#  (weakly) $(\alpha,\beta)$-prime hyperideals in commutative multiplicative
hyperrings

Mahdi Anbarloei  Department of Mathematics, Faculty of Sciences, Imam Khomeini
International University, Qazvin, Iran<EMAIL_ADDRESS>
###### Abstract.

Let $H$ be a commutative multiplicative hyperring and
$\alpha,\beta\in\mathbb{Z}^{+}$. A proper hyperideal $P$ of $H$ is called
(weakly) $(\alpha,\beta)$-prime if ($0\notin x^{\alpha}\circ y\subseteq P$ )
$x^{\alpha}\circ y\subseteq P$ for $x,y\in H$ implies $x^{\beta}\subseteq P$
or $y\in P$. In this paper, we aim to investigate (weakly)
$(\alpha,\beta)$-prime hyperideals and then we present some properties of
them.

###### Key words and phrases:

$(\alpha,\beta)$-prime, weakly $(\alpha,\beta)$-prime, $(\alpha,\beta)$-zero.

###### 2010 Mathematics Subject Classification:

Primary 20N20; Secondary 16Y99

##  1\. Introduction

Delving into the study of prime ideal generalizations has emerged as a
profoundly intriguing and groundbreaking pursuit in the realm of commutative
ring theory. In a recent study [22], Khashan and Celikel presented
$(\alpha,\beta)$-prime ideals which is an intermediate class between prime
ideals and $(\alpha,\beta)$-closed ideals. Let
$\alpha,\beta\in\mathbb{Z}^{+}$. A proper ideal $I$ of a commutative ring $R$
refers to an $(\alpha,\beta)$-prime ideal if for $x,y\in R$, $x^{\alpha}y\in
I$ implies either $x^{\beta}\in I$ or $y\in I$. Moreover, the authors
generalized this notion to weakly $(\alpha,\beta)$-prime ideals in [23]. A
proper ideal $I$ of $R$ is called a weakly $(\alpha,\beta)$-prime ideal if for
$x,y\in R$ with $0\neq x^{\alpha}y\in I$, then either $x^{\beta}\in I$ or
$y\in I$.

Several key concepts in modern algebra were expanded by extending their
underlying structures to hyperstructures. In 1934 [24], the French
mathematician F. Marty pioneered the notion of hyperstructures or
multioperations, where an operation yields a set of values rather than a
single value. Subsequently, numerous authors have contributed to the
advancement of this novel area of modern algebra [12, 13, 14, 15, 16, 25, 29,
35]. An importan type of the algebraic hyperstructures called the
multiplicative hyperring was introduced by Rota in 1982 [31]. In this
hyperstructure, the multiplication is a hyperoperation and the addition is an
operation. Multiplicative hyperrings are richly demonstrated and characterized
in [3, 4, 5, 21, 27, 28, 30, 34]. Dasgupta studied the prime and primary
hyperideals in multiplicative hyperrings in [17]. The idea of
$(\alpha,\beta)$-closed hyperideals in a multiplicative hyperring was proposed
in [6]. A proper hyperideal $P$ of a multiplicative hyperring $H$ is said to
be $(\alpha,\beta)$-closed if for $x\in H$ with $x^{\alpha}\subseteq P$, then
$x^{\beta}\subseteq P$. Motivated from this notion, the aim of this research
work is to introduce and study the notion of $(\alpha,\beta)$-prime
hyperideals in a commutative multiplicative hyperring. Several specific
results are given to illustrate the structures of the new notion. Furthermore,
we extend this notion to weakly $(\alpha,\beta)$-prime hyperideals. We present
some characterizations of weakly $(\alpha,\beta)$-prime hyperideals on
cartesian product of commutative multiplicative hyperring.

##  2\. Some basic definitions concerning multiplicative hyperrings

In this section we give some basic definitions and results which we need to
develop our paper. [19] A hyperoperation $``\circ"$ on nonempty set $I$ is a
mapping from $I\times I$ into $P^{*}(I)$ where $P^{*}(I)$ is the family of all
nonempty subsets of $I$. If $``\circ"$ is a hyperoperation on $I$, then
$(I,\circ)$ is called hypergroupoid. The hyperoperation on $I$ can be extended
to subsets of $I$. Let $I_{1},I_{2}$ be two subsets of $I$ and $a\in I$, then
$I_{1}\circ I_{2}=\cup_{a_{1}\in I_{1},a_{2}\in I_{2}}a_{1}\circ a_{2},$ and
$I_{1}\circ a=I_{1}\circ\\{a\\}.$ A hypergroupoid $(I,\circ)$ is called a
semihypergroup if $\cup_{v\in b\circ c}a\circ v=\cup_{u\in a\circ b}u\circ c$
for all $a,b,c\in I$ which means $\circ$ is associative. A semihypergroup $I$
is said to be a hypergroup if $a\circ I=I=I\circ a$ for all $a\in I$. A
nonempty subset $J$ of a semihypergroup $(I,\circ)$ is called a subhypergroup
if we have $a\circ J=J=J\circ a$ for all $a\in J$.

######  Definition 2.1.

[19] An algebraic structure $(H,+,\circ)$ is said to be commutative
multiplicative hyperring if

  * (1)

$(H,+)$ is a commutative group;

  * (2)

$(H,\circ)$ is a semihypergroup;

  * (3)

for all $a,b,c\in H$, we have $a\circ(b+c)\subseteq a\circ b+a\circ c$ and
$(b+c)\circ a\subseteq b\circ a+c\circ a$;

  * (4)

for all $a,b\in H$, we have $a\circ(-b)=(-a)\circ b=-(a\circ b)$;

  * (5)

for all $a,b\in H$, $a\circ b=b\circ a$.

If in (3) the equality holds then we say that the multiplicative hyperring is
strongly distributive. Let $(\mathbb{Z},+,\cdot)$ be the ring of integers.
Corresponding to every subset $X\in P^{\star}(\mathbb{Z})$ with $|X|\geq 2$,
there exists a multiplicative hyperring $(\mathbb{Z}_{X},+,\circ)$ with
$\mathbb{Z}_{X}=\mathbb{Z}$ and for any $a,b\in\mathbb{Z}_{X}$, $a\circ
b=\\{a.x.b\ |\ x\in X\\}$ [17].

######  Definition 2.2.

[1] An element $e$ in $H$ is an identity element if $a\in a\circ e$ for all
$a\in H$. Moreover, an element $e$ in $H$ is a scalar identity element if
$a=a\circ e$ for all $a\in H$.

Throughout this paper, $H$ denotes a commutative multiplicative hyperring with
identity 1.

######  Definition 2.3.

[19] A non-empty subset $A$ of $H$ is a hyperideal if

  * (i)

$a-b\in A$ for all $a,b\in A$;

  * (ii)

$r\circ a\subseteq A$ for all $a\in A$ and $r\in H$.

######  Definition 2.4.

[17] A proper hyperideal $A$ of $H$ is called a prime hyperideal if $a\circ
b\subseteq A$ for $a,b\in H$ implies that $a\in A$ or $b\in A$.

The intersection of all prime hyperideals of $H$ containing a hyperideal $A$
is called the prime radical of $A$, denoted by $rad(A)$. If the multiplicative
hyperring $H$ does not have any prime hyperideal containing $A$, we define
$rad(A)=H$. Let C be the class of all finite products of elements of $H$ that
is ${\bf C}=\\{r_{1}\circ r_{2}\circ...\circ r_{n}\ :\ r_{i}\in
G,n\in\mathbb{N}\\}\subseteq P^{\ast}(H)$. A hyperideal $A$ of $H$ is said to
be a C-hyperideal of $H$ if, for any $J\in{\bf C},A\cap J\neq\varnothing$
implies $J\subseteq A$. Let $A$ be a hyperideal of $H$. Then, $D\subseteq
rad(A)$ where $D=\\{r\in H\ |\ r^{n}\subseteq A\ for\ some\
n\in\mathbb{N}\\}$. The equality holds when $A$ is a C-hyperideal of $H$ (see
Proposition 3.2 in [17]). Recall that a hyperideal $A$ of $H$ is called a
strong C-hyperideal if for any $E\in\mathfrak{U}$, $E\cap A\neq\varnothing$,
then $E\subseteq A$, where $\mathfrak{U}=\\{\sum_{i=1}^{n}J_{i}\ |\
J_{i}\in{\bf C},n\in\mathbb{N}\\}$ and ${\bf C}=\\{r_{1}\circ
r_{2}\circ...\circ r_{n}\ |\ r_{i}\in H,n\in\mathbb{N}\\}$ (for more details
see [18]).

######  Definition 2.5.

[1] A proper hyperideal $A$ of $H$ is maximal in $H$ if for any hyperideal $B$
of $H$ with $A\subset B\subseteq H$ then $B=H$.

Also, we say that $H$ is a local multiplicative hyperring, if it has just one
maximal hyperideal.

######  Definition 2.6.

[1] Assume that $A$ and $B$ are hyperideals of $H$. We define $(A:B)=\\{x\in
H\ |\ x\circ B\subseteq A\\}$.

##  3\.  $(\alpha,\beta)$-prime hyperideals

In this section, we introduce the concept of $(\alpha,\beta)$-prime hyperideal
and give some basic properties of them.

######  Definition 3.1.

Let $\alpha,\beta\in\mathbb{Z}^{+}$. A proper hyperideal $P$ of $H$ is called
$(\alpha,\beta)$-prime if $x^{\alpha}\circ y\subseteq P$ for $x,y\in H$
implies $x^{\beta}\subseteq P$ or $y\in P$.

######  Example 3.2.

Consider the multiplicative hyperring $\mathbb{Z}_{A}$. Let $p$ is a prime
integer such that $A\cap\langle p\rangle=\varnothing$. Then $\langle p\rangle$
is an $(\alpha,\beta)$-prime for all $\alpha,\beta\in\mathbb{Z}^{+}$.

######  Remark 3.3.

Let $\alpha,\beta\in\mathbb{Z}^{+}$.

  * (i)

Every $(\alpha,\beta)$-prime hyperideal of $H$ is $(\alpha,\beta)$-closed.

  * (ii)

Every $(\alpha,\beta)$-prime hyperideal of $H$ is primary.

The following example shows that the converse part of statements in Remark 3.3
may not be true, in general.

######  Example 3.4.

Consider the multiplicative hyperring $(\mathbb{Z}_{X},+,\circ)$.

  * (i)

Let $X=\\{2,3\\}$. Then $P=\langle 6\rangle$ is an $(3,2)$-closed hyperideal
of $\mathbb{Z}$, but it is not $(3,2)$-prime as $2^{3}\circ 3\subseteq P$
where neither $2^{2}=\\{8,12\\}\nsubseteq P$ nor $3\notin P$.

  * (ii)

Let $X=\\{2,4\\}$. Then $P=\langle 8\rangle$ is a primary hyperideal of the
multiplicative hyperring $(\mathbb{Z},+,\circ)$. However, it is not
$(4,3)$-prime as the fact that $1^{4}\circ 4\subseteq P$ but
$1^{3}=\\{4,8,16\\}\nsubseteq P$ and $4\notin P$.

Our first theorem presents a characterization of $(\alpha,\beta)$-prime
hyperideals

######  Theorem 3.5.

Assume that $P\neq H$ is a hyperideal of $H$ and
$\alpha,\beta\in\mathbb{Z}^{+}$. Then the following are equivalent:

  * (i)

$P$ is an $(\alpha,\beta)$-prime hyperideal in $H$.

  * (ii)

$P=(P:x^{\alpha})$ such that $x^{\beta}\nsubseteq P$ for $x\in H$.

  * (iii)

If $x^{\alpha}\circ P^{\prime}\subseteq P$ for some hyperideal $P^{\prime}$ of
$H$ and $x\in H$, then $x^{\beta}\subseteq P$ or $P^{\prime}\subseteq P$.

###### Proof.

(i) $\Longrightarrow$ (ii) Suppose that $x^{\beta}\nsubseteq P$ for $x\in H$.
Take any $y\in(P:x^{\alpha})$. So we have $x^{\alpha}\circ y\subseteq P$.
Since $P$ is an $(\alpha,\beta)$-prime hyperideal in $H$ and
$x^{\beta}\nsubseteq P$, we get $y\in P$ which means $(P:x^{\alpha})\subseteq
P$. Since the inclusion $P\subseteq(P:x^{\alpha})$ always holds, we obtain
$P=(P:x^{\alpha})$.

(ii) $\Longrightarrow$ (iii) Let $P^{\prime}$ be a hyperideal of $H$ and $x\in
H$ such that $x^{\alpha}\circ P^{\prime}\subseteq P$. If $x^{\beta}\subseteq
P$, we are done. If $x^{\beta}\nsubseteq P$, we get the result that
$(P:x^{\alpha})=P$ by (ii), and so $P^{\prime}\subseteq P$.

(iii) $\Longrightarrow$ (i) Let $x^{\alpha}\circ y\subseteq P$ for $x,y\in H$.
Then $\langle x^{\alpha}\rangle\circ\langle y\rangle\subseteq\langle
x^{\alpha}\circ y\rangle\subseteq P$ by Proposition 2.15 in [17]. Put $\langle
y\rangle=P^{\prime}$. Hence we have $x^{\alpha}\circ P^{\prime}\subseteq P$.
From (iii) it follows that either $x^{\beta}\subseteq P$ or $y\in
P^{\prime}\subseteq P$, as needed. ∎

Recall from [17] that a hyperideal $A$ of $H$ refers to a principal hyperideal
if $A=\langle x\rangle$ for $x\in H$. A hyperring whose every hyperideal is
principal is called principal hyperideal hyperring.

######  Proposition 3.6.

Let $P\neq H$ be a hyperideal of a principal hyperideal hyperring $H$ and
$\alpha,\beta\in\mathbb{Z}^{+}$. Then the following statements are equivalent:

  * (i)

$P$ is an $(\alpha,\beta)$-prime hyperideal in $H$.

  * (ii)

If $P_{1}^{\alpha}\circ P_{2}\subseteq P$ for hyperideals $P_{1},P_{2}$ of
$H$, then $P_{1}^{\beta}\subseteq P$ or $P_{2}\subseteq P$.

  * (iii)

$P=(P:P_{1}^{\alpha})$ such that $P_{1}^{\beta}\nsubseteq P$ for every
hyperideal $P_{1}$ of $H$.

  * (iv)

If $P_{1}^{\alpha}\circ y\subseteq P$ where $y\in H$ and $P_{1}$ is a
hyperideal of $H$, then $P_{1}^{\beta}\subseteq P$ or $y\in P$.

###### Proof.

(i) $\Longrightarrow$ (ii) Let $P_{1}^{\alpha}\circ P_{2}\subseteq P$ for
hyperideals $P_{1},P_{2}$ of $H$. Since $H$ is a principal hyperideal
hyperring, there exists $x\in H$ such that $P_{1}=\langle x\rangle$ and so
$x^{\alpha}\circ P_{2}\subseteq P$. Since $P$ is an $(\alpha,\beta)$-prime
hyperideal in $H$, by Theorem 3.5 we conclude that $x^{\beta}\subseteq P$
which means $P_{1}^{\beta}=\langle x\rangle^{\beta}\subseteq\langle
x^{\beta}\rangle\subseteq P$ or $P_{2}\subseteq P$.

