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\title{Affine Semigroups of Maximal Projective Dimension}

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\author[O.~P.~Bhardwaj, K.~Goel, and I.~Sengupta]{Om Prakash Bhardwaj\thanks{\href{mailto:om.prakash@iitgn.ac.in}{om.prakash@iitgn.ac.in}}\addressmark{1}, Kriti Goel\thanks{\href{mailto:kritigoel.maths@gmail.com}{kritigoel.maths@gmail.com}}\addressmark{2}, \and Indranath Sengupta\thanks{\href{mailto:indranathsg@iitgn.ac.in}{indranathsg@iitgn.ac.in}}\addressmark{3}}

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\address{\addressmark{1}Discipline of Mathematics, IIT Gandhinagar, India \\ \addressmark{2}Discipline of Mathematics, IIT Gandhinagar, India \\ \addressmark{3}Discipline of Mathematics, IIT Gandhinagar, India}

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\received{24-11-2021}

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\abstract{A submonoid of $\mathbb{N}^d$ is of maximal projective dimension ($\mathrm{MPD}$) if the associated affine semigroup $k$-algebra has the maximum possible projective dimension. Such submonoids have a nontrivial set of pseudo-Frobenius elements. We generalize the notion of symmetric semigroups, pseudo-symmetric semigroups, and row-factorization matrices for pseudo-Frobenius elements of numerical semigroups to the case of $\mathrm{MPD}$-semigroups in $\mathbb{N}^d$.  
We prove that under suitable conditions these semigroups satisfy the generalized Wilf's conjecture. We prove that the generic nature of the defining ideal of the associated semigroup algebra of an $\mathrm{MPD}$-semigroup implies the uniqueness of the row-factorization matrix for each pseudo-Frobenius element. Further, we give a description of pseudo-Frobenius elements and row-factorization matrices of gluing of $\mathrm{MPD}$-semigroups. We prove that the defining ideal of gluing of $\mathrm{MPD}$-semigroups is never generic.}

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\keywords{MPD-semigroup, pseudo-Frobenius elements, row-factorization matrix, generic toric ideals}

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\begin{document}

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\section{Introduction}
Let $\mathbb{Z}$ and $\mathbb{N}$ denote the sets of integers and non-negative integers respectively. An affine semigroup $S$ is a finitely generated submonoid of $\mathbb{N}^d$ for some positive integer $d.$ When $d=1$, affine semigroups correspond to numerical semigroups. Equivalently, a submonoid $S$ of $\mathbb{N}$ is called a numerical semigroup if it has a finite complement in $\mathbb{N}.$ If $S \neq \mathbb{N},$ then the largest integer not belonging to $S$ is known as the Frobenius number of $S,$ denoted by $\mathrm{F}(S).$ Also, the finiteness of $\mathbb{N} \setminus S$ implies that there exists at least one element $f \in \mathbb{N} \setminus S$ such that $f + (S \setminus \{0\}) \subset S.$ These elements are called pseudo-Frobenius elements of the numerical semigroup $S$. The set of pseudo-Frobenius elements is denoted by $\mathrm{PF}(S).$ In particular, $\mathrm{F}(S)$ is a pseudo-Frobenius number. But, in general, this does not hold for affine semigroups in $\mathbb{N}^d.$ The study of pseudo-Frobenius elements in affine semigroups over $\mathbb{N}^d$ is studied in~\cite{pfelements}, where the authors consider the complement of the affine semigroup in its rational polyhedral cone. They give a necessary and a sufficient condition for the existence of pseudo-Frobenius elements using properties of the associated semigroup ring. Let $S$ be an affine semigroup and let $k$ be a field. The semigroup ring $k[S] = \bigoplus_{s \in S} k \ {\bf t}^s$ of $S$ is a $k$-subalgebra of the polynomial ring $k[t_1,\ldots,t_d],$ where $t_1,\ldots,t_d$ are indeterminates and ${\bf t}^s = \prod_{i=1}^{d} t_i^{s_i}$ for all $s = (s_1,\ldots,s_d) \in S.$ In~\cite{pfelements}, the authors prove that an affine semigroup $S$ has pseudo-Frobenius elements if and only if the length of the graded minimal free resolution of the corresponding semigroup ring is maximal. Affine semigroups having pseudo-Frobenius elements are called maximal projective dimension ($\mathrm{MPD}$) semigroups. The cardinality of the set of pseudo-Frobenius elements is called the type of $S.$ Note that this is not the Cohen--Macaulay type of $S$ since $\mathrm{MPD}$-semigroups, when $d \geq 2,$ are not Cohen--Macaulay.

