 4.9.

Let $S=\mathbb{k}\llbracket{H}\rrbracket$ be an almost Gorenstein ring of
minimal reduced type and $\operatorname{type}(S)\geqslant 2$. Then
$\operatorname{Ref}(S)$ is of finite type if and only if $S$ has one of the
following forms:

  1. (1)

$\mathbb{k}\llbracket{t^{3},t^{7},t^{11}}\rrbracket$,

  2. (2)

$\mathbb{k}\llbracket{t^{3},t^{8},t^{13}}\rrbracket$.

###### Proof.

Throughout the proof, we set
$B=\operatorname{End}_{S}(\mathfrak{n})=\mathbb{k}\llbracket{H^{*}}\rrbracket$.
By Theorem 4.4, we know that $\operatorname{Ref}(S)$ is of finite type if and
only if $\operatorname{CM}(B)$ is of finite type.

Assume $S$ is of the forms (1) or (2). The type of $S$ is 2 and the pseudo
Frobenius set is $\\{4,8\\}$ or $\\{5,10\\}$ respectively. Thus
$B=\mathbb{k}\llbracket{t^{3},t^{4}}\rrbracket$ or
$B=\mathbb{k}\llbracket{t^{3},t^{5}}\rrbracket$ respectively. These rings have
ADE hypersurface singularities and hence $\operatorname{CM}(B)$ is of finite
type by Theorem 4.1.

Conversely, assume that $\operatorname{Ref}(S)$ is of finite type. So
$\operatorname{CM}(B)$ is of finite type and hence $e(B)\leqslant 3$ by
Theorem 4.1(1). Since $s(S)=1$, we get that $e(S)\leqslant 3$ by Proposition
3.43(1). Since $\operatorname{type}(S)\geqslant 2$, it forces $e(S)=3$ and
$\operatorname{type}(S)=2$, i.e., $S$ has minimal multiplicity $3$. Hence,
$\operatorname{edim}(S)=3$, Let $H=\langle 3,b_{2},b_{3}\rangle$. We can
describe $S$ completely using [RGS09, Lemma 4.27]. We include the proof here
for convenience of the reader: since $S$ is almost Gorenstein of minimal
multiplicity, we have that $\operatorname{PF}(H)=\\{b_{2}-3,b_{3}-3\\}$ with
$b_{3}-3=2(b_{2}-3)$. Thus, taking $b_{2}=h+3$, we get that $b_{3}=2h+3$.
Further, since $b_{2}$ is not a multiple of $3$, $3$ does not divide $h$.
Finally notice that $s(S)=1$ implies that $b_{2}-3\leqslant(b_{3}-3+1)-3-2$.
This upon simplification gives $b_{3}\geqslant b_{2}+4$ which in turn implies
that $h\geqslant 4$. Thus $H=\langle 3,h+3,2h+3\rangle$ where $h\geqslant 4$,
$h$ is not divisible by $3$ and $\operatorname{PF}(H)=\\{h,2h\\}$. So
$B=\mathbb{k}\llbracket{t^{3},t^{h}}\rrbracket$. Since $\operatorname{CM}(B)$
is of finite type, $B$ has $ADE$ hypersurface singularities by Theorem 4.1.
Now the only possibilities for the defining equation of $B$ are coming from
$(E_{6})$ or $(E_{8})$ of [LW12, §3, Page 70,71]. Thus
$B=\mathbb{k}\llbracket{t^{3},t^{4}}\rrbracket,(h=4)$ or
$B=\mathbb{k}\llbracket{t^{3},t^{5}}\rrbracket,(h=5)$. Thus we have
$R=\mathbb{k}\llbracket{t^{3},t^{7},t^{11}}\rrbracket$ or
$R=\mathbb{k}\llbracket{t^{3},t^{8},t^{13}}\rrbracket$. ∎

######  Question 4.10.

Can we classify all numerical semigroup rings of minimal reduced type which
are not almost Gorenstein but $\operatorname{Ref}(S)$ is of finite type?

######  Proposition 4.11.

Let $S=\mathbb{k}\llbracket{H}\rrbracket$ be an almost Gorenstein ring of
maximal reduced type $2$ and $e(S)=3$. Then $\operatorname{CM}(S)$ (and hence
$\operatorname{Ref}(S)$) is of finite type.