(ii) $\Longrightarrow$ (iii) Let $y\in(P:P_{1}^{\alpha})$ and
$P_{1}^{\beta}\nsubseteq P$ for a hyperideal $P_{1}$ of $H$. Then $\langle
y\rangle\subseteq(P:P_{1}^{\alpha})$ as $(P:P_{1}^{\alpha})$ is a hyperideal
of $H$ by Theoremm 3.8 in [1], and so $P_{1}^{\alpha}\circ\langle
y\rangle\subseteq P$. By the hypothesis, we get $y\in\langle y\rangle\subseteq
P$ which implies $(P:P_{1}^{\alpha})\subseteq P$. The other containment is
clear.

(iii) $\Longrightarrow$ (iv) Let $P_{1}^{\alpha}\circ y\subseteq P$ and
$P_{1}^{\beta}\nsubseteq P$. Then we have $y\in(P:P_{1}^{\alpha})=P$, as
required.

(iv) $\Longrightarrow$ (i) Let $x^{\alpha}\circ y\subseteq P$ for $x,y\in H$.
We assume that $P_{1}=\langle x\rangle$. Hence we have $P_{1}^{\alpha}\circ
y\subseteq\langle x\rangle^{\alpha}\circ\langle y\rangle\subseteq\langle
x^{\alpha}\circ y\rangle\subseteq P$. By the assumption, we get the result
that $x^{\beta}\subseteq P_{1}^{\beta}\subseteq P$ or $y\in P$. Consequently,
$P$ is an $(\alpha,\beta)$-prime hyperideal in $H$. ∎

By Remark 3.3, every $(\alpha,\beta)$-prime $\mathcal{C}$-hyperideal of $H$ is
a primary hyperideal and so its radical is a prime hyperideal of $H$ by
Proposition 3.6 in [17]. Let $P$ be an $(\alpha,\beta)$-prime
$\mathcal{C}$-hyperideal of $H$. Then $P$ is referred as a
$Q$-$(\alpha,\beta)$-prime $\mathcal{C}$-hyperideal of $H$ where $rad(P)=Q$.

######  Theorem 3.7.

If $P$ is a $Q$-$(\alpha,\beta)$-prime $\mathcal{C}$-hyperideal of $H$ for
$\alpha,\beta\in\mathbb{Z}^{+}$, then $Q=\\{x\in H\ |\ x^{\beta}\subseteq
P\\}$.

###### Proof.

The inclusion $\\{x\in H\ |\ x^{\beta}\subseteq P\\}\subseteq Q$ holds. Assume
that $x\in Q=rad(P)$ and $n$ is the smallest positive integer with
$x^{n}\subseteq P$. Then we have $x^{\alpha}\circ x^{n-1}\subseteq P$. Take
any $y\in x^{n-1}$. Since $x^{\alpha}\circ y\subseteq P$ and $P$ is an
$(\alpha,\beta)$-prime hyperideal of $H$, we get $x^{\beta}\subseteq P$ or
$y\in P$. Let $y\in P$. Since $P$ is a $\mathcal{C}$-hyperideal of $H$ and
$y\in x^{n-1}$, we get the result that $x^{n-1}\subseteq P$, a contradiction.
Therefore we obtain $x^{\beta}\subseteq P$. Then $Q\subseteq\\{x\in H\ |\
x^{\beta}\subseteq P\\}$, so this completes the proof. ∎

######  Theorem 3.8.

Let $P$ be a $\mathcal{C}$-hyperideal of $H$ with an $i$-set and
$\alpha,\beta\in\mathbb{Z}^{+}$ such that $Q=\\{x\in H\ |\ x^{\beta}\subseteq
P\\}$ is a maximal hyperideal of $H$. Then $P$ is an
$Q$-$(\alpha,\beta)$-prime hyperideal.

###### Proof.

Assume that $x^{\alpha}\circ y\subseteq P$ with $x^{\beta}\nsubseteq P$ for
$x,y\in H$. Let $x^{\alpha}\subseteq Q$. Since $Q$ is a maximal hyperideal of
$H$, we get $x\in Q$ by Proposition 2.18 in [17]. This contradict by
$x^{\beta}\nsubseteq P$. Then $x^{\alpha}\nsubseteq Q$. Let $a\in x^{\alpha}$
such that $a\notin Q$. Then we have $\langle a,Q\rangle=H$. So, there exists
$m\in Q$ such that $1\in\langle a\rangle+m\subseteq\langle
x^{\alpha}\rangle+m$. Therefore $1\in(\langle
x^{\alpha}\rangle+m)^{\beta}\subseteq\Sigma_{i=0}^{\beta}\tbinom{\beta}{i}\langle
x^{\alpha}\rangle^{\beta-i}\circ m^{i}$ and so $y\in 1\circ
y\subseteq(\Sigma_{i=0}\tbinom{\beta}{i}\langle
x^{\alpha}\rangle^{\beta-i}\circ m^{i})\circ y\subseteq P$. Hence $P$ is an
$(\alpha,\beta)$-prime hyperideal. By Theorem 3.7, we conclude that $P$ is an
$Q$-$(\alpha,\beta)$-prime hyperideal. ∎

As an immediate consequence of the previous theorem, we have the following
result.

######  Corollary 3.9.

Let $Q$ is a maximal hyperideal of $H$, $\alpha,\beta\in\mathbb{Z}^{+}$ and
$n\leq\beta$. Then $Q^{n}$ is an $Q$-$(\alpha,\beta)$-prime hyperideal.

######  Theorem 3.10.

Assume that $P_{1},\cdots,P_{n}$ are $\mathcal{C}$-hyperideals of $H$ such
that for every $i\in\\{1,\cdots,n\\}$, $P_{i}$ is a
$Q$-$({\alpha}_{i},{\beta}_{i})$-prime hyperideal with
${\alpha}_{i},{\beta}_{i}\in\mathbb{Z}^{+}$. Then $\bigcap_{i=1}^{n}P_{i}$ is
a $Q$-$(\alpha,\beta)$-prime hyperideal in $H$ where
$\alpha\leq\min\\{{\alpha}_{1},\cdots,{\alpha}_{n}\\}$ and
$\beta\geq\max\\{{\beta}_{1},\cdots,{\beta}_{n}\\}$.

###### Proof.

Suppose that $x^{\alpha}\circ y\subseteq\bigcap_{i=1}^{n}P_{i}$ for $x,y\in H$
such that $y\notin\bigcap_{i=1}^{n}P_{i}$. This means $y\notin P_{t}$ for some
$t\in\\{1,\cdots,n\\}$. Since $P_{t}$ is a
$Q$-$({\alpha}_{t},{\beta}_{t})$-prime hyperideal in $H$ and
$x^{{\alpha}_{t}}\circ y\subseteq P_{t}$, we conclude that
$x^{{\beta}_{t}}\subseteq P_{t}$ which implies $x\in Q$. Then we get
$x^{{\beta}_{i}}\subseteq P_{i}$ for each $i\in\\{1,\cdots,n\\}$ by Theorem
3.7, and so $x^{\beta}\subseteq\bigcap_{i=1}^{n}P_{i}$ where
$\beta\geq\max\\{{\beta}_{1},\cdots,{\beta}_{n}\\}$. On the other hand, we
have $rad(\bigcap_{i=1}^{n}P_{i})=\bigcap_{i=1}^{n}rad(P_{i})=Q$ by
Proposition 3.3 in [17]. Consequently, $\bigcap_{i=1}^{n}P_{i}$ is a
$Q$-$(\alpha,\beta)$-prime hyperideal in $H$. ∎

Let $\mathfrak{C}(P)=\\{(\alpha,\beta)\in\mathbb{Z}^{+}\times\mathbb{Z}^{+}\
|\ P\text{ is ($\alpha$,$\beta$)-closed }\\}$ where $P$ is a proper hyperideal
of $H$. Then we get $\\{(\alpha,\beta)\in\mathbb{Z}^{+}\times\mathbb{Z}^{+}\
|\
1\leq\alpha\leq\beta\\}\subseteq\mathfrak{C}(P)\subseteq\mathbb{Z}^{+}\times\mathbb{Z}^{+}$.
Furthermore, $rad(P)=P$ if and only if
$\mathfrak{C}(P)=\mathbb{Z}^{+}\times\mathbb{Z}^{+}$ [6]. Now, let us define
$\mathfrak{L}(P)=\\{(\alpha,\beta)\in\mathbb{Z}^{+}\times\mathbb{Z}^{+}\ |\
P\text{ is ($\alpha$,$\beta$)-prime }\\}$. Let
$\mathfrak{L}(P)=\mathbb{Z}^{+}\times\mathbb{Z}^{+}$. Clearly, the hyperideal
$P$ is prime if and only if $(1,1)\in\mathfrak{L}(P)$.

######  Remark 3.11.

Let $P$ be a proper hyperideal of $H$ and $\alpha,\beta\in\mathbb{Z}^{+}$. If
$(\alpha,\beta)\in\mathfrak{L}(P)$, then
$(\alpha^{\prime},\beta^{\prime})\in\mathfrak{L}(P)$ for
$\alpha^{\prime},\beta^{\prime}\in\mathbb{Z}^{+}$ such that
$\alpha^{\prime}\leq\alpha$ and $\beta\leq\beta^{\prime}$ by being $P$ a
hyperideal.

######  Remark 3.12.

Suppose that $P$ is a proper hyperideal of $H$ and
$\alpha,\beta,\theta\in\mathbb{Z}^{+}$. If $(\alpha,\beta)\in\mathfrak{L}(P)$
and $(\beta,\theta)\in\mathfrak{C}(P)$, then
$(\alpha,\theta)\in\mathfrak{L}(P)$.

######  Theorem 3.13.

Assume that $P$ is a proper hyperideal of $H$ and
$\alpha,\beta\in\mathbb{Z}^{+}$. Then $(\alpha,\beta)\in\mathfrak{L}(P)$ if
and only if $(\alpha+1,\beta)\in\mathfrak{L}(P)$.

###### Proof.

$(\Longrightarrow)$ Let $(\alpha,\beta)\in\mathfrak{L}(P)$. Assume that
$x^{\alpha+1}\circ y\subseteq P$ for $x,y\in H$ such that $y\notin P$. Since
$\alpha+1\leq 2\alpha$, we have $(x^{2})^{\alpha}\circ y\subseteq P$. Since
$(\alpha,\beta)\in\mathfrak{L}(P)$ and $y\notin P$, we get
$x^{2\beta}\subseteq P$ which means $x\in rad(P)$. By Theorem 3.7, we conclude
that $x^{\alpha}\subseteq P$. $(\Longleftarrow)$ It follows from Remark 3.11.
∎

######  Theorem 3.14.

Let the zero hyperideal of $H$ be a $\mathcal{C}$-hyperideal and
$\alpha,\beta\in\mathbb{Z}^{+}$. Then every proper hyperideal of $H$ is
$(\alpha,\beta)$-prime if and only if $H$ has no non-trivial idempotents,
every prime $\mathcal{C}$-hyperideal of $H$ is maximal.

###### Proof.

Let every proper hyperideal of $H$ be $(\alpha,\beta)$-prime. Assume that $e$
is a non-trivial idempotent in $H$. Then $0\in e\circ(e-1)$ and so $0\in
e^{\alpha}\circ(e-1)$. Since the zero hyperideal of $H$ is a
$\mathcal{C}$-hyperideal, we have $e^{\alpha}\circ(e-1)=0$. Therefore we get
$e\in e^{\beta}=0$ as the zero hyperideal of $H$ is $(\alpha,\beta)$-prime and
$e\neq 1$. This is a contradiction and so $H$ has no non-trivial idempotents.
Suppose that $P$ is a prime $\mathcal{C}$-hyperideal that is not maximal. Then
we have $P\subset Q$ for some hyperideal $Q$ of $H$. Let $x\in Q-P$. Then
$x^{\alpha}\circ x\subseteq\langle x^{\alpha+1}\rangle$. This follows that
$x^{\beta}\subseteq\langle x^{\alpha+1}\rangle$ or $x\in\langle
x^{\alpha+1}\rangle$. This implies that $x^{\beta}\subseteq x^{\alpha+1}\circ
a$ for some $a\in H$ or $x\in x^{\alpha+1}\circ b$ for some $b\in H$. In the
first case, $0\in x^{\beta}-x^{\alpha+1}\circ a$. Since $P$ is a
$\mathcal{C}$-hyperideal, we get $x^{\beta}-x^{\alpha+1}\circ a\subseteq P$.
From $x^{\beta}\circ(1-x^{\alpha-\beta+1}\circ r)\subseteq
x^{\beta}-x^{\alpha+1}\circ a$ it follows that
$x^{\beta}\circ(1-x^{\alpha-\beta+1}\circ r)\subseteq P$. Since $P$ is a prime
hyperideal and $x^{\beta}\nsubseteq P$, we get $1-x^{\alpha-\beta+1}\circ
r\subseteq P\subset Q$ and so $1\in Q$, a contradiction. In the second case,
we get $0\in(x-x^{\alpha+1}\circ b)\cap P$. Then we get the result that
$x\circ(1-x^{\alpha}\circ b)\subseteq x-x^{\alpha+1}\circ b\subseteq P$. Thus
we have $1-x^{\alpha}\circ b\subseteq P\subset Q$ because $x\nsubseteq P$.
This follows that $1\in Q$ which is a contradiction. Consequently, every prime
$\mathcal{C}$-hyperideal of $H$ is maximal. ∎

We say that a hyperideal $P$ of $H$ is of maximum length $\beta$ if for every
ascending chain $P=P_{0}\subseteq P_{1}\subseteq P_{2}\subseteq\cdots$ of
hyperideals of $H$, $\beta$ is the largest integer with
$P_{\beta}=P_{\beta+1}=\cdots$.

######  Theorem 3.15.

Assume that $\alpha,\beta\in\mathbb{Z}^{+}$ and $P$ is a strong
$\mathcal{C}$-hyperideal of $H$ of maximum length $\beta$. If $P$ is
irreducible in $H$, then $P$ is an $(\alpha,\beta)$-prime hyperideal.