One of the widely studied class of numerical semigroups is symmetric semigroups --- numerical semigroups of type 1. The motivation to study these semigroups comes from the work E. Kunz, who proved that a one-dimensional analytically irreducible Noetherian local ring is Gorenstein if and only if its value semigroup is symmetric. In other words, a numerical semigroup is symmetric if and only if the associated semigroup ring is Gorenstein. Symmetric numerical semigroups have odd Frobenius number. A numerical semigroup $S$ is called pseudo-symmetric if $\mathrm{F}(S)$ is even and $\mathrm{PF}(S) = \{ \mathrm{F}(S)/2, \mathrm{F}(S) \}.$ We study a generalization of these notions to the case of $\mathrm{MPD}$-semigroups. Let $S$ be an $\mathrm{MPD}$-semigroup and let $\mathrm{cone}(S)$ denote the rational polyhedral cone of $S.$ Set $\mathcal{H}(S) = (\mathrm{cone}(S) \setminus S) \cap \mathbb{N}^d.$ For a fixed term order $\prec$ on $\mathbb{N}^d$, we define the Frobenius element as $\mathrm{F}(S)_{\substack{\\\prec}} = \max_{\substack{\\\prec}} \mathcal{H}(S).$ Note that in the case of $\mathrm{MPD}$-semigroups, Frobenius elements may not always exist, with respect to any term order. But if there is a term order $\prec$ such that $\mathrm{F}(S)_{\substack{\\\prec}}$ exists and $|\mathrm{PF}(S)|=1,$ then we say that $S$ is a $\prec$-symmetric semigroup. Further, if $\mathrm{PF}(S) = \{ \mathrm{F}(S)_{\substack{\\\prec}}/2, \mathrm{F}(S)_{\substack{\\\prec}} \},$ then we say that $S$ is a $\prec$-pseudo-symmetric semigroup. 

Any affine semigroup $S$ has a unique minimal generating set whose cardinality is known as the embedding dimension of $S$, and it is denoted by $e(S).$ In 1978, Wilf proposed a conjecture related to the Diophantine Frobenius Problem that claims that the inequality 
\[ \mathrm{F}(S)+1 \leq e(S) \cdot |\{ s \in S \mid s < \mathrm{F}(S) \}| \] 
is true for every numerical semigroup. While this conjecture still remains open, a potential extension of Wilf's conjecture to affine semigroups is studied in~\cite{Wilfconjecture}. We prove that $\prec$-symmetric and $\prec$-pseudo-symmetric semigroups satisfy the generalized Wilf's conjecture under suitable assumptions.

In~\cite{moscariello}, A.~Moscariello introduced the notion of a row-factorization ($\mathrm{RF}$) matrices associated to the pseudo-Frobenius elements of a numerical semigroup. He used this object to investigate the type of almost symmetric semigroups of embedding dimension four and prove a conjecture given by T.~Numata in~\cite{numata}, which states that the type of an almost symmetric semigroup of embedding dimension four is at most three. In recent years, $\mathrm{RF}$-matrices have been studied by K.~Eto and J.~Herzog--K.~Watanabe in (\cite{eto2017,etoGeneric,herzogWatanabe}). We extend the definition of $\mathrm{RF}$-matrices of the pseudo-Frobenius elements to the setting of $\mathrm{MPD}$-semigroups. We also give a description of pseudo-Frobenius elements and their $\mathrm{RF}$-matrices in a glued $\mathrm{MPD}$-semigroup. For an affine semigroup $S,$ let $G(S)$ denote the group generated by $S$ in $\mathbb{Z}^d.$ Recall that an affine semigroup $S$ is said to be a gluing if there exists a non-trivial partition of its minimal generating set, $A_1 \amalg A_2,$ and $d \in \langle A_1 \rangle \cap \langle A_2 \rangle$ such that $G(\langle A_1 \rangle) \cap G(\langle A_2 \rangle) = d \mathbb{Z}.$  

Let $k$ be a field and $S = \langle a_1, \ldots, a_n \rangle$ be a finitely generated submonoid of $\mathbb{N}^d.$ Then the semigroup ring $k[S] = k[{\bf t}^{a_1}, \ldots, {\bf t}^{a_n}]$ of $S$ can be represented as a quotient
of a polynomial ring using a canonical surjection $\pi \colon k[x_1,\ldots,x_n] \rightarrow k[S]$ given by $\pi(x_i) = {\bf t}^{a_i}$ for all $i=1,\ldots,n.$ The kernel of this $k$-algebra homomorphism $\pi$, denoted by $I_S$, is a toric ideal, called defining ideal of $k[S],$ and the ring $k[S]$ is called a toric ring. A toric ideal is called generic if it has a minimal generating set consisting of binomials of full support. The notion of genericity of lattice ideal is introduced by I.~Peeva and B.~Sturmfels in~\cite{peeva-sturmfels}. The authors give a minimal free resolution, namely the Scarf complex, for the toric ring when the defining toric ideal is generic. The generic toric ideals are further studied in~\cite{peeva-gasharov}, where the authors prove that  if $I_S$ is generic toric ideal, then it has a unique minimal system of generators of indispensable binomials up to the sign of binomials. In~\cite{etoGeneric}, K.~Eto gave a necessary and sufficient condition for the defining ideal $I_S$ of a numerical semigroup ring $k[S]$ to be generic. For $\mathrm{MPD}$-semigroups, we give a necessary condition for the generic nature of the defining ideal using $\mathrm{RF}$-matrices. In this article we will be giving an overview of results, detailed proofs can be seen in~\cite{op-kriti-ISG}.