###### Proof.

Since $s(S)=2=\operatorname{type}(R)$ and $e(S)=3$, we get that $S$ is of
minimal multiplicity. Since $S$ is almost Gorenstein, the same argument as in
the second part of Theorem 4.9 shows that $H=\langle 3,h+3,2h+3\rangle$ where
$3$ does not divide $h$. Again, $\operatorname{PF}(H)=\\{h,2h\\}$ and since
$S$ is of maximal reduced type, by Corollary 3.10 we get that $h+3\geqslant
2h+1$, i.e., $h\leqslant 2$. Hence, $H=\langle 3,4,5\rangle$ or $H=\langle
3,5,7\rangle$. So $\operatorname{CM}(S)$ is of finite type in both cases by
Theorem 4.6 and consequently, $\operatorname{Ref}(S)$ is of finite type. ∎

######  Theorem 4.12.

Let $S$ be a non-Gorenstein almost Gorenstein ring of maximal reduced type.
Let $B=\operatorname{End}_{S}(\mathfrak{n})$ and assume that $e(B)\leqslant
2$. Then $\operatorname{Ref}(S)$ is of finite type if and only if $S$ has one
of the following forms.

  1. (1)

$\mathbb{k}\llbracket{t^{b_{1}},t^{b_{1}+1},\ldots,t^{2b_{1}-1}}\rrbracket$
with $b_{1}\geqslant 3$,

  2. (2)

$\mathbb{k}\llbracket{t^{b_{1}},t^{b_{1}+2},t^{b_{1}+3},\ldots,t^{2b_{1}-1},t^{2b_{1}+1}}\rrbracket$
with $b_{1}\geqslant 3$,

###### Proof.

By Theorems 4.9 and 4.2, $e(B)\leqslant 3$ is necessary. Now the assumption on
$e(B)$ implies that $S$ is of minimal multiplicity by [GMP13, Theorem 5.1].
So, if
$S=\mathbb{k}\llbracket{t^{b_{1}},t^{b_{2}},\ldots,t^{b_{n}}}\rrbracket$, then
$b_{2}\geqslant c$ by Corollary 3.10.

It is easy to check that for the given forms of $S$, $\operatorname{CM}(B)$ is
of finite type and hence $\operatorname{Ref}(S)$ is of finite type. The fact
that the given forms are almost Gorenstein of maximal reduced type follows
from [Nar13, Theorem 2.4] and Corollary 3.10.

We discuss the converse in separate cases. Since $S$ is non-Gorenstein,
$b_{1}=e(S)\geqslant 3$.

  1. (1)

Assume $e(B)=1$. So $B=\mathbb{k}\llbracket{t}\rrbracket$. Since
$B=\mathfrak{n}:\mathfrak{n}=S:\mathfrak{n}$, then $B=\overline{S}$ if and
only if $S:\mathfrak{n}=\overline{S}$ if and only if
$S:(S:\mathfrak{n})=S:\overline{S}=\mathfrak{C}$ if and only if
$\mathfrak{n}=\mathfrak{C}$ where the last part follows from [DMS23,
Proposition 2.7 and Corollary 3.2]. Thus,
$S=\mathbb{k}\llbracket{t^{b_{1}},t^{b_{1}+1},\ldots,t^{2b_{1}-1}}\rrbracket$.

  2. (2)

Assume $e(B)=2$. Since
$\operatorname{PF}(H)=\\{b_{2}-b_{1},\ldots,b_{n}-b_{1}\\}$, Remark 3.42 shows
that $b_{2}=b_{1}+2=c$. Thus,
$S=\mathbb{k}\llbracket{t^{b_{1}},t^{b_{1}+2},t^{b_{1}+3},\ldots,t^{2b_{1}-1},t^{2b_{1}+1}}\rrbracket$.

∎

######  Theorem 4.13.

Let $S$ be a non-Gorenstein almost Gorenstein ring of maximal reduced type.
Let $B=\operatorname{End}_{S}(\mathfrak{n})$ and assume that $e(B)=3$. Then
$\operatorname{Ref}(S)$ is of finite type if and only if
$S=\mathbb{k}\llbracket{t^{b_{1}},t^{b_{1}+1},t^{b_{1}+3},t^{b_{1}+4},\cdots,t^{2b_{1}-1}}\rrbracket,b_{1}\geqslant
4$