###### Proof.

Let $x^{\alpha}\circ y\subseteq P$ for $x,y\in H$. Consider the ascending
chain $P=P_{0}\subseteq P_{1}\subseteq P_{2}\subseteq\cdots$ where
$P_{i}=\\{b\in H\ |\ x^{i}\circ b\subseteq P\\}$. By the assumption we have
$P_{\beta}=P_{\beta+1}=\cdots$. Put $I=P+\langle x^{\beta}\rangle$ and
$J=P+\langle y\rangle$. Then we have $P\subseteq I\cap J$. Let $a\in I\cap J$.
Then there exist $a_{1},a_{2}\in H$ and $p_{1},p_{2}\in P$ such that
$a\in(p_{1}+x^{\beta}\circ a_{1})\cap(p_{2}+y\circ a_{2}).$ It follows that
$a=p_{1}+x_{1}=p_{2}+x_{2}$ for some $x_{1}\in x^{\beta}\circ a_{1}$ and
$x_{2}\in y\circ a_{2}$. Therefore we get $x_{1}-x_{2}\in(x^{\beta}\circ
a_{1}-y\circ a_{2})\cap P$ and so $x^{\beta}\circ a_{1}-y\circ a_{2}\subseteq
P$ as $P$ is a strong $\mathcal{C}$-hyperideal of $H$. Since $(x^{\beta}\circ
a_{1}-y\circ a_{2})\circ x^{\alpha}\subseteq x^{\alpha+\beta}\circ a_{1}\
-x^{\alpha}\circ y\circ a_{2}$, we obtain $x^{\alpha+\beta}\circ a_{1}\
-x^{\alpha}\circ y\circ a_{2}\subseteq P$ which implies $x^{\alpha+\beta}\circ
a_{1}\subseteq P$. This means $a_{1}\in P_{\alpha+\beta}=P_{\beta}$ and so
$x^{\beta}\circ a_{1}\subseteq P$. Then we conclude that $a=p_{1}+x_{1}\in P$
which means $I\cap J\subseteq P$ and so $P=I\cap J$. By the hypothesis, we get
$P=I$ which implies $x^{\beta}\subseteq P$ or $P=J$ which means $y\in P$. Thus
$P$ is an $(\alpha,\beta)$-prime hyperideal. ∎

Let $S$ be a non-empty subset of $H$ with scalar identity $1$. Recall from [1]
that $S$ is a multiplicative closed subset (briefly, MCS), if $S$ is closed
under the hypermultiplication and $S$ contains $1$. [26] Consider the set
$(H\times S/\sim)$ of equivalence classes, being denoted by $S^{-1}H$, such
that $(x_{1},t_{1})\sim(x_{2},t_{2})$ if and only if there exists $t\in S$
with $t\circ t_{1}\circ x_{2}=t\circ t_{2}\circ x_{1}$. The equivalence class
of $(x,t)\in H\times S$ is denoted by $\frac{x}{t}$. The set $S^{-1}H$ is a
multiplicative hyperring where the operation $\oplus$ and the multiplication
$\odot$ are defined by

$\hskip
28.45274pt\frac{x_{1}}{t_{1}}\oplus\frac{x_{2}}{t_{2}}=\frac{t_{1}\circ
x_{2}+t_{2}\circ x_{1}}{t_{1}\circ t_{2}}=\\{\frac{a+b}{c}\ |\ a\in t_{1}\circ
x_{2},b\in t_{2}\circ x_{1},c\in t_{1}\circ t_{2}\\}$

$\hskip 28.45274pt\frac{x_{1}}{t_{1}}\odot\frac{x_{2}}{t_{2}}=\frac{x_{1}\circ
a_{2}}{t_{1}\circ t_{2}}=\\{\frac{a}{b}\ |\ a\in x_{1}\circ x_{2},b\in
t_{1}\circ t_{2}\\}$

The localization map $\pi:H\longrightarrow S^{-1}H$, defined by
$x\mapsto\frac{x}{1}$, is a homomorphism of hyperrings. Also, if $A$ is a
hyperideal of $H$, then $S^{-1}A$ is a hyperideal of $S^{-1}H$ [26].

######  Theorem 3.16.

Assume that $P$ is a $\mathcal{C}$-hyperideal of $H$ and $S$ a MCS such that
$P\cap S=\varnothing$. If $P$ is an $(\alpha,\beta)$-prime hyperideal of $H$,
then $S^{-1}P$ is an $(\alpha,\beta)$-prime hyperideal of $S^{-1}H$.

###### Proof.

Let
$\underbrace{\frac{x}{t_{1}}\odot\cdots\odot\frac{x}{t_{1}}}_{\alpha}\odot\frac{y}{t_{2}}=\frac{x^{\alpha}\circ
y}{t_{1}^{\alpha}\circ t_{2}}\subseteq S^{-1}P$ for some
$\frac{x}{t_{1}},\frac{y}{t_{2}}\in S^{-1}H$. So we get
$\frac{a}{t}\in\frac{x^{\alpha}\circ y}{t_{1}^{\alpha}\circ t_{2}}$ for every
$a\in x^{\alpha}\circ y=\underbrace{x\circ\cdots\circ x}_{\alpha}\circ y$ and
$t\in t_{1}^{\alpha}\circ t_{2}=\underbrace{t_{1}\circ\cdots\circ
t_{1}}_{\alpha}\circ t_{2}$. Hence $\frac{a}{t}=\frac{a^{\prime}}{t^{\prime}}$
for some $a^{\prime}\in P$ and $t^{\prime}\in S$. Then there exists $s\in S$
such that $s\circ a\circ t^{\prime}=s\circ a^{\prime}\circ t$. This implies
$s\circ a\circ t^{\prime}\subseteq P$. Since $a\in x^{\alpha}\circ y$, we
conclude that $s\circ a\circ t^{\prime}\subseteq s\circ x^{\alpha}\circ y\circ
t^{\prime}$. Since $P$ is a $\mathcal{C}$-hyperideal of $H$, we get $s\circ
x^{\alpha}\circ y\circ t^{\prime}\subseteq P$ and then $s^{\alpha}\circ
x^{\alpha}\circ y\circ{t^{\prime}}^{\alpha}=(s\circ x\circ
t^{\prime})^{\alpha}\circ y\subseteq P$. Take any $z\in s\circ x\circ
t^{\prime}$. Since $z^{\alpha}\circ y\subseteq P$ and $P$ is an
$(\alpha,\beta)$-prime hyperideal of $H$, we have either $z^{\beta}\subseteq
P$ or $y\in P$. In first possibily, we get $(s\circ x\circ
t^{\prime})^{\beta}\subseteq P$ as $P$ is a $\mathcal{C}$-hyperideal of $H$
and $z^{\beta}\subseteq(s\circ x\circ t^{\prime})^{\beta}$. Hence
$\frac{x^{\beta}}{t_{1}^{\beta}}=\frac{s^{\beta}\circ
x^{\beta}\circ{t^{\prime}}^{\beta}}{s^{\beta}\circ
t_{1}^{\beta}\circ{t^{\prime}}^{\beta}}\subseteq S^{-1}P$ which means
$\underbrace{\frac{x}{t_{1}}\odot\cdots\odot\frac{x}{t_{1}}}_{\beta}\subseteq
S^{-1}P$ or $\frac{y}{t_{2}}\in S^{-1}P$. This shows that $S^{-1}P$ is an
$(\alpha,\beta)$-prime hyperideal of $S^{-1}H$. ∎

Assume that $(H,+,\circ)$ is a multiplicative hyperring and $x$ is an
indeterminate. Then $(H[x],+,\diamond)$ is a polynomail multiplicative
hyperring such that $ux^{n}\diamond vx^{m}=(u\circ v)x^{n+m}$ [11].

######  Theorem 3.17.

Let $\alpha,\beta\in\mathbb{Z}^{+}$. If $P$ is an $(\alpha,\beta)$-prime
hyperideal of $(H,+,\circ)$, then $P[x]$ is an $(\alpha,\beta)$-prime
hyperideal of $(H[x],+,\diamond)$.

###### Proof.

Suppose that $u(x)^{\alpha}\diamond v(x)\subseteq P[x]$. Without loss of
generality, we may assume that $u(x)=ax^{n}$ and $v(x)=bx^{m}$ for $a,b\in H$.
Hence $a^{\alpha}\circ bx^{\alpha n+m}\subseteq P[x]$. This means
$a^{\alpha}\circ b\subseteq P$. Since $P$ is an $(\alpha,\beta)$-prime
hyperideal of $(H,+,\circ)$, we get $a^{\beta}\subseteq P$ or $b\in P$ which
implies $u(x)^{\beta}={(ax^{n})}^{\beta}=a^{\beta}x^{\beta\cdot n}\subseteq P$
or $v(x)=bx^{m}\in P[x]$. Consequently, $P[x]$ is an $(\alpha,\beta)$-prime
hyperideal of $(H[x],+,\diamond)$. ∎

In view of Theorem 3.17, we have the following result.

######  Corollary 3.18.

Assume that $P$ is an $(\alpha,\beta)$-prime hyperideal of $H$. Then $P[x]$ is
an $(\alpha,\beta)$-prime hyperideal of $H[x]$.

Assume that $H$ is a multiplicative hyperring. Then the set of all
hypermatrices of $H$ is denoted by $M_{m}(H)$. Let $A=(A_{ij})_{m\times
m},B=(B_{ij})_{m\times m}\in P^{\star}(M_{m}(H))$. Then $A\subseteq B$ if and
only if $A_{ij}\subseteq B_{ij}$[1].

######  Theorem 3.19.

Suppose that $P$ is a hyperideal of $H$ and $\alpha,\beta\in\mathbb{Z}^{+}$.
If $M_{m}(P)$ is an $(\alpha,\beta)$-prime hyperideal of $M_{m}(H)$, then $P$
is an $(\alpha,\beta)$-prime hyperideal of $H$.

###### Proof.

Let $x^{\alpha}\circ y\subseteq P$ for some $x,y\in H$. Then we have

$\begin{pmatrix}x^{\alpha}\circ y&0&\cdots&0\\\ 0&0&\cdots&0\\\
\vdots&\vdots&\ddots\vdots\\\ 0&0&\cdots&0\end{pmatrix}\subseteq M_{m}(P).$

Since $M_{m}(P)$ is an $(\alpha,\beta)$-prime hyperideal of $M_{m}(H)$ and

$\begin{pmatrix}x^{\alpha}\circ y&0&\cdots&0\\\ 0&0&\cdots&0\\\
\vdots&\vdots&\ddots\vdots\\\
0&0&\cdots&0\end{pmatrix}=\underbrace{\begin{pmatrix}x&0&\cdots&0\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots\vdots\\\
0&0&\cdots&0\end{pmatrix}\circ\cdots\circ\begin{pmatrix}x&0&\cdots&0\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots\vdots\\\
0&0&\cdots&0\end{pmatrix}}_{\alpha}\circ\begin{pmatrix}y&0&\cdots&0\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots\vdots\\\ 0&0&\cdots&0\end{pmatrix},$

we get the result that

$\underbrace{\begin{pmatrix}x&0&\cdots&0\\\ 0&0&\cdots&0\\\
\vdots&\vdots&\ddots\vdots\\\
0&0&\cdots&0\end{pmatrix}\circ\cdots\circ\begin{pmatrix}x&0&\cdots&0\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots\vdots\\\
0&0&\cdots&0\end{pmatrix}}_{\beta}=\begin{pmatrix}x^{\beta}&0&\cdots&0\\\
0&0&\cdots&0\\\ \vdots&\vdots&\ddots\vdots\\\
0&0&\cdots&0\end{pmatrix}\subseteq M_{m}(P)$

or

$\begin{pmatrix}y&0&\cdots&0\\\ 0&0&\cdots&0\\\ \vdots&\vdots&\ddots\vdots\\\
0&0&\cdots&0\end{pmatrix}\in M_{m}(P).$

Then we conclude that $x^{\beta}\subseteq P$ or $y\in P$. Consequently, $P$ is
an $(\alpha,\beta)$-prime hyperideal of $H$. ∎

Recall from [19] that a mapping $\psi$ from the multiplicative hyperring
$(H_{1},+_{1},\circ_{1})$ into the multiplicative hyperring
$(H_{2},+_{2},\circ_{2})$ refers to a hyperring good homomorphism if
$\psi(a+_{1}b)=\psi(a)+_{2}\psi(b)$ and
$\psi(a\circ_{1}b)=\psi(a)\circ_{2}\psi(b)$ for all $a,b\in H_{1}$.

######  Theorem 3.20.

Assume that $H_{1}$ and $H_{2}$ are two multiplicative hyperrins,
$\psi:H_{1}\longrightarrow H_{2}$ a hyperring good homomorphism and
$\alpha,\beta\in\mathbb{Z}^{+}$.

  * (i)

If $P_{2}$ is an $(\alpha,\beta)$-prime hyperideal of $H_{2}$, then
$\psi^{-1}(P_{2})$ is an $(\alpha,\beta)$-prime hyperideal of $H_{1}$.

  * (ii)

If $\psi$ is surjective and $P_{1}$ is a an $(\alpha,\beta)$-prime
$\mathcal{C}$-hyperideal of $H_{1}$ with $Ker(\psi)\subseteq P_{1}$ , then
$\psi(P_{1})$ is an $(\alpha,\beta)$-prime hyperideal of $H_{2}$.

###### Proof.

(i) Let $x^{\alpha}\circ_{1}x_{1}\subseteq\psi^{-1}(P_{2})$ for some
$x,x_{1}\in H_{1}$. Then we have
$\psi(x^{\alpha}\circ_{1}x_{1})=\psi(x)^{\alpha}\circ_{2}\psi(x_{1})\subseteq
P_{2}$ as $\psi$ is a hyperring good homomorphism. Since $P_{2}$ is an
$(\alpha,\beta)$-prime hyperideal of $H_{2}$, we have
$\psi(x^{\beta})=(\psi(x))^{\beta}\subseteq P_{2}$ which means
$x^{\beta}\subseteq\psi^{-1}(P_{2})$ or $\psi(x_{1})\in P_{2}$ which implies
$x_{1}\in\psi^{-1}(P_{2})$. Consequently, $\psi^{-1}(P_{2})$ is an
$(\alpha,\beta)$-prime hyperideal of $H_{1}$.

(ii) Let $y^{\alpha}\circ_{2}y_{1}\subseteq\psi(P_{1})$ for $y,y_{1}\in
H_{2}$. Then $\psi(x)=y$ and $\psi(x_{1})=y_{1}$for some $x,x_{1}\in H_{1}$
because $\psi$ is surjective. Hence
$\psi(x^{\alpha}\circ_{1}x_{1})=\psi(x)^{\alpha}\circ_{2}\psi(x_{1})\subseteq\psi(P_{1})$.
Now, pick any $a\in x^{\alpha}\circ_{1}x_{1}$. Then
$\psi(a)\in\psi(x^{\alpha}\circ_{1}x_{1})\subseteq\psi(P_{1})$ and so there
exists $b\in P_{1}$ such that $\psi(a)=\psi(b)$. Then we have $\psi(a-b)=0$
which means $a-b\in Ker(\psi)\subseteq P_{1}$ and so $a\in P_{1}$. Therefore
$x^{\alpha}\circ_{1}x_{1}\subseteq P_{1}$ as $P_{1}$ is a
$\mathcal{C}$-hyperideal. Since $P_{1}$ is an $(\alpha,\beta)$-prime
hyperideal of $H_{1}$, we obtain $x^{\beta}\subseteq P_{1}$ or $x_{1}\in
P_{1}$. This implies that $y^{\beta}=\psi(x^{\beta})\subseteq\psi(P_{1})$ or
$y_{1}=\psi(x_{1})\in\psi(P_{1})$. Thus $\psi(P_{1})$ is an
$(\alpha,\beta)$-prime hyperideal of $H_{2}$. ∎

Now, we have the following result.

######  Corollary 3.21.

Let $P_{1}$ and $P_{2}$ be two hyperideals of $H$ with $P_{1}\subseteq P_{2}$
and $\alpha,\beta\in\mathbb{Z}^{+}$. Then $P_{2}$ is an $(\alpha,\beta)$-prime
hyperideal of $H$ if and only if $P_{2}/P_{1}$ is an $(\alpha,\beta)$-prime
hyperideal of $H/P_{1}$.

###### Proof.

Consider the homomorphism $\pi:H\longrightarrow H/P_{1}$ defined by
$\pi(x)=x+P_{1}$. Then the claim follows from Theorem 3.20 as $\pi$ is a good
epimorphism. ∎

##  4\. weakly $(\alpha,\beta)$-prime hyperideals

######  Definition 4.1.

Assume that $P$ is a proper hyperideal of $H$ and
$\alpha,\beta\in\mathbb{Z}^{+}$. $P$ is said to be a weakly
$(\alpha,\beta)$-prime hyperideal if $0\notin x^{\alpha}\circ y\subseteq P$
for $x,y\in H$ implies that $x^{\beta}\subseteq P$ or $y\in P$.