\section{Pseudo-Frobenius elements}

Let $S$ be a finitely generated submonoid of $\mathbb{N}^d$, say minimally generated by $a_1, \ldots, a_n \subseteq \mathbb{N}^d.$ Such submonoids are called affine semigroups. Consider the cone of $S$ in $\mathbb{Q}^d_{\geq 0}$,
\[
\mathrm{cone}(S) := \left\lbrace \sum_{i=1}^n \lambda_i a_i  \mid \lambda_i \in \mathbb{Q}_{\geq 0}, i = 1, \ldots, n \right\rbrace 
\]
and set $\mathcal{H}(S) := (\mathrm{cone}(S) \setminus S) \cap \mathbb{N}^d.$ An element $f \in \mathcal{H}(S)$ is called a pseudo-Frobenius element of $S$ if
	$f + s \in S$ for all $s \in S \setminus \{0\}.$ The set of pseudo-Frobenius elements of $S$ is denoted by $\mathrm{PF}(S).$ In particular,
	\[ \mathrm{PF}(S) = \{ f \in \mathcal{H}(S) \mid f + a_j \in S, \text{ for all } j \in [1,n] \}.  \] 


Observe that the set $\mathrm{PF}(S)$ may be empty. Therefore, in this article, we consider such class of rings where $\mathrm{PF}(S)$ is non-empty. 
Let $k$ be a field. The semigroup ring $k[S]$ of $S$ is a $k$-subalgebra of the polynomial ring $k[t_1,\ldots,t_d].$ In other words, $k[S] = k[{\bf t}^{a_1}, \ldots, {\bf t}^{a_n}]$, where ${\bf t}^{a_i} = t_1^{a_{i1}} \cdots t_d^{a_{id}}$ for $a_i = (a_{i1},\ldots,a_{id})$ and for all $i=1,\ldots,n.$ Set $R = k[x_1,\ldots,x_n]$ and define a map $\pi \colon R \rightarrow k[S]$ given by $\pi(x_i) = {\bf t}^{a_i}$ for all $i=1,\ldots,n.$ Set $\deg x_i = a_i$ for all $i=1,\ldots,n.$ Observe that $R$ is a multi-graded ring and that $\pi$ is a degree preserving surjective $k$-algebra homomorphism. We denote by $I_S$ the kernel of $\pi.$ Then $I_S$ is a homogeneous ideal, generated by binomials, called the defining ideal of $S$. Note that a binomial $\phi = \prod_{i=1}^n x_i^{\alpha_i} - \prod_{j=1}^n x_j^{\beta_j} \in I_S$ if and only if $\sum_{i=1}^n \alpha_i a_i = \sum_{j=1}^n \beta_j a_j.$ With respect to this grading, $\deg \phi = \sum_{i=1}^n \alpha_i a_i.$

We say that $S$ satisfies the {\bf maximal projective dimension ($\mathrm{MPD}$)} if $\mathrm{pdim}_R k[S] = n-1.$ Equivalently, $\mathrm{depth}_R k[S]=1.$ In \cite[Theorem 6]{pfelements}, the authors proved that $S$ is a $\mathrm{MPD}$-semigroup if and only if $\mathrm{PF}(S) \neq \emptyset$. In particular, they prove that if $S$ is a $\mathrm{MPD}$-semigroup then $b \in S$ is the $S$-degree of the $(n-2)$th minimal syzygy of $k[S]$ if and only if $ b \in \left\lbrace a + \sum_{i=1}^n a_i  \mid a \in \mathrm{PF}(S)\right\rbrace$. Moreover, $\mathrm{PF}(S)$ has finite cardinality.

\begin{example} Let $S = \langle (2,11), (3,0), (5,9), (7,4) \rangle$. The minimal free resolution of $k[S],$ as a module over $R = k[x_1,\ldots,x_4],$ is 
	\[ 0 \rightarrow R(-(81,93)) \oplus R(-(94,82)) \rightarrow R^6 \rightarrow R^5 \rightarrow R \rightarrow k[S] \rightarrow 0. \]
	In particular, the degrees of minimal generators of the third syzygy modules are $(81,93)$, $(94,82)$. Therefore, $S$ has two pseudo-Frobenius elements $(64,69)$ and $(77,58)$.
\end{example}
Let us recall the definition of gluing \cite[Theorem 1.4]{rosales97}. Let $S \subseteq \mathbb{N}^d$ be an affine semigroup and $G(S)$ be the group spanned by $S$, that is, $G(S) =\{ a-b \in \mathbb{Z}^d \mid a, b \in S\}$. Let $A$ be the minimal generating system of $S$ and $A = A_1 \amalg A_2$ be a nontrivial partition of $A$ . Let $S_i$ be the submonoid of $\mathbb{N}^d$ generated by $A_i, i \in {1, 2}$. Then $S = S_1 + S_2.$ We say that $S$ is the gluing of $S_1$ and $S_2$ by $d$ if $d \in S_1 \cap S_2$ and, $G(S_1 ) \cap G(S_2) = d\mathbb{Z}$. If $S$ is a gluing of $S_1$ and $S_2$ by $d,$ we write $S = S_1 +_d S_2$.

\begin{theorem}\label{PFGluing}
	Let $S$ be an affine semigroup such that $S = S_1 +_d S_2,$ where $S_1$ and $S_2$ are $\mathrm{MPD}$-semigroups. Then $S$ is a $\mathrm{MPD}$-semigroup and
	\begin{align*}
		\mathrm{PF}(S)=\{f+g+d \mid f \in \mathrm{PF}(S_1), \ g\in \mathrm{PF}(S_2) \}.
	\end{align*} 
\end{theorem}