###### Proof.

We know $\operatorname{CM}(B)$ is of finite type by Theorem 4.4. Assume
$\operatorname{edim}(B)=2$, i.e., $B$ is Gorenstein but not of minimal
multiplicity. Also, $S$ is of minimal multiplicity by [GMP13, Theorem 5.4].
Since $\operatorname{CM}(B)$ is of finite type, the only possibilities of $B$
are $\mathbb{k}\llbracket{t^{3},t^{4}}\rrbracket$ and
$\mathbb{k}\llbracket{t^{3},t^{5}}\rrbracket$ as in the argument of Theorem
4.9. If $e(S)=3$, then the proof of Proposition 4.11 shows that $H=\langle
3,4,5\rangle$ or $\langle 3,5,7\rangle$. But for either of these rings,
$e(B)=2$, a contradiction. Hence $e(S)\geqslant 4$. Since $S$ is of maximal
reduced type, $b_{2}\geqslant c$. Also,
$\operatorname{PF}(H)=\\{b_{2}-b_{1},\ldots,b_{n}-b_{1}\\}$. Since $3\in
v(B)$, we must have $b_{2}=b_{1}+3$. If $c<b_{2}$, then $c=b_{1}$ and hence
$b_{2}=b_{1}+1$, contradicting our implication that $b_{2}=b_{1}+3$. Thus
$c=b_{2}$. Using the fact that $S$ is of minimal multiplicity, we see that
$S=\mathbb{k}\llbracket{t^{b_{1}},t^{b_{1}+3},t^{b_{1}+4},\ldots,t^{2b_{1}-1},t^{2b_{1}+1},t^{2b_{1}+2}}\rrbracket$
with $b_{1}\geqslant 4$. However,
$\operatorname{PF}(H)=\\{3,4,\ldots,b_{1}-1,b_{1}+1,b_{1}+2\\}$. This is not
almost Gorenstein since $3+b_{1}+1\neq b_{1}+2$ by Theorem 3.14.

Thus $e(B)=3=\operatorname{edim}(B)$, i.e., $B$ is of minimal multiplicity.
Then by Theorem 4.6, $B=\mathbb{k}\llbracket{t^{3},t^{4},t^{5}}\rrbracket$ or
$B=\mathbb{k}\llbracket{t^{3},t^{5},t^{7}}\rrbracket$. Moreover, $S$ cannot be
of minimal multiplicity and hence $b_{1}=e(S)\geqslant 4$ (because
$\operatorname{type}(S)\geqslant 2$). Since $t^{3}\in B$,
$3\in\operatorname{PF}(H)$. Thus $b_{i}=b_{1}+3$ for some $i\geqslant 2$.
Since $b_{1}\geqslant 4$ and $3\in\operatorname{PF}(H)$, we have $3\geqslant
c-b_{1}$ as $S$ has maximal reduced type. If $3>c-b_{1}$, then $c-b_{1}$ is in
the valuation semigroup of $S$ which is a contradiction as $1\leqslant
c-b_{1}<3<b_{1}$. Notice that the first inequality is because of the fact that
$S$ is not of minimal multiplicity and hence $c>b_{1}$. Thus $c-b_{1}=3$ or
$c=b_{1}+3$ and hence $c-1=b_{1}+2$ is not a member of the valuation semigroup
of $S$. Thus the only choice of $S$ with these properties is
$S=\mathbb{k}\llbracket{t^{b_{1}},t^{b_{1}+1},t^{b_{1}+3},t^{b_{1}+4},\ldots,t^{2b_{1}-1}}\rrbracket,b_{1}\geqslant
4$. Moreover, notice that
$\operatorname{PF}(H)=\\{3,4,5,\ldots,b_{1}-1,b_{1}+2\\}$. Hence $S$ is almost
Gorenstein by Theorem 3.14. In other words $v(B)=\langle 3,4,5\rangle$ or
$B=\mathbb{k}\llbracket{t^{3},t^{4},t^{5}}\rrbracket$. Interestingly, notice
that $B$ cannot take the form
$\mathbb{k}\llbracket{t^{3},t^{5},t^{7}}\rrbracket$ as mentioned in the
beginning of this paragraph.

Conversely, if
$S=\mathbb{k}\llbracket{t^{b_{1}},t^{b_{1}+1},t^{b_{1}+3},t^{b_{1}+4},\ldots,t^{2b_{1}-1}}\rrbracket,b_{1}\geqslant
4$, then $v(B)=\langle 3,4,5\rangle$ and hence
$B=\mathbb{k}\llbracket{t^{3},t^{4},t^{5}}\rrbracket$. Thus
$\operatorname{CM}(B)$ is of finite type (by Theorem 4.6), equivalently
$\operatorname{Ref}(S)$ is of finite type. ∎

######  Question 4.14.

Can we classify all numerical semigroup rings of maximal reduced type which
are not almost Gorenstein but $\operatorname{Ref}(S)$ is of finite type?

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