######  Example 4.2.

Consider the ring $(\mathbb{Z}_{8},\oplus,\odot)$ where $\bar{x}\oplus\bar{y}$
and $\bar{x}\odot\bar{y}$ are remainder of $\frac{x+y}{8}$ and $\frac{x\cdot
y}{8}$, respectively, where $+$ and $\cdot$ are ordinary addition and
multiplication for all $\bar{x},\bar{y}\in\mathbb{Z}_{8}$. Define the
hyperoperation
$\bar{x}\circ\bar{y}=\\{\overline{xy},\overline{2xy},\overline{3xy},\overline{4xy},\overline{5xy},\overline{6xy},\overline{7xy}\\}$.
Then the hyperideal $Q=\\{\bar{0},\bar{4}\\}$ of
$(\mathbb{Z}_{8},\oplus,\circ)$ is weakly $(3,1)$-prime but it is not
$(3,1)$-prime.

######  Theorem 4.3.

Assume that the zero hyperideal of $H$ is a $\mathcal{C}$-hyperideal such that
its radical is prime. If $P$ is a weakly $(\alpha,\beta)$-prime
$\mathcal{C}$-hyperideal of $H$ for $\alpha,\beta\in\mathbb{Z}^{+}$, then
$rad(P)$ is prime. In particular, $x^{\beta}\subseteq P$ for each $x\in
rad(P)-rad(0)$.

###### Proof.

Let $x\circ y\subseteq rad(P)$ for $x,y\in H$. Then there exists
$n\in\mathbb{Z}^{+}$ such that $x^{n}\circ y^{n}\subseteq P$. Assume that
$a\in x^{n}$ and $b\in y^{n}$, so $a^{\alpha}\circ b\subseteq P$. If $0\in
a^{\alpha}\circ b$, then $a^{\alpha}\circ b=0$ as the zero hyperideal of $H$
is a $\mathcal{C}$-hyperideal. Since radical of the hyperideal is prime, we
have $a\in rad(0)$ or $b\in rad(0)$. This means $x\in rad(0)$ or $y\in
rad(0)$. Then we get the result that $x\in rad(P)$ or $y\in rad(P)$. Assume
that $0\notin a^{\alpha}\circ b$. Since $P$ is a weakly $(m,n)$-prime
hyperideal of $H$, we get either $a^{\beta}\subseteq P$ or $b\in P$. Since $P$
is a $\mathcal{C}$-hyperideal, $a^{\beta}\subseteq x^{\beta n}$ or $b\in
y^{n}$, we have $x^{\beta n}\subseteq P$ which implies $x\in rad(P)$ or
$y^{n}\subseteq P$ which means $y\in rad(P)$, as needed. Now, take any $x\in
rad(P)-rad(0)$. Assume that $n$ is the least positive integer with
$x^{n}\subseteq P$. Since $x\notin rad(0)$, we conclude that $0\notin
x^{\alpha}\circ x^{n-1}$. Let $y\in x^{n-1}$. So $0\notin x^{\alpha}\circ
y\subseteq P$. Since $P$ is a weakly $(m,n)$-prime hyperideal and $y\notin P$,
we get $x^{\beta}\subseteq P$, as required. ∎

######  Theorem 4.4.

If every proper hyperideal of $H$ is weakly $(\alpha,\beta)$-prime such that
$\alpha,\beta\in\mathbb{Z}^{+}$ and $\alpha\geq\beta$, then every prime
$\mathcal{C}$-hyperideal of $H$ is maximal.

###### Proof.

Suppose that $I$ is a prime $\mathcal{C}$-hyperideal such that it is not
maximal. Let $J$ be a proper hyperideal such that $I\subset J$. Take any $x\in
J-I$. Put $P=\langle x^{\alpha+1}\rangle$. Therefore we have $0\notin
x^{\alpha}\circ x\subseteq P$. By the hypothesis, we have $x^{\beta}\subseteq
P$ or $x\in P$. In the first possibilty, we obtan $x^{\beta}\subseteq
x^{\alpha+1}\circ r$ for some $r\in H$ which means $0\in
x^{\beta}-x^{\alpha+1}\circ r\cap I$. Then $x^{\beta}-x^{\alpha+1}\circ
r\subseteq I$, also $x^{\beta}\circ(1-x^{\alpha-\beta+1}\circ r)\subseteq
x^{\beta}-x^{\alpha+1}\circ r$. Hence
$x^{\beta}\circ(1-x^{\alpha-\beta+1}\circ r)\subseteq I$. Since
$x^{\beta}\nsubseteq I$ and $I$ is a prime hyperideal, we get
$1-x^{\alpha-\beta+1}\circ r\subseteq I\subset J$ which means $1\in J$, a
contradiction. In the second possibilty, we get a contradiction by a similar
argument. Consequenntly, every prime $\mathcal{C}$-hyperideal of $H$ is
maximal. ∎

Let $P$ be a weakly $(\alpha,\beta)$-prime $\mathcal{C}$-hyperideal of $H$ and
$x,y\in H$. We say that $(x,y)$ is an $(\alpha,\beta)$-zero of $P$ if $0\in
x^{\alpha}\circ y$, $x^{\beta}\nsubseteq P$ and $y\notin P$.

######  Proposition 4.5.

Let $P$ be a weakly $(\alpha,\beta)$-prime $\mathcal{C}$-hyperideal of $H$ and
and $(x,y)$ be an $(\alpha,\beta)$-zero of $P$ where
$\alpha,\beta\in\mathbb{Z}^{+}$. Then the following hold:

  * (i)

$0\in(x+a)^{\alpha}\circ y$ for all $a\in P$.

  * (ii)

$0\in x^{\alpha}\circ(y+a)$ for all $a\in P$.

  * (iii)

If the hyperideal zero of $H$ is a strong $\mathcal{C}$-hyperideal, then
$x^{\alpha}\circ a=0$ for all $a\in P$.

###### Proof.

(i) Let $(x,y)$ be an $(\alpha,\beta)$-zero of $P$ and
$0\notin(x+a)^{\alpha}\circ y$ for some $a\in P$. Since $0\in x^{\alpha}\circ
y$ and $P$ is a $\mathcal{C}$-hyperideal of $H$, we conclude that
$x^{\alpha}\circ y\subseteq P$. Therefore $0\notin(x+a)^{\alpha}\circ
y\subseteq x^{\alpha}\circ
y+\Sigma_{i=1}^{\alpha}\tbinom{\alpha}{i}x^{\alpha-i}\circ a^{i}\circ
y\subseteq P.$ Since $P$ is a weakly $(\alpha,\beta)$-prime hyperideal of $H$
and $y\notin P$, we get $(x+a)^{\beta}\subseteq P$. On the other hand, since
$(x,y)$ is an $(\alpha,\beta)$-zero of $P$ and $x^{\beta}\nsubseteq P$, we get
the result that $(x+a)^{\beta}\nsubseteq P$ which is a contradiction. Thus
$0\in(x+a)^{\alpha}\circ y$ for all $a\in P$.

(ii) Let $0\notin x^{\alpha}\circ(y+a)$ for some $a\in P$. Hence $0\notin
x^{\alpha}\circ(y+a)\subseteq x^{\alpha}\circ y+x^{\alpha}\circ a\subseteq P$
as $P$ is a $\mathcal{C}$-hyperideal of $H$. Since $P$ is a weakly
$(\alpha,\beta)$-prime hyperideal of $H$ and $x^{\beta}\nsubseteq P$, we
obtain $y+a\in P$ and so $y\in P$ which is a contradiction. Consequently,
$0\in x^{\alpha}\circ(y+a)$ for all $a\in P$.

(iii) Assume that $x^{\alpha}\circ a\neq 0$ for some $a\in P$. Then there
exists $u\in x^{\alpha}\circ a$ such that $u\neq 0$. By (ii) we have $0\in
x^{\alpha}\circ(y+a)$. Since the hyperideal zero of $H$ is a strong
$\mathcal{C}$-hyperideal and $0\in x^{\alpha}\circ(y+a)\subseteq
x^{\alpha}\circ y+x^{\alpha}\circ a$, we get $x^{\alpha}\circ
y+x^{\alpha}\circ a=0$. Moreover, since $0\in x^{\alpha}\circ y$ and $0\neq
u\in x^{\alpha}\circ a$, we get $u=0+u\in x^{\alpha}\circ y+x^{\alpha}\circ a$
which is a contradiction. Hence $x^{\alpha}\circ a=0$ for all $a\in P$. ∎

Recall from [1] that an element $x\in G$ is nilpotent if there exists an
integer $t$ such that $0\in x^{t}$. The set of all nilpotent elements of $G$
is denoted by $\Upsilon$.

######  Theorem 4.6.

Let $P$ be a weakly $(\alpha,\beta)$-prime $\mathcal{C}$-hyperideal of a
strongly distributive multiplicative hyperring $H$ and $(x,y)$ be an
$(\alpha,\beta)$-zero of $P$ where $\alpha,\beta\in\mathbb{Z}^{+}$. Then

  * (i)

If the hyperideal zero of $H$ is a strong $\mathcal{C}$-hyperideal, then
$x\circ a\subseteq\Upsilon$ for all $a\in P$.

  * (ii)

If the hyperideal zero of $H$ is a $\mathcal{C}$-hyperideal, $y\circ
a\subseteq\Upsilon$ for all $a\in P$.

###### Proof.

(i) Since $x^{\alpha}\circ a=0$ for all $a\in P$, by Proposition 4.5 (3), we
get the result that $x\circ a\subseteq\Upsilon$.

(ii) Take any $a\in P$. Since $0\in(x+a)^{\alpha}\circ y$ by Proposition
4.5(1) and the hyperideal zero of $H$ is a $\mathcal{C}$-hyperideal, we get
$((x+a)\circ y)^{\alpha}=0$. This means that $(x+a)\circ y\subseteq\Upsilon$.
Also, since $0\in x^{\alpha}\circ y$ and the hyperideal zero of $H$ is a
$\mathcal{C}$-hyperideal, we have $(x\circ y)^{\alpha}=0$ which implies
$x\circ y\subseteq\Upsilon$. Since $H$ is a strongly distributive
multiplicative hyperring, we have $a\circ y=(x+a)\circ y-x\circ
y\subseteq\Upsilon$, as needed. ∎

######  Theorem 4.7.

Let $\\{P_{i}\\}_{i\in I}$ be a family of weakly $(\alpha,\beta)$-prime
hyperideals of $H$ and $D(P_{i})=\\{x\in H\ |\ x^{\beta}\subseteq P_{i}\\}$
for all $i\in I$ where $\alpha,\beta\in\mathbb{Z}^{+}$. If $D(P_{i})=D(P_{j})$
for all $i,j\in I$, then $\bigcap_{i\in I}P_{i}$ is a weakly
$(\alpha,\beta)$-prime hyperideal of $H$.

###### Proof.

Assume that $0\notin x^{\alpha}\circ y\subseteq\bigcap_{i\in I}P_{i}$ for
$x,y\in H$ but $y\notin\bigcap_{i\in I}P_{i}$. Therefore we conclude that
$y\notin P_{j}$ for some $j\in I$. Since $P_{j}$ is a weakly
$(\alpha,\beta)$-prime hyperideal of $H$ and $0\notin x^{\alpha}\circ
y\subseteq P_{j}$, we get the result that $x^{\beta}\subseteq P_{j}$. This
implies that $x\in D(P_{j})$ and so $x\in D(P_{i})$ for all $i\in I$ by the
hypothesis. Then $x^{\beta}\subseteq\bigcap_{i\in I}P_{i}$. This shows that
$\bigcap_{i\in I}P_{i}$ is a a weakly $(\alpha,\beta)$-prime hyperideal of
$H$. ∎

Let $I$ be a finite sum of finite products of elements of $H$. Consider the
relation $\gamma$ on a multiplicative hyperring $H$ defined as $x\gamma y$ if
and only if $\\{x,y\\}\subseteq I$, namely, $x\gamma y$ if and only if
$\\{x,y\\}\subseteq\sum_{j\in J}\prod_{i\in I_{j}}z_{i}$ for some
$z_{1},...,z_{n}\in H$ and $I_{j},J\subseteq\\{1,...,n\\}$. $\gamma^{\ast}$
denotes the transitive closure of $\gamma$. The relation $\gamma^{\ast}$ is
the smallest equivalence relation on $H$ such that the set of all equivalence
classes, i.e., the quotient $G/\gamma^{\ast}$, is a fundamental ring. Assume
that $\Sigma$ is the set of all finite sums of products of elements of $H$. We
can rewrite the definition of $\gamma^{\ast}$ on $H$, namely,
$x\gamma^{\ast}y$ if and only if there exist $z_{1},...,z_{n}\in H$ such that
$z_{1}=x,z_{n+1}=y$ and $u_{1},...,u_{n}\in\Sigma$ where
$\\{z_{i},z_{i+1}\\}\subseteq u_{i}$ for $1\leq i\leq n$. Suppose that
$\gamma^{\ast}(x)$ is the equivalence class containing $x\in H$. Define
$\gamma^{\ast}(x)\oplus\gamma^{\ast}(y)=\gamma^{\ast}(z)$ for every
$z\in\gamma^{\ast}(x)+\gamma^{\ast}(y)$ and
$\gamma^{\ast}(x)\odot\gamma^{\ast}(y)=\gamma^{\ast}(w)$ for every
$w\in\gamma^{\ast}(x)\circ\gamma^{\ast}(y)$. Then
$(H/\gamma^{\ast},\oplus,\odot)$ is a ring called a fundamental ring of $H$
[36].

######  Theorem 4.8.

Assume that $P$ is a hyperideal of $H$. Then $P$ is a weakly
$(\alpha,\beta)$-prime hyperideal of $(H,+,\circ)$ if and only if
$P/\gamma^{\ast}$ is a weakly $(\alpha,\beta)$-prime ideal of
$(H/\gamma^{\ast},\oplus,\odot)$.

###### Proof.

Let $0\neq\underbrace{x\odot\cdots\odot x}_{\alpha}\odot y\in P/\gamma^{\ast}$
for some $x,y\in H/\gamma^{\ast}$. Hence we have $x=\gamma^{\ast}(a)$ and
$y=\gamma^{\ast}(b)$ for some $a,b\in H$. This means that
$\underbrace{x\odot\cdots\odot x}_{\alpha}\odot
y=\underbrace{\gamma^{\ast}(a)\odot\cdots\odot\gamma^{\ast}(a)}_{\alpha}\odot\gamma^{\ast}(b)=\gamma^{\ast}(a^{\alpha}\circ
b)$. Since $\gamma^{\ast}(0)\neq\gamma^{\ast}(a^{\alpha}\circ b)\in
P/\gamma^{\ast}$, we get $0\notin a^{\alpha}\circ b\subseteq P$. Since $P$ is
a weakly $(\alpha,\beta)$-prime hyperideal of $H$, we get the result that
$a^{\beta}\subseteq P$ or $b\in P$. This implies that
$\underbrace{x\odot\cdots\odot
x}_{\beta}=\underbrace{\gamma^{\ast}(a)\odot\cdots\odot\gamma^{\ast}(a)}_{\beta}=\gamma^{\ast}(a^{\beta})\in
P/\gamma^{\ast}$ or $y=\gamma^{\ast}(a)\in P/\gamma^{\ast}$. Consequently,
$P/\gamma^{\ast}$ is a weakly $(\alpha,\beta)$-prime ideal of
$H/\gamma^{\ast}$. ∎

Let $(H_{1},+_{1},\circ_{1})$ and $(H_{2},+_{2},\circ_{2})$ be two
multiplicative hyperrings with nonzero identity. The set $H_{1}\times H_{2}$
with the operation $+$ and the hyperoperation $\circ$ defined as

$(x_{1},x_{2})+(y_{1},y_{2})=(x_{1}+_{1}y_{1},x_{2}+_{2}y_{2})$

$(x_{1},x_{2})\circ(y_{1},y_{2})=\\{(x,y)\in H_{1}\times H_{2}\ |\ x\in
x_{1}\circ_{1}y_{1},y\in x_{2}\circ_{2}y_{2}\\}$
is a multiplicative hyperring [34]. Now, we present some characterizations of
weakly $(\alpha,\beta)$-prime hyperideals on cartesian product of commutative
multiplicative hyperring.