Recall that a gluing two numerical semigroups is defined as follows: Let $H_1$ and $H_2$ be two numerical semigroups minimally generated by $n_1,\ldots,n_r$ and $n_{r+1},\ldots,n_e$ respectively. Let $\lambda \in H_1 \setminus \{ n_1,\ldots,n_r \}$ and $\mu \in H_2 \setminus \{ n_{r+1}, \ldots, n_e \}$ be such that $\gcd(\lambda,\mu)=1.$ We say that $S = \langle \mu n_1, \ldots, \mu n_r, \lambda n_{r+1}, \ldots, \lambda n_{e} \rangle$ is a gluing of $H_1$ and $H_2.$
In other words, if $S$ is as in the definition above, then $S = S_1 +_{\mu\lambda} S_2,$ where $S_1 = \langle \mu n_1, \ldots, \mu n_r \rangle$ and $S_2 = \langle \lambda n_{r+1}, \ldots, \lambda n_{e} \rangle.$ Since $\mathrm{PF}(S_1) = \mu \mathrm{PF}(H_1)$ and $\mathrm{PF}(S_2) = \lambda \mathrm{PF}(H_2)$, the following result now follows from the Theorem \ref{PFGluing}.

\begin{theorem}[{\cite[Proposition 6.6]{nari}}]
	If $S = \langle \mu H_1, \lambda H_2 \rangle$, then
	\[ \mathrm{PF}(S) = \{ \mu f + \lambda g + \mu \lambda \mid f \in \mathrm{PF}(H_1), g \in \mathrm{PF}(H_2) \}. \]
\end{theorem}

On $\mathcal{H}(S),$ we define a relation : ${\bf x} \leq {\bf y} $ if ${\bf y} - {\bf x} \in S.$ It is a partial order (reflexive, anti-symmetric and transitive) on $\mathcal{H}(S)$. In \cite[Proposition 2.19]{numerical}, the authors proved that if $S$ is a numerical semigroup, then $\mathrm{PF}(S) = \text{Maximals}_{\leq} (\mathbb{Z} \setminus S).$ In other words, $x \in \mathbb{Z} \setminus S$ if and only if $f - x \in S$ for some $f \in \mathrm{PF}(S).$ However, in the case of $\mathrm{MPD}$-semigroups over $\mathbb{N}^d,$ $d \geq 2,$ we observe that this result is not true.

\begin{example}	Let $S$ be the semigroup generated by the columns of the following matrix
	\begin{align*} 
		\left(
		\begin{tabular}{cccccccccccc}
			18 & 18 & 4 & 20 & 23 & 8 & 11 & 11 & 10 & 14 & 7 & 7 \\
			9 & 3 & 1 & 8 & 10 & 3 & 5 & 2 & 3 & 3 & 2 & 3 
		\end{tabular}
		\right).
	\end{align*}
	Then $S$ is a $\mathrm{MPD}$-semigroup and $\mathrm{PF}(S) = \{ (13,4) \}$ (see \cite[Example 5]{pfelements}). Observe that $(15,7) \in \mathrm{cone}(S) \setminus S$ but $(13,4) - (15,7)  = (-2,-3) \notin S.$ 
\end{example}

Observe that if $\mathcal{H}(S)$ is a finite set and $\mathcal{H}(S) \neq \emptyset$, then $\mathrm{PF}(S) \neq \emptyset.$ In particular, if $\mathcal{H}(S)$ is finite, then the following result holds.

\begin{theorem}  \label{MaxPseudoFrob}
	Let $\mathcal{H}(S)$ be a non-empty finite set. Then
	\begin{enumerate}
\item $\mathrm{PF}(S) = \mathrm{Maximals}_{\leq} \ \mathcal{H}(S).$
\item Let ${\bf x} \in \mathbb{N}^d.$ Then ${\bf x} \in \mathcal{H}(S)$ if and only if $f-{\bf x} \in S$ for some $f \in \mathrm{PF}(S).$
	\end{enumerate}
		
\end{theorem}


\section{$\prec$-symmetric semigroups}

Let $\prec$ be a term order on $\mathbb{N}^d.$ Then $\mathrm{F}(S)_{\substack{\\\prec}} = \mathrm{max}_{\substack{\\\prec}} \mathcal{H}(S)$, if it exists, is called the Frobenius element of $S$ with respect to the term order $\prec.$ In particular, 
	\[ \mathrm{F}(S) = \{ \mathrm{F}(S)_{\substack{\\\prec}} = \mathrm{max}_{\prec} \mathcal{H}(S) \mid \; \prec \text{ is a term order } \}. \]
We write $\mathrm{F}(S)$ for the set of Frobenius elements of $S.$ Note that Frobenius elements may not exist. However, if $| \mathcal{H}(S) | < \infty$, then Frobenius elements do exist. Also, from \cite[Lemma 12]{pfelements}, we have that every Frobenius element is a pseudo-Frobenius element, \textit{i.e.}, $\mathrm{F}(S) \subseteq \mathrm{PF}(S).$

In the case $\mathrm{F}(S) \neq \emptyset$, we fix a term order $\prec$ such that $\mathrm{F}(S)_{\substack{\\\prec}} = \mathrm{max}_{\substack{\\\prec}} \mathcal{H}(S) \in \mathrm{F}(S).$
	\begin{enumerate}
		\item If $|\mathrm{PF}(S)|=1$ and $\mathrm{PF}(S) = \{\mathrm{F}(S)_{\substack{\\\prec}}\}$, then $S$ is called a {\bf $\prec$-symmetric} semigroup.
		
		\item Put $\mathrm{PF'}(S) = \mathrm{PF}(S) \setminus \{ \mathrm{F}(S)_{\substack{\\\prec}} \}.$ If $\mathrm{PF'}(S) \neq \emptyset$ and if for any $g \in \mathrm{PF'}(S)$, $\mathrm{F}(S)_{\substack{\\\prec}} - g \in \mathrm{PF'}(S)$, we say that $S$ is {\bf $\prec$-almost symmetric}. Further, if $|\mathrm{PF}(S)|=2$, then $S$ is called {\bf $\prec$-pseudo-symmetric}. In this case, $\mathrm{PF}(S) = \{ \mathrm{F}(S)_{\substack{\\\prec}}, \mathrm{F}(S)_{\substack{\\\prec}}/2 \}.$ 
	\end{enumerate}
Observe that when $S$ is $\prec$-pseudo-symmetric, for some term order $\prec,$ then $\mathrm{F}(S)_{\substack{\\\prec}}$ has all even coordinates. Hereafter $\mathrm{F}(S)_{\substack{\\\prec}}$ is even means that $\mathrm{F}(S)_{\substack{\\\prec}}$ has all even coordinates.
\begin{example}
The semigroup $S_1 = \langle (0,1),(3,0),(5,0),(1,3),(2,3) \rangle$ is a $\prec$-symmetric semigroup as $\mathrm{PF}(S_1) = \{(7,2)\}$ and $(7,2) = \mathrm{max}_{\prec} \mathcal{H}(S_1)$, where $\prec$ is a graded lexicographic order.\\
	The semigroup $S_2 = \langle (0,1),(3,0),(4,0),(1,4),(5,0),(2,7) \rangle$ is $\prec$-almost symmetric semigroup of type $2$ as $\mathrm{PF}(S_2) = \{(1,3),(2,6)\}$ and $(2,6) = \mathrm{max}_{\prec} \mathcal{H}(S_2)$, where $\prec$ is a graded lexicographic order. In particular, $S_2$ is a $\prec$-pseudo-symmetric semigroup.
\end{example}

For numerical semigroups, the concept of symmetric and pseudo-symmetric numerical semigroups is also characterized using elements in the gap set, $\mathbb{Z} \setminus S$ (see \cite[Proposition 4.4]{numerical}). In order to attempt such characterization in the case of $\mathrm{MPD}$-semigroups, we assume that $\mathcal{H}(S)$ is a finite set. Let $S$ be a $\mathrm{MPD}$-semigroup. If $\mathcal{H}(S)$ is a non-empty finite set, then $S$ is said to be a $\mathcal{C}$-semigroup, where $\mathcal{C}$ denotes the cone of the semigroup. The concept of $\mathcal{C}$-semigroups is introduced in~\cite{Wilfconjecture} and several properties of these semigroups are investigated in~\cite{cSemigroup}. When $S$ is a $\mathcal{C}$-semigroup, we give a characterization of $\prec$-symmetric and $\prec$-pseudo-symmetric semigroups.

\begin{theorem}\label{precsymmetricity}
	Let $S$ be a $\mathcal{C}$-semigroup and let $\mathrm{F}(S)_{\substack{\\\prec}}$ denote the Frobenius element of $S$ with respect to an order $\prec.$ 
	Then $S$ is a $\prec$-symmetric semigroup if and only if for each $g \in \mathrm{cone}(S) \cap \mathbb{N}^d$ we have: \[ g \in S \text{ if and only if } \mathrm{F}(S)_{\substack{\\\prec}} - g \notin S. \]
\end{theorem}

\begin{theorem}\label{precPseudosymmetricity}
	Let $S$ be a $\mathcal{C}$-semigroup and let $\mathrm{F} (S)_{\substack{\\\prec}}$ denote the Frobenius element of $S$ with respect to an order $\prec.$ Then $S$ is a $\prec$-pseudo-symmetric semigroup if and only if $\mathrm{F}(S)_{\substack{\\\prec}}$ is even, and for each $g \in \mathrm{cone}(S) \cap \mathbb{N}^d$ we have: 
	\[ g \in S \text{ if and only if } \mathrm{F}(S)_{\substack{\\\prec}} - g \notin S \text{ and } g \neq \mathrm{F}(S)_{\substack{\\\prec}}/2. \]
\end{theorem}