######  Proposition 4.9.

Let $(H_{1},+_{1},\circ_{1})$ and $(H_{2},+_{2},\circ_{2})$ be two
multiplicative hyperrings with scalar identities $1_{H_{1}}$ and $1_{H_{2}}$,
respectively, $P$ a proper nonzero hyperideal of $H_{1}\times H_{2}$, and
$\alpha,\beta\in\mathbb{Z}^{+}$. If $P$ is weakly $(\alpha,\beta)$-prime, then
it has one of the following cases:

  * (i)

$P=P_{1}\times H_{2}$ such that $P_{1}$ is an $(\alpha,\beta)$-prime
hyperideal of $H_{1}$.

  * (ii)

$P=H_{1}\times P_{2}$ such that $P_{2}$ is an $(\alpha,\beta)$-prime
hyperideal of $H_{2}$.

###### Proof.

$\Longrightarrow$ Suppose that $P=P_{1}\times P_{2}$ is a nonzero weakly
$(\alpha,\beta)$-prime hyperideal of $H_{1}\times H_{2}$ such that $P_{1}$ and
$P_{2}$ are hyperideals of $H_{1}$ and $H_{2}$, respectively. Let us assume
$P_{1}$ and $P_{2}$ are proper, and $P_{1}\neq 0$. Take any $0\neq x\in
P_{1}$. Therefore we have
$(0,0)\notin(1_{H_{1}},0)^{\alpha}\circ(x,1_{H_{2}})\subseteq P_{1}\times
P_{2}$. Since $P$ is a nonzero weakly $(\alpha,\beta)$-prime hyperideal of
$H_{1}\times H_{2}$, we get the result that $(1_{H_{1}},0)^{\beta}\subseteq
P_{1}\times P_{2}$ or $(x,1_{H_{2}})\in P_{1}\times P_{2}$. It follows that
$1_{H_{1}}\in P_{1}$ or $1_{H_{2}}\in P_{2}$. Then $P_{1}=H_{1}$ or
$P_{2}=H_{2}$. This is a contradiction. Let us consider $P_{1}$ is proper and
$P_{2}=H_{2}$. Now, we shows that $P_{1}$ is an $(\alpha,\beta)$-prime
hyperideal of $H_{1}$. Let $x^{\alpha}\circ_{1}y\subseteq P_{1}$ for $x,y\in
H_{1}$. Hence we get
$(0,0)\notin(x,1_{H_{2}})^{\alpha}\circ(y,1_{H_{2}})\subseteq P_{1}\times
H_{2}$ and so we have $(x,1_{H_{2}})^{\beta}\subseteq P_{1}\times H_{2}$ or
$(y,1_{H_{2}})\in P_{1}\times H_{2}$. This implies that $x^{\beta}\subseteq
P_{1}$ or $y\in P_{1}$. Similarly, it can be seen that if $P_{1}=H_{1}$ and
$P_{2}$ is a proper hyperideal of $H_{2}$, then $P_{2}$ is
$(\alpha,\beta)$-prime. ∎

######  Theorem 4.10.

Assume that $H=H_{1}\times\cdots\times H_{n}$ where $H_{1},\cdots,H_{n}$ are
commutative multiplicative hyperrings, $P$ a proper nonzero hyperideal of $H$
and $\alpha,\beta\in\mathbb{Z}^{+}$. Then the following are equivalent.

  * (i)

$P$ is a weakly $(\alpha,\beta)$-prime hyperideal of $H$.

  * (ii)

$P=H_{1}\times\cdots\times P_{i}\times\cdots\times H_{n}$ such that $P_{i}$ is
an $(\alpha,\beta)$-prime hyperideal of $H_{i}$ for some
$i\in\\{1,\cdots,n\\}$.

  * (iii)

$P$ is an $(\alpha,\beta)$-prime hyperideal of $H$.

###### Proof.

(i) $\Longrightarrow$ (ii) Let $P=P_{1}\times\cdots\times P_{n}$ is a weakly
$(\alpha,\beta)$-prime hyperideal of $H$. We use the induction on $n$. If
$n=2$, then the claim is true by Proposition 4.9. Let the claim be true for
$n-1$. Assume that $I=P_{1}\times\cdots\times P_{n-1}$. So $P=I\times P_{n}$.
Then we conclude that $I$ is an $(\alpha,\beta)$-prime hyperideal of
$H_{1}\times\cdots\times H_{n-1}$ and $P_{n}=H_{n}$ or
$I=H_{1}\times\cdots\times H_{n-1}$ and $P_{n}$ is an $(\alpha,\beta)$-prime
hyperideal of $H_{n}$ by Proposition 4.9. In the first possibility, we obtain
$I=H_{1}\times\cdots\times P_{i}\times\cdots\times H_{n-1}$ such that $P_{i}$
is an $(\alpha,\beta)$-prime hyperideal of $H_{i}$ by induction hypothesis and
$P_{n}=H_{n}$. This shows that $P=H_{1}\times\cdots\times
P_{i}\times\cdots\times H_{n-1}\times H_{n}$ such that $P_{i}$ is an
$(\alpha,\beta)$-prime hyperideal of $H_{i}$. In the second possibility, we
have $I_{i}=H_{i}$ for all $i\in\\{1,\cdots,n-1\\}$ and $P_{n}$ is an
$(\alpha,\beta)$-prime hyperideal of $H_{n}$. It follows that
$P=H_{1}\times\cdots\times H_{n-1}\times P_{n}$ where $P_{n}$ is is an
$(\alpha,\beta)$-prime hyperideal of $H_{n}$.

(ii) $\Longrightarrow$ (iii) Without loss of generality, we assume that
$P_{1}$ is an $(\alpha,\beta)$-prime hyperideal of $H_{1}$ and $P_{i}=H_{i}$
for all $i\neq 1$. Let
$(x_{1},x_{2},\cdots,x_{n})^{\alpha}\circ(y_{1},y_{2},\cdots,y_{n})\subseteq
P_{1}\times H_{2}\times\cdots\times H_{n}$ such that
$(y_{1},y_{2},\cdots,y_{n})\notin P_{1}\times H_{2}\times\cdots\times H_{n}$.
This implies that $x_{1}^{\alpha}\circ_{1}y_{1}\subseteq P_{1}$ and
$y_{1}\notin P_{1}$. Since $P_{1}$ is an $(\alpha,\beta)$-prime hyperideal of
$H_{1}$, we get $x_{1}^{\beta}\subseteq P_{1}$. It follows that
$(x_{1},x_{2},\cdots,x_{n})^{\beta}\subseteq P_{1}\times
H_{2}\times\cdots\times H_{n}$ and this completes the proof.

(iii) $\Longrightarrow$ (i) It is straightforward. ∎

## References

  * [1] R. Ameri, A. Kordi, S. Hoskova-Mayerova, Multiplicative hyperring of fractions and coprime hyperideals, An. St. Univ. ovidious Constanta, 25(2017), No. 1, 5-23.
  * [2] R. Ameri, M. Norouzi, On commutative hyperrings, Int. Journal of Algebraic Hyperstructures and its Applications,  1(1) (2014) 45-58.
  * [3] R. Ameri, A. Kordi, On regular multiplicative hyperrings, European Journal of Pure and Applied Mathematics, 9(4)(2016) 402-418.
  * [4] R. Ameri, A. Kordi, Clean multiplicative hyperrings, Italian Journal of Pure and Applied Mathematics, (35) (2015) 625-636.
  * [5] M. Anbarloei, A generalization of prime hyperideals, Journal of Algebraic Systems, 8 (1) (2020) 113-127.
  * [6] M. Anbarloei, (weakly) $(s,n)$-closed hyperideals in commutative multiplicative hyperrings, Journal of algebraic systems, (2024), https://doi.org/10.22044/JAS.2024.13889.1780.
  * [7] D.F. Anderson, A. Badawi. On n-absorbing ideals of commutative rings, Comm. Algebra, 39 (5) (2011) 1646-1672.
  * [8] D.F. Anderson, A. Badawi, On $(m,n)$-closed ideals of commutative rings, Journal of Algebra and Its Applications, 16(1)(2017) 1-21.
  * [9] E. Ay, G. Yesilot, D. Sonmez, $\delta$-Primary hyperideals on commutative hyperrings, International Journal Mathematics and Mathematical Sciences (2017): Article ID 5428160, 4 pages.
  * [10] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75 (2007) 417-429.
  * [11] R.P. Ciampi, R. Rota, Polynomials over multiplicative hyperrings, J. Discrete Math. Sci. Cryptogr., 6 (2-3) (2003) 217–225.
  * [12] P. Corsini, Prolegomena of hypergroup theory, Second ed., Aviani Editore, 1993.
  * [13] P. Corsini, V. Leoreanu, Applications of hyperstructures theory, Adv. Math., Kluwer Academic Publishers, 2003.
  * [14] J. Chvalina, J., S. Krehlik, M. Novak, Cartesian composition and the problem of generalizing the MAC condition to quasi- multiautomata, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 24(3) (2016) 79-100.
  * [15] I. Cristea, S. Jancic-Rasovic, Compositions hyperrings, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 21(2) (2013) 81-94.
  * [16] I. Cristea, Regularity of intuitionistic fuzzy relations on hypergroupoids, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 22(1) (2014) 105-119.
  * [17] U. Dasgupta, On prime and primary hyperideals of a multiplicative hyperring, Annals of the Alexandru Ioan Cuza University-Mathematics, LVIII (1) (2012) 19-36.
  * [18] U. Dasgupta, On certain classes of hypersemirings, PhD Thsis, University of Calcutta, 2012.
  * [19] B. Davvaz, V. Leoreanu-Fotea, Hyperring theory and applications, International Academic Press, USA, 2007.
  * [20] F. Farzalipour, P. Ghiasvand, Semiprime hyperideals in multiplicative hyperrings, 51th Annual Iranian Mathematics conference, 1 (2021) 801-804.
  * [21] L. Kamali Ardekani, B. Davvaz, Differential multiplicative hyperring, Journal of Algebraic Systems, 2 (1) (2014) 21-35.
  * [22] H.A. Khashan, E.Y. Celikel, $(m,n)$-prime ideals of commutative rings, Preprints 2024, 2024010472. https://doi.org/10.20944/preprints202401.0472.v1
  * [23] H.A. Khashan, E.Y. Celikel, On weakly $(m,n)$-prime ideals of commutative rings, Bull. Korean Math. Soc., In press.
  * [24] F. Marty, Sur une generalization de la notion de groupe, In: 8iem Congres des Mathematiciens Scandinaves, Stockholm, (1934) 45-49.
  * [25] C. G. Massouros, On the theory of hyperrings and hyperfields, Algebra i Logika, 24 (1985) 728-742.
  * [26] A.A. Mena, I. Akray, $n$-absorbing $I$-prime hyperideals in multiplicative hyperrings, Journal of Algebraic Systems, 12 (1) (2024) 105-121.
  * [27] P. Ghiasvand, F. Farzalipour, On $S$-prime hyperideals in multiplicative hyperrings, journal of Algebraic Hyperstructures and Logical Algebras, 2 (2) (2021) 25-34.
  * [28] P. Ghiasvand, F. Farzalipour, 1-Absorbing prime avoidance theorem in multiplicative hyperrings, Palestine Journal of Mathematics, 12(1) (2023) 900-908.
  * [29] M. Novak, $n$-ary hyperstructures constructed from binary quasi-orderer semigroups, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 22 (3) (2014) 147-168.
  * [30] R. Procesi, R. Rota, On some classes of hyperstructures, Combinatorics Discrete Math., 208/209 (1999) 485-497.
  * [31] R. Rota, Sugli iperanelli moltiplicativi, Rend. Di Math., Series VII, 4 (1982) 711-724.
  * [32] E. Sengelen Sevim, B.A. Ersoy, B. Davvaz, Primary hyperideals of multiplicative hyperrings, Eurasian Bulletin of Mathematics, 1(1) (2018) 43-49.
  * [33] M.K. Sen, U. Dasgupta, Some aspects of $G_{H}$-rings, An. Stiint. Univ. ”Al. I. Cuza” Iasi. Mat., 56 (2010) 253-272.
  * [34] G. Ulucak, On expansions of prime and 2-absorbing hyperideals in multiplicative hyperrings, Turkish Journal of Mathematics, 43 (2019) 1504-1517.
  * [35] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press, Inc., Palm Harber, USA, 1994.
  * [36] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press Inc., Florida, (1994).


Single inclusive particle production in pA collisions at forward rapidities:
beyond the hybrid model

Tolga Altinoluk

Theoretical Physics Division, National Centre for Nuclear Research, Pasteura
7, Warsaw 02-093, Poland

Néstor Armesto111Speaker.

Instituto Galego de Física de Altas Enerxías IGFAE, Universidade de Santiago
de Compostela, 15782 Santiago de Compostela, Galicia-Spain

Alexander Kovner

Physics Department, University of Connecticut, 2152 Hillside Road, Storrs, CT
06269, USA

Michael Lublinsky

Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105,
Israel

> In this contribution we reconsider the calculation at next-to-leading order
> of forward inclusive single hadron production in $pA$ collisions within the
> hybrid approach. We conclude that the proper framework to compute this cross
> section beyond leading order is not collinear factorization as assumed so
> far, but the TMD factorized framework.

> PRESENTED AT
>
>
>
>
> DIS2023: XXX International Workshop on Deep-Inelastic Scattering and Related
> Subjects,
> Michigan State University, USA, 27-31 March 2023
>

##  1 Introduction

In the Color Glass Condensate (CGC), single inclusive particle production in
the forward rapidity region in collisions between a dilute projectile, $p$,
and a dense target, $A$, is usually addressed in the hybrid approach. The
projectile is described through standard collinear parton densities. The
projectile partons then scatter on the color field of the target modelled
through averages of Wilson lines, to produce the final parton which
subsequently hadronizes through standard collinear fragmentation functions.

The hybrid approach was formulated at the lowest order (LO) in [1]. A partial
next-to-leading order (NLO) calculation was performed in [2], while the full
NLO result appeared in [3, 4]. In this result, a collinear subtraction
resulted in the DGLAP evolution of parton densities and fragmentation
functions, while a rapidity cut-off to regularize the soft divergencies was
required that led to the JIMWLK evolution equation, or its mean field version
- the BK equation, for the averages of Wilson lines.

A numerical implementation of the NLO results in [5, 6] showed that the cross
section seemed to drop very fast with increasing transverse momentum and
rather quickly became negative. Later, in [7] a restriction on the lifetime of
the fluctuations of the projectile wavefunction (Ioffe time) was introduced
that provided a soft cut-off and made evident the existence of additional NLO
terms (resembling those in the BK evolution equation), which was also found in
[8] coming from kinematic restrictions. Such considerations alleviated the
negativity problem but did not solve it completely. Further developments have
been threshold [9, 10, 11] and Sudakov [11] resummations. While these
modifications resulted in improved fits to data, they seem rather ad hoc and
there is no guarantee that they ensure positivity of the cross section at
large transverse momentum.