\subsection{Extended Wilf's conjecture}
Let $S$ be a $\mathcal{C}$-semigroup. Define the Frobenius number of $S$ as $\mathcal{N}(\mathrm{F}(S)_{\substack{\\\prec}}) = | \mathcal{H}(S)| + | \{ g \in S \mid g \prec \mathrm{F}(S)_{\substack{\\\prec}}\}|.$ Observe that if $S$ is a numerical semigroup, then $\mathcal{N}(\mathrm{F}(S)_{\substack{\\\prec}}) = \mathrm{F}(S),$ the Frobenius number of $S.$ For a numerical semigroup $S,$ Wilf proposed a conjecture related to the Diophantine Frobenius Problem that claims that the inequality 
\[ \mathrm{F}(S)+1 \leq e(S) \cdot |\{ s \in S \mid s < \mathrm{F}(S) \}| \] 
is true. While this conjecture still remains open, a potential extension of Wilf's conjecture to affine semigroups is studied in~\cite{Wilfconjecture}.\\

\begin{conjecture} [{\bf Extension of Wilf's conjecture} {\cite[Conjecture 14]{Wilfconjecture}}]
	Let $S$ be a $\mathcal{C}$-semigroup. The extended Wilf's conjecture is
	\[ | \{ g \in S \mid g \prec \mathrm{F}(S)_{\substack{\\\prec}} \} | \cdot e(S) \geq \mathcal{N}(\mathrm{F}(S)_{\substack{\\\prec}}) + 1, \]
	where $e(S)$ denotes the embedding dimension of $S.$
\end{conjecture}

The following result gives a different characterization of $\prec$-symmetric and $\prec$-pseudo-symmetric $\mathcal{C}$-semigroups when $\mathrm{cone}(S) = \mathbb{N}^d.$ On $\mathrm{cone}(S),$ we define a relation $\leq_c$ as follows: $g \leq_c f$ if $g_i \leq f_i$ for all $i \in [1,d].$

\begin{theorem}\label{precsymmetricCardH(S)}
	Let $S$ be a $\mathcal{C}$-semigroup such that $\mathrm{cone}(S) = \mathbb{N}^d.$ Then
	\begin{enumerate}
		\item $S$ is $\prec$-symmetric if and only if $| \mathcal{H}(S)| = | \{ g \in S \mid g \leq_c \mathrm{F}(S)_{\substack{\\\prec}}\}|.$
		
		\item $S$ is $\prec$-pseudo-symmetric if and only if $| \mathcal{H}(S) \setminus \{ \mathrm{F}(S)_{\substack{\\\prec}}/2 \}| = | \{ g \in S \mid g \leq_c \mathrm{F}(S)_{\substack{\\\prec}}\}|$ and $\mathrm{F}(S)_{\substack{\\\prec}}$ is even. 
	\end{enumerate}  
\end{theorem}
Note that if $S$ is an affine semigroup of $\mathbb{N}^d,$ where $d \geq 2,$ then the semigroup ring $k[S]$ is Cohen--Macaulay when $e(S)=2.$ Since our affine semigroups are $\mathrm{MPD}$, we may assume that $e(S) \geq 3.$

\begin{theorem}  \label{Wilf}
	In the above characterizations of $\prec$-symmetric and $\prec$-pseudo-symmetric semigroups, the extended Wilf's conjecture holds.
\end{theorem}

\section{RF-matrices and generic toric ideals}


Let $S = \langle a_1, \ldots, a_n \rangle$ be a $\mathrm{MPD}$-semigroup in $\mathbb{N}^d,$ minimally generated by  $ a_1, \ldots, a_n.$ We recall the notion of row-factorization matrix ($\mathrm{RF}$-matrix), introduced by A.~Moscariello in~\cite{moscariello}.

\begin{definition} Let $f \in \mathrm{PF}(S).$ An $n \times n$ matrix $M = (m_{ij})$ is an $\mathrm{RF}$-matrix of $f$ if
	$m_{ii} = -1$ for every $i$, $m_{ij} \in \mathbb{N}$ if $i \neq j$ and for every $i = 1, \ldots ,n$, $\sum_{j=1}^n m_{ij} a_j = f.$
\end{definition}	

While $\mathrm{RF}$ matrices were defined for pseudo-Frobenius elements in numerical semigroups, we observe that the above definition holds in the case of pseudo-Frobenius elements in $\mathrm{MPD}$-semigroups over $\mathbb{N}^d.$ Note that an $\mathrm{RF}$-matrix of $f$ need not be uniquely determined. Thus, the notation $\mathrm{RF}(f)$ will denote one of the possible $\mathrm{RF}$-matrices of $f.$

\begin{lem} \label{RFBinomial}
	Let $m_1, \ldots, m_n$ be the row vectors of $\mathrm{RF}(f)$, and set $m_{(ij)} = m_i - m_j$ for all $i, j$ with $1 \leq i < j \leq n.$ Then $\phi_{ij} = {\bf x}^{m_{(ij)}^+} - {\bf x}^{m_{(ij)}^-} \in I_S$ for all $i < j.$
\end{lem}

The binomials $\phi_{ij}$ defined in Lemma {\rm \ref{RFBinomial}} are called $\mathrm{RF}(f)$-relations. We call a binomial relation $\phi \in I_S$ an \textbf{$\mathrm{RF}$-relation} if it is an $\mathrm{RF}(f)$-relation for some $f \in \mathrm{PF}(S)$.