In this contribution we revisit this NLO calculation following the formulation
in [7]. We work in a frame in which the target is highly evolved, finding no
need of additional BK evolution of the LO dipoles for physically motivated
choices of factorization scales. We avoid any collinear or soft subtraction as
previously done and conclude that the correct framework to resum the large
logarithms that appear is not collinear but transverse momentum dependent
(TMD) factorization for the projectile PDFs and for the FFs. We will present
here the results for the $q\to q\to H$ channel only. The complete results for
all channels and full details can be found in [12].

##  2 Transverse momentum distributions

First of all we proceed to define our TMD PDFs222The definition of TMD FFs is
completely analogous and, together with the generalization for gluons and
several quark and antiquark species, can be found in [12].. Several
definitions of TMDs are used in the literature, differing predominantly in the
way one treats the soft resolution scale, see, e.g., [13, 14], in addition to
the process dependence of TMDs which requires, for different observables, the
inclusion of different Wilson line factors in their definition, and to the
nonperturbative aspects of TMDs. Here we use the simple intuitive perturbative
definition, which is perfectly adequate for our purposes and resembles that
used in the parton branching method [15, 16].

The unpolarized quark TMD PDF is defined perturbatively through its relation
with the collinear PDFs as

$\displaystyle x{\cal
T}_{q}(x,k^{2};k^{2};\xi_{0})=\frac{g^{2}}{(2\pi)^{3}}\frac{N_{c}}{2}\int_{\xi_{0}}^{1}d\xi\frac{1+(1-\xi)^{2}}{\xi}\,\frac{x}{1-\xi}\,f^{q}_{k^{2}}\left(\frac{x}{1-\xi}\right)\frac{1}{k^{2}}\
.$ (1)

This closely resembles the known perturbative relation between TMD and
collinear PDFs in the large $k$ region [13, 14]. Here $\xi$ denotes the
momentum fraction taken by the emitted gluon, the soft divergence in the gluon
emission is regulated by the cutoff $\xi_{0}$, and the third argument in the
TMD is the transverse resolution (factorization) scale.

The factorization scale dependence of the TMD PDFs is then given by the DGLAP-
like equation

$\displaystyle x{\cal T}_{q}(x,k^{2};\mu^{2};\xi_{0})$ $\displaystyle=$
$\displaystyle\theta(\mu^{2}-k^{2})\,\Bigg{[}x{\cal
T}_{q}(x,k^{2};k^{2};\xi_{0})$ $\displaystyle\hskip
0.0pt-\frac{g^{2}}{(2\pi)^{3}}\frac{N_{c}}{2}\int_{k^{2}}^{\mu^{2}}\frac{\pi
dl^{2}}{l^{2}}\,\int_{\xi_{0}}^{1}d\xi\frac{1+(1-\xi)^{2}}{\xi}x\,{\cal
T}_{q}\left(x,k^{2};l^{2};\xi_{0}\right)\Bigg{]}.$

This equation can be interpreted in the following way: increasing the
transverse resolution means that the number of quarks at a fixed transverse
momentum decreases due to DGLAP splittings into quark-gluon pairs with higher
longitudinal momentum given by the resolution scale. Note that we regulate the
soft divergence in gluon emissions by introducing the cut-off on momentum
fraction, $\xi_{0}$333The definition of the longitudinal cutoff we use follows
our earlier approach [7], where we have limited the life time of the
fluctuations by the Ioffe time cutoff. The resolution $\xi_{0}$ then depends
on the virtuality $l$ of the gluon in the splitting as
$\xi_{0}(l)=l^{2}/(xs_{0})$, where $s_{0}$ is the Ioffe cutoff parameter..

With these definitions, the collinear quark PDF, related to the quark TMD PDF
via

$\displaystyle xf^{q}_{\mu^{2}}(x)=\int_{0}^{\mu^{2}}\pi dk^{2}\,x{\cal
T}_{q}(x,k^{2};\mu^{2};\xi_{0}),$ (3)

satisfies the DGLAP evolution equations.

##  3 The $q\to q\to H$ channel

We start from the expressions derived in light-cone perturbation theory in [7]
prior to any collinear or soft subtraction. Here we just outline the results
and provide the final expressions; full details can be found in [12].

First, the real contribution can be reorganized into terms that yield the TMD
PDFs and FFs as defined in (1), plus a genuine NLO remainder which shows no
large soft or collinear logarithmic contribution.

Second, the virtual contribution can be manipulated to give the evolution of
the TMD PDFs and FFs in the LO contribution as provided in (2) (expanded to
order $g^{2}$), plus another NLO remainders which, for our choice of scales,
see below, also show no large soft or collinear logarithmic contributions.

Working at large number of colors $N_{c}$, assuming translational invariance
of the target averages of the ensembles of Wilson lines characterizing the
target, and neglecting pieces that are either power suppressed or ${\cal
O}(\alpha_{s}^{2})$, our final expression for the quark channel, for
production of a hadron with rapidity $\eta$ and transverse momentum $p$, reads

$\displaystyle\frac{d{\sigma}^{q\to q\to
H}}{d^{2}pd\eta}=S_{\perp}\int_{x_{F}}^{1}\frac{d\zeta}{\zeta^{2}}\int
d\xi\int d^{2}l\int d^{2}k\ {\cal
F}_{H}^{q}\left(\zeta,l^{2};\mu^{2};\xi_{0}=\frac{\zeta\mu^{2}}{x_{F}s_{0}}\right)$
(4) $\displaystyle\hskip 56.9055pt\times\frac{x_{F}}{\zeta(1-\xi)}{\cal
T}_{q}\left(\frac{x_{F}}{\zeta(1-\xi)},k^{2};\mu^{2};\xi_{0}=\frac{\zeta\mu^{2}}{x_{F}s_{0}}\right)\,{\cal
P}(\xi,\zeta;k+l;p,s_{0},\mu^{2}).$

In this expression, $S_{\perp}$ is the transverse overlap area in the
collision, ${\cal F}_{H}^{q}$ the TMD FF of a quark into a hadron $H$, and
${\cal T}_{q}$ the TMD PDF of a quark in the projectile. The production
probability reads

$\displaystyle{\cal
P}(\xi,\zeta;k+l;p,s_{0},\mu^{2})=\delta(\xi)\Bigg{\\{}\int_{q}s\left(-(k+l)+\frac{p}{\zeta}\right)\left[1-\frac{(k+l)\cdot
q}{q^{2}}\right]s\left(-q+\frac{p}{\zeta}\right)$ $\displaystyle\hskip
22.76228pt-2\frac{g^{2}}{(2\pi)^{3}}S_{\perp}\frac{N_{c}}{2}\Bigg{(}\int_{q}\int_{\mu^{2}}^{\left(q-\frac{1}{\zeta}p\right)^{2}}\frac{d^{2}m}{m^{2}}\int_{m^{2}\zeta/(x_{F}s_{0})}^{1}d\lambda\,\,\frac{1+(1-\lambda)^{2}}{\lambda}s\left(\frac{p}{\zeta}\right)\,s\left(q\right)$
(5)
$\displaystyle\hskip-22.76228pt+\int_{m}\int_{0}^{1}d\lambda\,\frac{1+(1-\lambda)^{2}}{\lambda}\int_{q}s\left(\frac{p}{\zeta}\right)\,s(q)\,\left[\frac{\frac{p}{\zeta}(1-\lambda)-q-m}{(\frac{p}{\zeta}(1-\lambda)-q-m)^{2}}-\frac{\frac{p}{\zeta}-q-m}{(\frac{p}{\zeta}-q-m)^{2}}\right]\cdot\frac{m}{m^{2}}\Bigg{)}\Bigg{\\}}$
$\displaystyle\hskip-22.76228pt+\frac{g^{2}}{(2\pi)^{3}}\,\frac{N_{c}}{2}\int_{m}\frac{1+(1-\xi)^{2}}{\xi}\theta\left(\xi-\frac{m^{2}\zeta}{x_{F}s_{0}}\right)$
$\displaystyle\hskip-22.76228pt\times\int_{q}s(m)s(q)\left[\frac{p/\zeta-m}{(p/\zeta-m)^{2}}-\frac{p/\zeta-(1-\xi)m}{(p/\zeta-(1-\xi)m)^{2}}\right]\cdot\left[\frac{p/\zeta-q}{(p/\zeta-q)^{2}}-\frac{p/\zeta-(1-\xi)q}{(p/\zeta-(1-\xi)q)^{2}}\right].$

As discussed in detail in [12], our expectation for the factorization scales
for TMD PDFs and FFs are, respectively,

$\displaystyle\mu^{2}_{T}={\rm
max}\left\\{k^{2},q^{2},Q_{s}^{2},\left(\frac{p}{\zeta}\right)^{2}\right\\}\approx{\rm
max}\left\\{(k+q)^{2},Q_{s}^{2},\left(\frac{p}{\zeta}\right)^{2}\right\\},$
(6) $\displaystyle\mu^{2}_{F}=[(q+k)-p/\zeta]^{2}\approx{\rm
max}\left\\{(q+k)^{2},(p/\zeta)^{2}\right\\},$

with $Q_{s}$ the saturation scale of the target, $\zeta$ the momentum fraction
in the fragmentation function, $k$ the transverse momentum in the TMD PDF and
$q$ the transverse momentum exchanged with the target. On the other hand, the
explicit calculations show that the main contribution to particle production
arises from the kinematic regions $(k+q)^{2}\sim(p/\zeta)^{2}$ and
$(k+q)^{2}\sim Q_{s}^{2}$. Thus both factorization scales become approximately
$Q_{s}$ and $|p|/\zeta$ for $|p|\ll Q_{s}$ and $|p|\gg Q_{s}$, respectively.
Therefore we expect the physically relevant choice to be

$\displaystyle\mu_{T}^{2}=\mu^{2}_{F}=\mu^{2}={\rm
max}\left\\{Q_{s}^{2},\left(p/\zeta\right)^{2}\right\\}.$ (7)

##  4 Conclusions

We have revisited the calculation of single inclusive hadron production in
$pA$ at forward rapidities at NLO within the hybrid approach. We have shown
that beyond leading order the collinear resummation paradigm does not hold in
this approach. In order to properly resum large transverse logarithms at NLO
one needs to work within the TMD factorization framework.

The need to introduce TMDs is in fact quite clear intuitively. At high
transverse momentum of the observed hadron, the naive parton model picture of
a low transverse momentum projectile parton that scatters with high momentum
transfer off the target breaks down. We showed that this is one of the
mechanisms for producing the high momentum hadron in the final state, but not
the only one. The additional mechanism discussed in [2], which amounts to the
projectile parton acquiring large transverse momentum due to perturbative
splittings in the projectile wave function, is at least equally important and
should eventually guarantee the positivity of the cross section at large
transverse momentum. It is clear that collinear factorization is physically
inappropriate for the description of such a process. We have shown in the
present paper that the proper way to account for it, in the sense of proper
resummation of large transverse logarithms associated with perturbative
splittings in the projectile wave function, is to view the parton as coming
from the projectile TMD PDF with large transverse momentum.

Also, we have identified for the first time the additional need of including
the hard perturbative fragmentation of a low transverse momentum parton
emerging from the scattering with the target. The naive collinear framework
assumes that no hard momentum is produced in the fragmentation process. We
showed here that this is not the case for the observable at hand, and that
this process needs to be taken into account in order to resum the large
logarithms.

## Acknowledgements

Special thanks are due to Guillaume Beuf who participated in early stages of
this work. NA has received financial support from Xunta de Galicia (Centro
singular de investigación de Galicia accreditation 2019-2022), from the
European Union ERDF, and from the Spanish Research State Agency under project
PID2020-119632GB-I00. AK is supported by the NSF Nuclear Theory grant 2208387.
ML is supported by the Binational Science Foundation grant #2021789. This
material is based upon work supported by the U.S. Department of Energy, Office
of Science, Office of Nuclear Physics through the Saturated Glue (SURGE)
Topical Collaboration. This work has been performed in the framework of the
European Research Council project ERC-2018-ADG-835105 YoctoLHC and the MSCA
RISE 823947 ”Heavy ion collisions: collectivity and precision in saturation
physics” (HIEIC), and has received funding from the European Union’s Horizon
2020 research and innovation program under grant agreement No. 824093.

## References

  * [1] A. Dumitru, A. Hayashigaki and J. Jalilian-Marian, Nucl. Phys. A 765 (2006), 464-482 doi:10.1016/j.nuclphysa.2005.11.014 [arXiv:hep-ph/0506308 [hep-ph]].
  * [2] T. Altinoluk and A. Kovner, Phys. Rev. D 83, 105004 (2011) doi:10.1103/PhysRevD.83.105004 [arXiv:1102.5327 [hep-ph]].
  * [3] G. A. Chirilli, B. W. Xiao and F. Yuan, Phys. Rev. Lett. 108 (2012), 122301 doi:10.1103/PhysRevLett.108.122301 [arXiv:1112.1061 [hep-ph]].
  * [4] G. A. Chirilli, B. W. Xiao and F. Yuan, Phys. Rev. D 86 (2012), 054005 doi:10.1103/PhysRevD.86.054005 [arXiv:1203.6139 [hep-ph]].
  * [5] A. M. Stasto, B. W. Xiao and D. Zaslavsky, Phys. Rev. Lett. 112 (2014) no.1, 012302 doi:10.1103/PhysRevLett.112.012302 [arXiv:1307.4057 [hep-ph]].
  * [6] A. M. Staśto, B. W. Xiao, F. Yuan and D. Zaslavsky, Phys. Rev. D 90 (2014) no.1, 014047 doi:10.1103/PhysRevD.90.014047 [arXiv:1405.6311 [hep-ph]].
  * [7] T. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky, Phys. Rev. D 91, no.9, 094016 (2015) doi:10.1103/PhysRevD.91.094016 [arXiv:1411.2869 [hep-ph]].
  * [8] K. Watanabe, B. W. Xiao, F. Yuan and D. Zaslavsky, Phys. Rev. D 92, no.3, 034026 (2015) doi:10.1103/PhysRevD.92.034026 [arXiv:1505.05183 [hep-ph]].
  * [9] H. Y. Liu, Z. B. Kang and X. Liu, Phys. Rev. D 102, no.5, 051502 (2020) doi:10.1103/PhysRevD.102.051502 [arXiv:2004.11990 [hep-ph]].
  * [10] B. W. Xiao and F. Yuan, Phys. Lett. B 788, 261-269 (2019) doi:10.1016/j.physletb.2018.10.067 [arXiv:1806.03522 [hep-ph]].
  * [11] Y. Shi, L. Wang, S. Y. Wei and B. W. Xiao, Phys. Rev. Lett. 128 (2022) no.20, 202302 doi:10.1103/PhysRevLett.128.202302 [arXiv:2112.06975 [hep-ph]].
  * [12] T. Altinoluk, N. Armesto, A. Kovner and M. Lublinsky, [arXiv:2307.14922 [hep-ph]].
  * [13] J. Collins, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 32 (2011), 1-624 Cambridge University Press, 2013, ISBN 978-1-107-64525-7, 978-1-107-64525-7, 978-0-521-85533-4, 978-1-139-09782-6
  * [14] R. Boussarie, M. Burkardt, M. Constantinou, W. Detmold, M. Ebert, M. Engelhardt, S. Fleming, L. Gamberg, X. Ji and Z. B. Kang, et al. [arXiv:2304.03302 [hep-ph]].
  * [15] F. Hautmann, H. Jung, A. Lelek, V. Radescu and R. Zlebcik, JHEP 01, 070 (2018) doi:10.1007/JHEP01(2018)070 [arXiv:1708.03279 [hep-ph]].
  * [16] A. B. Martinez, [arXiv:2307.06704 [hep-ph]].


# Lattice-Assisted Molecular Excitations in Frustrated Pyrochlores

Shang Gao<EMAIL_ADDRESS>Department of Physics, University of Science and
Technology of China, Hefei, Anhui 230026, People’s Republic of China

###### Abstract

Resonance-like molecular excitations have been discovered in a series of
pyrochlore antiferromagnets, yet their origins and relationships with the
coexisting magnon excitations remain a puzzle. Here, by incorporating the
spin-lattice coupling and the consequent four-spin interactions through the
site-phonon model, we accomplish a unified description of the molecular and
magnon excitations, revealing the molecular modes as statistical
approximations of the dispersive magnon excitations. Our work demonstrates a
general approach to understand exotic spin dynamics in magnetoelastic systems.

###### pacs:

Introduction. Spin wave theory establishes a general framework for the
description of collective spin excitations in ordered magnets, yet the
applicability of this textbook theory is seemingly challenged by a family of
frustrated pyrochlores Lee  _et al._ (2000, 2002); Chung  _et al._ (2005);
Matsuda  _et al._ (2007); Tomiyasu  _et al._ (2008); Ji  _et al._ (2009);
Matsuda  _et al._ (2010); Tomiyasu  _et al._ (2011a, b); Okamoto  _et al._
(2013); Tomiyasu  _et al._ (2014, 2013); Gao  _et al._ (2018); He  _et al._
(2021); Gen  _et al._ (2023); Kamazawa  _et al._ (2003); Dronova  _et al._
(2024). In these compounds, spin dynamics is dominated by resonance-like
excitations with a structural factor similar to that of free molecules, while
the spin wave excitations, if discernable, only exhibit relatively weak
intensity. In the absence of a unified description, the excitation spectra in
these frustrated pyrochlores are often mechanically separated into the
dispersive and resonant parts, so that the latter can be understood as
excitations of free molecules including hexamers Lee  _et al._ (2002);
Tomiyasu  _et al._ (2008); He  _et al._ (2021), heptamers Tomiyasu  _et al._
(2008, 2013); Gao  _et al._ (2018), or their derivatives Tomiyasu  _et al._
(2011a); Dronova  _et al._ (2024). However, such a compromised solution raises
a serious question: Are these molecules really existing or just phantoms of a
more fundamental theory?

Recent diffuse neutron scattering studies on MgCr2O4 shed lights on how to
address this question Bai  _et al._ (2019). Through analysis of the short-
range spin correlations at temperatures above the long-range order (LRO)
transition, it is found that the wavevector dependence of the scattering
intensity, which was previously explained through free molecules Tomiyasu  _et
al._ (2008, 2013); Gao  _et al._ (2018), can be reproduced by a lattice
Heisenberg model with couplings up to the third neighbors Conlon and Chalker
(2010); Bai  _et al._ (2019). This observation indicates the peripheral nature
of the molecular pictures in the paramagnetic phase. In the ordered state,
however, the puzzling dynamics is still left untouched, since the proposed
Heisenberg model produces no resonant excitations below the LRO transition
temperature, $T_{N}$.

The failure of the Heisenberg model may lie in the neglected lattice degree of
freedom (DOF) Lee  _et al._ (2000); Sushkov  _et al._ (2005); Chung  _et al._
(2005); Matsuda  _et al._ (2007); Kant  _et al._ (2009); Nilsen  _et al._
(2015); Rossi  _et al._ (2019). The importance of the lattice DOF is first
established through the analysis of the 1/2-magnetization plateau that is
often observed in this family of frustrated pyrochlores Ueda  _et al._ (2005a,
b); Miyata  _et al._ (2011); Tsurkan  _et al._ (2017); Gen  _et al._ (2023).
Assuming independent modulations in the bonding distances, a bond-phonon model
has been proposed Tchernyshyov  _et al._ (2002); Penc  _et al._ (2004), which
leads to effective biquadratic interactions over the neighboring spins. Spin
wave calculations following this oversimplified bond-phonon model indeed
generate a resonant mode, yet the calculated structural factors, as discussed
below, are completely different from those of the target molecules.

Here we show that under the random phase approximation (RPA), the site-phonon
model Bergman  _et al._ (2006); Wang and Vishwanath (2008); Aoyama and
Kawamura (2016, 2019), a more physical spin-lattice model that assumes
independent atomic vibrations, provides an elegant yet encompassing
explanation for the molecular-like dynamics in frustrated pyrochlores.
Applications of this approach on both sides of $T_{N}$ are demonstrated for
two model materials, MgCr2O4 and LiGaCr4O8, which represent the regular and
breathing pyrochlore-lattice compounds, respectively.

Site-phonon model. The site-phonon model on a regular pyrochlore lattice can
be described as

${\cal H}=\sum_{\langle
ij\rangle}J\left(|{\bm{r}}^{(0)}_{ij}+{\bm{u}}_{i}-{\bm{u}}_{j}|\right){\bm{S}}_{i}\cdot{\bm{S}}_{j}+\frac{c}{2}\sum_{i}|{\bm{u}}_{i}|^{2}.$
(1)

In this model, the exchange interaction $J$ between the nearest-neighboring
spins $\bm{S}_{i}$ at sites $\bm{r}_{i}$ depends on their relative distance
$\bm{r}_{ij}=\bm{r}_{ij}^{(0)}+\bm{u}_{i}-\bm{u}_{j}$, where the atomic
displacements $\bm{u}_{i}$ from the original positions $\bm{r}_{i}^{(0)}$ are
penalized by an elastic energy proportional to a positive constant $c$. In the
first order of atomic displacements, the exchange strength can be expanded as
$J\left(|{\bm{r}}^{(0)}_{ij}+{\bm{u}}_{i}-{\bm{u}}_{j}|\right)\simeq\bar{J}+(dJ/dr)|_{r=|\bm{r}_{ij}^{(0)}|}\bm{\hat{e}}_{ij}\cdot(\bm{u}_{i}-\bm{u}_{j})$
with $\bar{J}=J(|\bm{r}_{ij}^{(0)}|)$ and
$\bm{\hat{e}}_{ij}=\bm{r}_{ij}^{(0)}/|\bm{r}_{ij}^{(0)}|$. This approximation
allows the lattice DOF to be integrated out through energy minimization Aoyama
and Kawamura (2016). Assuming antiferromagnetic coupling $J>0$ with $dJ/dr<0$,
the optimal atomic displacement can be solved as

$\bm{u}^{*}_{i}=\sqrt{\frac{\bar{J}\,b}{c}}\sum_{j\in\textrm{N}(i)}\bm{\hat{e}}_{ij}(\bm{S}_{i}\cdot\bm{S}_{j})\text{,}$
(2)

where $\mathrm{N}(i)$ denotes the nearest neighbors of site $i$ and
$b=(1/cJ)[(dJ/dr)|_{r=|\bm{r}_{ij}^{(0)}|}]^{2}$ is a positive parameter that
measures the strength of the spin-lattice coupling. The Hamiltonian in Eq.
(1), after integrating out the lattice DOF, becomes
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{\textrm{SL}}$ with

${\cal H}_{0}=\bar{J}\sum_{\langle
ij\rangle}\bm{S}_{i}\cdot\bm{S}_{j}\textrm{,}\qquad{\cal H}_{\rm
SL}=-\frac{c}{2}\sum_{i}|\bm{u}_{i}^{*}|^{2}\textrm{,}$ (3)

where the spin-lattice term $\mathcal{H}_{\textrm{SL}}$ can be expressed
through four-spin interactions following Eq. (2)

$\displaystyle{\cal H}_{\rm SL}=$ $\displaystyle-\bar{J}\,b\,\sum_{\langle
i,j\rangle}\left({\bm{S}}_{i}\cdot{\bm{S}}_{j}\right)^{2}$
$\displaystyle-\frac{\bar{J}\,b}{2}\sum_{i}\sum_{j\neq
k\in\mathrm{N}(i)}{\bm{\hat{e}}}_{ij}\cdot{\bm{\hat{e}}}_{ik}\,\left({\bm{S}}_{i}\cdot{\bm{S}}_{j}\right)\left({\bm{S}}_{i}\cdot{\bm{S}}_{k}\right).\qquad$
(4)

The second term in $H_{\textrm{SL}}$ produces effective couplings among the
second- and third-neighboring spins Aoyama and Kawamura (2016, 2019). As a
comparison, the effective spin Hamiltonian of the bond-phonon model contains
only the first term in $H_{\textrm{SL}}$ Penc  _et al._ (2004); Tchernyshyov
_et al._ (2002), limiting the four-spin interactions to the nearest neighbors.

The ground state of the site-phonon model with antiferromagnetic nearest-
neighbor interactions, $J_{1}$, on a pyrochlore lattice has been studied by
classical Monte Carlo simulations Aoyama and Kawamura (2016). Two different
collinear magnetic orders with propagation vectors
$\bm{q}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ and $(1,1,0)$ have been
observed in the regime of $b>0.25$ and $b<0.25$, respectively Aoyama and
Kawamura (2016). In the following, we will show that a weak spin-lattice
coupling with $b=0.1\sim 0.2$ is sufficient to explain the molecular-like
dynamics in frustrated pyrochlores.

Numerical methods. Staring from an initial spin configuration at time $t=0$,
the time evolution of classical spins in real space are solved using the
equation of motion Moessner and Chalker (1998); Samarakoon  _et al._ (2017);
Yan  _et al._ (2018); Pohle  _et al._ (2021)

$\frac{d\bm{S}_{i}}{dt}=\frac{i}{\hbar}[\bm{S}_{i},\mathcal{H}]=\bm{H}_{i}\times\bm{S}_{i}\
\textrm{,}$ (5)

where $\bm{H}_{i}$ is the molecular field at site $\bm{r}_{i}$. For the site-
phonon model in Eq. (3), the contribution of $\mathcal{H}_{0}$ to the
molecular field is straightforward following the conventional Landau-Lifshitz-
Gilbert equation Lakshmanan (2011)

$\bm{H}^{0}_{i}=\bar{J}\sum_{j\in\mathrm{N}(i)}\bm{S}_{j}\ \textrm{.}$ (6)

The contribution of the four-spin Hamiltonian, $\mathcal{H}^{\textrm{SL}}$,
can be treated using the random phase approximation. For the first term in Eq.
(4),

$\bm{H}^{\textrm{SL(1)}}_{i}\simeq-\bar{J}\,b\sum_{j\in\mathrm{N}(i)}\bar{S}_{ij}(t)\bm{S}_{j}\
\textrm{,}$ (7)

where $\bar{S}_{ij}(t)=\bm{S}_{i}(t)\cdot\bm{S}_{j}(t)$ with its time
dependence explicitly shown. The second term in Eq. (4) involves three spins,
therefore its contribution to the molecular field depends on the relative
positions of the sites $i$, $j$, and $k$:

$\displaystyle\bm{H}^{\textrm{SL(2)}}_{i}\simeq$
$\displaystyle-\frac{\bar{J}\,b}{2}\sum_{j\neq
k\in\mathrm{N}(i)}{\bm{\hat{e}}}_{ij}\cdot{\bm{\hat{e}}}_{ik}\,\bar{S}_{ij}(t){\bm{S}}_{k}$
(8) $\displaystyle-\bar{J}\,b\sum_{\begin{subarray}{c}j\in\mathrm{N}(i),\\\
k\in\mathrm{N}(j),k\neq
i\end{subarray}}{\bm{\hat{e}}}_{ji}\cdot{\bm{\hat{e}}}_{jk}\,\bar{S}_{jk}(t){\bm{S}}_{j}\
\textrm{.}$ (9)

In a total molecular field
$\bm{H}_{i}=\bm{H}_{i}^{0}+\bm{H}_{i}^{\textrm{SL(1)}}+\bm{H}_{i}^{\textrm{SL(2)}}$,
the equations of motion form a set of linear differential equations that can
be solved numerically. In our simulations, initial spin configurations were
generated from classical Monte Carlo simulations on a $10\times 10\times 10$
or $80\times 4\times 4$ supercell of conventional cubic cells after $1\times
10^{5}\sim 5\times 10^{5}$ sweeps of single-spin Metropolis updates. Five
overrelaxation sweeps were applied after each Monte Carlo sweep to reduce
autocorrelation. For the dynamics calculations, spin configurations were
evolved in a step size $\tau=1/2J$ for a number of steps $N_{t}=400\sim 800$
using Verner’s “Most Efficient” 7/6 Runge-Kutta method Verner (2010). The
dynamic response function, $S(\bm{Q},\omega)$, was then calculated through the
fast Fourier transform (FFT)

$S(\bm{Q},\omega)=\frac{1}{\sqrt{N_{t}}N}\sum_{i,j}^{N}e^{i\bm{q}(\bm{r}_{i}-\bm{r}_{j})}\sum_{n}^{N_{t}}e^{i\omega
n\tau}\langle\bm{S}_{i}(0)\cdot\bm{S}_{j}(n\tau)\rangle\ \textrm{,}$ (10)

where $N$ is the total number of spins. A Parzen function with a halfwidth of
$N_{t}/2$ was applied over the time evolution to reduce numerical artifacts
Kühner and White (1999). For better statistics, $S(\bm{Q},\omega)$ were
averaged over $50\sim 100$ independent runs. Following the discussions in Ref.
Pohle  _et al._ (2021), a classical statistical factor of
$\omega/(2k_{\textrm{B}}T)$ was applied over $S(\bm{Q},\omega)$ to compare
with the experimental data.

Figure 1: Dynamic response function, $S(\bm{Q},\omega)$, on a regular
pyrochlore lattice calculated at $T=0.2J$ using (a) the $J_{1}$-only
Heisenberg model and (b) the site-phonon model with $b=0.2$. Constant energy
slices of the $S(\bm{Q},\omega)$ for the site-phonon model are integrated in
the energy ranges of (c) [0, 0.4], (d) [1.4 1.6], (e) [2.15, 2.35], and (f)
[2.9, 3.1] in units of $J$.  Figure 2: Comparison of the low-temperature
$S(\bm{Q},\omega)$ for (a) the bond-phonon model and (b) the site-phonon model
on a regular pyrochlore lattice with $b=0.2$ and $T=0.1J$ in both models.
Constant energy slices of the $S(\bm{Q},\omega)$ for the bond-phonon model are
integrated in the energy ranges of (c) [0.75, 1.25] and (d) [1.75 2.25].
Similar slices for the site-phonon model are presented in panels (e) and (f).

Molecular excitations above $T_{N}$. Figures 1(a) and (b) compare the dynamic
response function, $S(\bm{Q},\omega)$, for the $J_{1}$ Heisenberg model with
only the nearest-neighboring bilinear couplings $J_{1}=J$ and the site-phonon
model with $\bar{J}=J$ and $b=0.2$, calculated at temperatures above the LRO
transition temperature $T_{N}$. For the latter model, $T_{N}$ is found to be
at $\sim 1.6J$. The symmetry lines in the plots follow those of the reported
inelastic neutron scattering (INS) spectra for MgCr2O4 Bai  _et al._ (2019),
of which the spin correlations have been thoroughly characterized Tomiyasu
_et al._ (2008, 2013); Gao  _et al._ (2018); Bai  _et al._ (2019); Nassar  _et
al._ (2024). To enable a direct comparison with the experimental data Bai  _et
al._ (2019), the magnetic form factor of the Cr3+ ions is applied over the
calculated $S(\bm{Q},\omega)$. For the convenience of comparison, Table 1 in
the Appendix summarizes the previously reported experimental and calculated
results that are related to each figures in our work. As revealed in Figs.
1(a) and (b), the site-phonon model and the $J_{1}$-only model exhibit similar
excitation spectra at energy transfers higher than $\sim 1.5J$. At lower
energy transfers, the spin-lattice coupling strongly modulates the intensity
distribution, leading to a spectra that is very similar to the experimental
observations in MgCr2O4 Bai  _et al._ (2019).