\begin{example}
	Let $S = \langle (2,11),(3,0),(5,9),(7,4) \rangle.$ Then $\mathrm{PF}(S) = \{(64,69),(77,58)\}$ and the $\mathrm{RF}$-matrices $\mathrm{RF}(64,69), \mathrm{RF}(77,58)$ are respectively
	\begin{align*}
		\begin{bmatrix}
    		-1 &  4 & 8 & 2     \\
		    0  &  -1 & 5 & 6   \\
		    6 & 12 & -1 & 3\\
		    5 & 17 & 2 & -1
		\end{bmatrix}
		\text{ and }
		\begin{bmatrix}
		    -1  &  4 & 5 & 6      \\
		    0  &  -1   & 2 & 10  \\
		    5 & 17 & -1  & 3 \\
		    4 & 22 & 2 & -1
		\end{bmatrix}. 
	\end{align*}
	Then $I_S$ is minimally generated by $\mathrm{RF}$-relations as $I_S = \langle \phi_1,\ldots, \phi_5 \rangle,$ where
	\begin{align*}
		\phi_1:= x_2^5x_3^3 &- x_1x_4^4, \ \phi_2:= x_1^6x_2^{13}-x_3^6x_4^3, \ \phi_3:= x_1^5x_2^{18}-x_3^3x_4^7, \\
		&\phi_4:= x_1^4x_2^{23}-x_4^{11}, \ \phi_5:= x_3^9-x_1^7x_2^8x_4.
	\end{align*}
	We see that $\phi_1,\phi_2,\phi_3,\phi_4$ are $\mathrm{RF}(77,58)$-relations and $\phi_5$ is a $\mathrm{RF}(64,69)$-relation.
\end{example}


\begin{lem}\label{RFGluing}
	Let $S = \langle a_1,\ldots,a_n \rangle$ be a $\mathrm{MPD}$-semigroup. Suppose $S = S_1 +_d S_2,$ where $S_1 = \langle a_1,\ldots,a_e \rangle$ and $S_2 = \langle a_{e+1},\ldots,a_n \rangle.$ Let $h \in \mathrm{PF}(S).$ Then by Theorem \ref{PFGluing}, there exist $f \in \mathrm{PF}(S_1)$ and $g \in \mathrm{PF}(S_2)$ such that $h = f+g+d.$ Since $f \in \mathrm{PF}(S_1),$ $f+d \in S_1.$ Write $f+d = \sum_{j=1}^e m_j a_j.$ Similarly, as $g+d \in S_2$, write $g+d = \sum_{j=e+1}^n m_j a_j.$ Hence the matrix
	\begin{align*}
	\left[
		\begin{array}{c | c}
			\mathrm{RF}(f) & B \\ 
			\hline 
			C & \mathrm{RF}(g) 
		\end{array}
		\right] ,
	\end{align*}
	where each row of the matrix $B$ is $(m_{e+1},\ldots,m_n)$ and each row of matrix $C$ is $(m_1,\ldots,m_e)$, serves as an $\mathrm{RF}$-matrix for $h.$
\end{lem}

Let $S = \langle a_1,\ldots,a_n \rangle \subseteq \mathbb{N}^d$ be an affine semigroup and $I_S \subset k[x_1,\ldots,x_n]$ be the defining ideal of the semigroup ring $k[S].$ For a given vector $a  = (a_1,\ldots,a_d) \in \mathbb{N}^d$, the support of $a$ is defined as
\[
\mathrm{supp}(a) = \{ i \mid i \in [1,d],\ a_i \neq 0\}.
\]
For a monomial ${\bf x}^u$, define $\mathrm{supp} ({\bf x}^u) =\mathrm{supp}(u)$ and for a binomial ${\bf x}^u - {\bf x}^v$, define $\mathrm{supp} ({\bf x}^u - {\bf x}^v) =\mathrm{supp}(u) \cup \mathrm{supp}(v)$. In~\cite{peeva-sturmfels}, Peeva and Sturmfels defined that a toric ideal $I_S \subset k[x_1,\ldots,x_n]$ is called \textbf{generic} if it is minimally generated by the binomials of full support. A binomial ${\bf x}^u - {\bf x}^v$ is called indispensable if every system of binomial generators of $I_S$ contains ${\bf x}^u - {\bf x}^v$ or ${\bf x}^v - {\bf x}^u.$  Using \cite[Theorem 3.1]{thomaindispensableArxiv} it follows that if $I_S$ is generic toric ideal, then it has a unique minimal system of generators $B(I_S)$ of indispensable binomials up to the sign of binomials. 

If $x = \sum_{j=1}^n m_ja_j$ is the unique expression for $x$ in $S,$ then we say $x$ has unique factorization in $S.$ In other words, if $x = \sum_{j=1}^n m_ja_j = \sum_{j=1}^n m_j'a_j$ are two factorizations of $x$ in $S,$ then $m_j = m_j'$ for all $j \in [1,n].$ We denote the set of such elements by $\mathrm{UF}(S)$.
We define a partial order $\leq_S$ on $S$ as $ x \leq_S y$ if and only if $y - x \in S$. We denote the set of minimal elements of $S \setminus \mathrm{UF}(S)$ with respect to $\leq_S$ by $\mathrm{min}(S \setminus \mathrm{UF}(S))$. For $0 \neq x \in S,$ we define a subset $\mathrm{Ap}(S, x)$ of $S$ relative to $x$ as
	\begin{align*}
		\mathrm{Ap}(S, x) = \{ y \in S \mid y-x \in \mathcal{H}(S) \}.
	\end{align*}

\begin{theorem}
Let $S = \langle a_1,\ldots,a_n \rangle$ be a $\mathrm{MPD}$-semigroup. If $ \bigcup_{j=1}^n \mathrm{Ap}(S,a_j) \subseteq \mathrm{UF}(S),$ then $\mathrm{RF}(f)$ is unique for all $f \in \mathrm{PF}(S)$. Moreover, if $\mathcal{H}(S)$ is finite, then the converse is also true.
\end{theorem}

\begin{theorem}\label{genericRFunique}
Let $S$ be a $\mathrm{MPD}$-semigroup. If $I_S$ is generic, then $\mathrm{RF}(f) = (m_{ij})$ is unique for each $f \in \mathrm{PF}(S)$ and $m_{ij} \neq m_{i'j}$ for all $i \neq i'$.
\end{theorem}

\begin{example} Let $S = \langle (20,0), (24,1), (1,25), (0,31) \rangle.$ Then
	\begin{align*}
		\mathrm{PF}(S) =
		\left\lbrace
		\begin{array}{c}
			(223,4445), (271,3145), (319,1845), (559,1256), (799,667), \\ 
			(1375,567), (1951,467), (2527,367), (3103,267)
		\end{array}
		\right\rbrace,
	\end{align*} 
	and $I_S$ is generated by 
\begin{align*}
	\left\lbrace
	\begin{array}{c}
	x_2^{24}x_3^4-x_1^{29}x_4^4, \ x_1^{12}x_3^{24}-x_2^{11}x_4^{19}, \ x_2^{13}x_3^{28}-x_1^{17}x_4^{23}, \ x_1^{41}x_3^{20}-x_2^{35}x_4^{15}, \ x_2^2x_3^{52}-x_1^5x_4^{42}, \\[2mm] x_1^{70}x_3^{16}-x_2^{59}x_4^{11}, \ x_1^7x_3^{76}-x_2^9x_4^{61}, \ x_1^{99}x_3^{12}-x_2^{83}x_4^7, \ x_1^{128}x_3^8-x_2^{107}x_4^3, \ x_2^{131}-x_1^{157}x_3^4x_4, \\[2mm] x_1^2x_3^{128}-x_2^7x_4^{103}, \ x_3^{180}-x_1^3x_2^5x_4^{145}
	\end{array}
	\right\rbrace.
\end{align*}
Hence, $I_S$ is generic and $\mathrm{RF}$-matrices for the elements of $\mathrm{PF}(S)$ are
\begin{align*}
		\begin{bmatrix}
    		-1 &  8 & 51 & 102     \\
		    6  &  -1 & 127 & 41   \\
		    4 & 6 & -1 & 144\\
		    1 & 1 & 179 & -1
		\end{bmatrix},
		\begin{bmatrix}
		    -1  &  10 & 51 & 60      \\
		    11  &  -1   & 75 & 41  \\
		    4 & 8 & -1  & 102 \\
		    6 & 1 & 127 & -1
		\end{bmatrix},
		 \begin{bmatrix}
    		-1 &  12 & 51 & 18     \\
		    16  &  -1 & 23 & 41   \\
		    4 & 10 & -1 & 60\\
		    11 & 1 & 75 & -1
		\end{bmatrix},
		\end{align*} 	
	\begin{align*}
		\begin{bmatrix}
			-1 &  23 & 27 & 18     \\
			28  &  -1 & 23 & 22   \\
			16 & 10 & -1 & 41\\
			11 & 12 & 51 & -1
		\end{bmatrix},
		\begin{bmatrix}
    		-1 &  34 & 3 & 18     \\
		    40  &  -1 & 23 & 3  \\
		    28 & 10 & -1 & 22\\
		    11 & 23 & 27 & -1
		\end{bmatrix},
        \begin{bmatrix}
    		-1 &  58 & 3 & 14     \\
		    69  &  -1 & 19 & 3   \\
		    28 & 34 & -1 & 18\\
		    40 & 23 & 23 & -1
		\end{bmatrix},
		\begin{bmatrix}
		    -1  &  82 & 3 & 10      \\
		    98  &  -1   & 15 & 3  \\
		    28 & 58 & -1  & 14 \\
		    69 & 23 & 19 & -1
		\end{bmatrix},
		\end{align*} 
		\begin{align*}
		\begin{bmatrix}
			-1 &  106 & 3 & 6     \\
			127  &  -1 & 11 & 3   \\
			28 & 82 & -1 & 10\\
			98 & 23 & 15 & -1
		\end{bmatrix}
		\text{ and }
		\begin{bmatrix}
    		-1 &  130 & 3 & 2     \\
		    156  &  -1 & 7 & 3   \\
		    28 & 106 & -1 & 6\\
		    127 & 23 & 11 & -1
		\end{bmatrix} \text{respectively}.
	\end{align*}
Moreover, these matrices are unique and no two entries in a column of each matrix are same.
\end{example}	

Using Theorem \ref{PFGluing}, Lemma \ref{RFGluing} and Theorem \ref{genericRFunique}, we prove the following result:
\begin{theorem}
Let $n \geq 3$ and $S = \langle a_1,\ldots,a_n \rangle$ be a gluing of $\mathrm{MPD}$-semigroups. Then $I_S$ is not generic.
\end{theorem}

\acknowledgements{Experiments with the computer algebra softwares \textsc{Macaulay2}~\cite{M2} and \textsc{GAP}~\cite{GAP4} have provided numerous valuable insights.
}

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