Figures 1(c-f) present the constant energy slices of $S(\bm{Q},\omega)$ for
the site-phonon model. The quasielastic slice shown in Fig. 1(c) reproduces
the experimental diffuse neutron scattering pattern Lee  _et al._ (2002);
Tomiyasu  _et al._ (2013); Bai  _et al._ (2019), suggesting that the further-
neighbor exchange couplings considered in the previous fits Bai  _et al._
(2019) may arise from the spin-lattice couplings Bergman  _et al._ (2006);
Aoyama and Kawamura (2016, 2019). Slices at higher energy transfers at $E\sim
1.5J$, $2.25J$, and $3.0J$ shown in Figs. 1(d-f) also reproduce the
experimental observations that were previously ascribed to the heptamer
excitations Tomiyasu  _et al._ (2013). Since the site-phonon model and the
$J_{1}$-only model share similar $S(\bm{Q},\omega)$ at $E\gtrsim 1.5J$, it can
be concluded that the variations of the structural factors in MgCr2O4 at
$E\sim 3J$ and $4J$, observed at temperatures both above and below $T_{N}$,
should arise from dispersive spin waves rather than free heptamers.

Figure 3: Dynamic response function $S(\bm{Q},\omega)$ for the site-phonon
model on a regular pyrochlore lattice along the ($\it{h}$,1,0) direction.
Results in the left and right panels are calculated at $T=0.1J$ and $0.13J$,
respectively, both being below $T_{N}\sim 0.16J$.  Figure 4: Powder-averaged
$S(\bm{Q},\omega)$ for the site-phonon model on a breathing pyrochlore lattice
calculated at $T=1$ K (a) and 50 K (b). The coupling strengths are $J=10.4$
meV and $J^{\prime}=6.2$ meV as proposed for LiGaCr4O8, with additional spin-
lattice couplings of $b=0.1$ on both $J$ and $J^{\prime}$ bonds. The magnetic
form factor of the Cr3+ ions is applied to compare with the experimental INS
spectra.

Molecular excitations below $T_{N}$. As the failure of the $J_{1}$-only model
at temperatures below $T_{N}$ is already known Tchernyshyov  _et al._ (2002),
Fig. 2 compares the low-temperature $S(\bm{Q},\omega)$ of the site-phonon
model to that of the bond-phonon model Tchernyshyov  _et al._ (2002); Penc
_et al._ (2004). Due to finite size effects, sharp flat modes appear in both
spectra at $E\lesssim 0.5J$. These sharp modes can be suppressed by increasing
the size of the supercell (see Fig. 3 for calculations using a $80\times
4\times 4$ supercell). Similar to the previous calculations Tchernyshyov  _et
al._ (2002), a resonance-like mode is observed at $E\sim J$ for the bond-
phonon model. However, the flatness of the resonance mode and its constant
energy slices shown in Figs. 2(c) and (d) deviate strongly from the
experimental observations Tomiyasu  _et al._ (2008, 2013); Gao  _et al._
(2018); Nassar  _et al._ (2024). As a contrast, the site-phonon model produces
a resonance mode at $E\sim J$ that is highly dispersive, which is again in
agreement with the experimental observations Gao  _et al._ (2018); Nassar  _et
al._ (2024). Due to this dispersion, the slices at $E\sim J$ and $2J$, shown
in Figs. 2(e) and (f), respectively, are strongly modulated compared to those
of the bond-phonon model. The intensity modulation reproduces all the details
of the experimental observations that were previously ascribed to molecular
excitations Tomiyasu  _et al._ (2008, 2013); Gao  _et al._ (2018).

Even the splitting of the resonance mode at $E\sim J$, which was recently
established in experiments for MgCr2O4 Gao  _et al._ (2018); Nassar  _et al._
(2024), is reproduced by the site-phonon model Tchernyshyov  _et al._ (2002);
Penc  _et al._ (2004). Figure 3 compares the $S(\bm{Q},\omega)$ for the site-
phonon model along the $(h,1,0)$ direction at $T=0.1J$ and $0.13J$, both being
below $T_{N}$. The shape of the resonance mode is well reproduced, including
the tip at $E\sim 1.5J$ at wavevector transfer (2,1,0) and the band splitting
at integer $h$. Comparison between the $T=0.1J$ and $0.13J$ data reveals that
the magnon band softening at higher temperatures enlarges the hollow inside
the split resonance modes around $(2,1,0)$. This observation helps explain the
recent experimental discovery that the band splitting in MgCr2O4 becomes
enhanced in high pressure Nassar  _et al._ (2024), since the stiffness of
magnon bands may be modulated in a similar way as through thermal
fluctuations.

Now we are ready to answer the question on the reality of the free molecules.
Previous analysis based on the bond-phonon model suggests that the hexamer
excitations may represent a type of string modes on a distorted lattice
Tchernyshyov  _et al._ (2002). However, this picture no longer holds for the
site-phonon model as its magnetic order does not allow hexamer-like string
excitations Aoyama and Kawamura (2016). Considering that molecular-like
excitations emerge only after introducing dispersion to the resonance modes as
discussed for Fig. 2, we can conclude that molecules in frustrated pyrochlores
are not real, but statistical approximations for the dispersive spin wave
excitations that are modulated by the spin-lattice coupling at the
characteristic energies.

Breathing pyrochlore lattice. The site-phonon model can be immediately
generalized to the breathing pyrochlore lattice, on which the neighboring
tetrahedra are of two different sizes and therefore two different exchange
strengths $J$ and $J^{\prime}$ are expected Aoyama and Kawamura (2019); Szabó
and Nilsen (2024). Figure 4 compares the powder-averaged dynamic response
function of the site-phonon model with $J=10.4$ meV and $J^{\prime}=6.2$ meV,
together with a spin-lattice coupling of $b=0.1$ on both types of tetrahedra.
The strengths of the exchange couplings are chosen for a direct comparison
with the experimental data for LiGaCr4O4, in which the coexistence of spin
waves and molecular excitations have been experimentally observed but not
explained He  _et al._ (2021). At $T=1$ K in the ordered phase, a resonance-
like mode at $\sim 6$ meV, together with a wavevector dependence that was
previously attributed to free hexamers He  _et al._ (2021), can be reproduced
by the site-phonon model. At $T=50$ K in the paramagnetic phase, the resonance
mode collapses to the elastic line, while its wavevector dependence stays
almost unchanged, all in agreement with the experiments He  _et al._ (2021).

Conclusion and discussion. By extending the site-phonon model to the dynamical
regime, we successfully explain the coexistence of spin waves and molecular
excitations in a series of frustrated pyrochlores. Through our analysis, the
molecular excitations are established as statistical approximations to
dispersive spin wave excitations rather than real entities.

The application of the site-phonon model in the dynamical regime shall be
generalized to a large family of magnetoelastic materials, including the model
triangular-lattice compounds NiGa2S4 Nakatsuji  _et al._ (2005); Stock  _et
al._ (2010); Valentine  _et al._ (2020) and $\alpha$-CaCr2O4 Toth  _et al._
(2012), where the lattice DOF has been shown to be important yet the spin
dynamics are not fully understood. On a broader scope, the effective four-spin
interactions, especially those involving three spin sites, may also arise from
multiple hopping of electrons Wysocki  _et al._ (2011); Paul  _et al._ (2020)
or multipolar interactions Chen  _et al._ (2010) even in the absence of a
strong spin-lattice couplings. A better understanding of their origins and
effects, which are far less explored compared to those of the two-spin
interactions, can be crucial to comprehend the rich spin correlated states in
the strongly correlated electronic systems.

###### Acknowledgements.

The author acknowledges helpful discussions with Oksana Zaharko, Tom Fennell,
Christian Rüegg, Joe Paddison, Xiaojian Bai, Yuan Li, Gang Chen, Erxi Feng,
James Jun He, and Jyong-Hao Chen. The author thanks the generous host at the
Interdisciplinary Center for Theoretical Study at USTC. Numerical calculations
in this paper were partly performed on the supercomputing system in the
Supercomputing Center of USTC. This project was funded by the National Science
Foundation of China (NSFC) under the Grant No. 12374152.

*

##  Appendix A Comparisons with the previous experimental data and molecular
models

For the convenience of comparison, Table 1 lists the previously reported
experimental and calculated molecular spectra that are related to each figures
in our work.

Table 1:  Previously reported experimental and calculated spectra that are related to each figures in our work. Our Work |  Experiments |  Molecular models
---|---|---
Fig. 1(b) |  Fig. 2(a) in Bai  _et al._ (2019) |  Fig. 2(a) in Bai  _et al._ (2019)
Fig. 1(c) |  Fig. 3 in Lee  _et al._ (2002) |  Fig. 3 in Lee  _et al._ (2002)
|  Fig. 1(d) in Bai  _et al._ (2019) |  -
Figs. 1(d)-(f) |  Fig. 1(d) in Tomiyasu  _et al._ (2013) |  Fig. 1(f) in Tomiyasu  _et al._ (2013)
Fig. 1(e) |  Fig. 5(d) in Gao  _et al._ (2018) |  Fig. 10 in Gao  _et al._ (2018)
Fig. 2(a) |  - |  Fig. 5 in Tchernyshyov  _et al._ (2002)
Fig. 2(b) |  Fig. 6(b) in Gao  _et al._ (2018) |  -
Figs. 2(e),(f) |  Figs. 5(b),(c) in Gao  _et al._ (2018) |  Fig. 10 in Gao  _et al._ (2018)
|  Figs. 2(a),(c) in Tomiyasu  _et al._ (2008) |  Figs. 2(e),(g) in Tomiyasu  _et al._ (2008)
|  Fig. 1(e) in Tomiyasu  _et al._ (2013) |  Fig. 1(f) in Tomiyasu  _et al._ (2013)
Fig. 3 |  Fig. 7(a) in Gao  _et al._ (2018) |  -
|  Fig. 5 in Nassar  _et al._ (2024) |  -
Fig. 4(a) |  Fig. 3(a) in He  _et al._ (2021) |  Fig. 3(d) in He  _et al._ (2021)
Fig. 4(b) |  Fig. 1(a) in He  _et al._ (2021) |  -

## References

  * Lee  _et al._ (2000) S.-H. Lee, C. Broholm, T. H. Kim, W. Ratcliff, and S.-W. Cheong, Phys. Rev. Lett. 84, 3718 (2000).
  * Lee  _et al._ (2002) S. H. Lee, C. Broholm, W. Ratcliff, G. Gasparovic, Q. Huang, T. H. Kim, and S. W. Cheong, Nature 418, 856 (2002).
  * Chung  _et al._ (2005) J.-H. Chung, M. Matsuda, S.-H. Lee, K. Kakurai, H. Ueda, T. J. Sato, H. Takagi, K.-P. Hong, and S. Park, Phys. Rev. Lett. 95, 247204 (2005).
  * Matsuda  _et al._ (2007) M. Matsuda, H. Ueda, A. Kikkawa, Y. Tanaka, K. Katsumata, Y. Narumi, T. Inami, Y. Ueda, and S. H. Lee, Nat. Phys. 3, 397 (2007).
  * Tomiyasu  _et al._ (2008) K. Tomiyasu, H. Suzuki, M. Toki, S. Itoh, M. Matsuura, N. Aso, and K. Yamada, Phys. Rev. Lett. 101, 0177401 (2008).
  * Ji  _et al._ (2009) S. Ji, S.-H. Lee, C. Broholm, T. Y. Koo, W. Ratcliff, S.-W. Cheong, and P. Zschack, Phys. Rev. Lett. 103, 037201 (2009).
  * Matsuda  _et al._ (2010) M. Matsuda, K. Ohoyama, S. Yoshii, H. Nojiri, P. Frings, F. Duc, B. Vignolle, G. L. J. A. Rikken, L.-P. Regnault, S.-H. Lee, H. Ueda, and Y. Ueda, Phys. Rev. Lett. 104, 047201 (2010).
  * Tomiyasu  _et al._ (2011a) K. Tomiyasu, H. Ueda, M. Matsuda, M. Yokoyama, K. Iwasa, and K. Yamada, Phys. Rev. B 84, 035115 (2011a).
  * Tomiyasu  _et al._ (2011b) K. Tomiyasu, M. K. Crawford, D. T. Adroja, P. Manuel, A. Tominaga, S. Hara, H. Sato, T. Watanabe, S. I. Ikeda, J. W. Lynn, K. Iwasa, and K. Yamada, Phys. Rev. B 84, 054405 (2011b).
  * Okamoto  _et al._ (2013) Y. Okamoto, G. J. Nilsen, J. P. Attfield, and Z. Hiroi, Phys. Rev. Lett. 110, 097203 (2013).
  * Tomiyasu  _et al._ (2014) K. Tomiyasu, K. Iwasa, H. Ueda, S. Niitaka, H. Takagi, S. Ohira-Kawamura, T. Kikuchi, Y. Inamura, K. Nakajima, and K. Yamada, Phys. Rev. Lett. 113, 236402 (2014).
  * Tomiyasu  _et al._ (2013) K. Tomiyasu, T. Yokobori, Y. Kousaka, R. I. Bewley, T. Guidi, T. Watanabe, J. Akimitsu, and K. Yamada, Phys. Rev. Lett. 110, 077205 (2013).
  * Gao  _et al._ (2018) S. Gao, K. Guratinder, U. Stuhr, J. S. White, M. Mansson, B. Roessli, T. Fennell, V. Tsurkan, A. Loidl, M. Ciomaga Hatnean, G. Balakrishnan, S. Raymond, L. Chapon, V. O. Garlea, A. T. Savici, A. Cervellino, A. Bombardi, D. Chernyshov, C. Rüegg, J. T. Haraldsen, and O. Zaharko, Phys. Rev. B 97, 134430 (2018).
  * He  _et al._ (2021) Z. He, Y. Gu, H. Wo, Y. Feng, D. Hu, Y. Hao, Y. Gu, H. C. Walker, D. T. Adroja, and J. Zhao, Phys. Rev. Lett. 127, 147205 (2021).
  * Gen  _et al._ (2023) M. Gen, A. Ikeda, K. Aoyama, H. O. Jeschke, Y. Ishii, H. Ishikawa, T. Yajima, Y. Okamoto, X. Zhou, D. Nakamura, S. Takeyama, K. Kindo, Y. H. Matsuda, and Y. Kohama, Proc. Natl. Acad. Sci. USA 120, e2302756120 (2023).
  * Kamazawa  _et al._ (2003) K. Kamazawa, Y. Tsunoda, H. Kadowaki, and K. Kohn, Phys. Rev. B 68, 024412 (2003).
  * Dronova  _et al._ (2024) M. G. Dronova, V. Petříček, Z. Morgan, F. Ye, D. M. Silevitch, and Y. Feng, Phys. Rev. B 109, 064421 (2024).
  * Bai  _et al._ (2019) X. Bai, J. A. M. Paddison, E. Kapit, S. M. Koohpayeh, J.-J. Wen, S. E. Dutton, A. T. Savici, A. I. Kolesnikov, G. E. Granroth, C. L. Broholm, J. T. Chalker, and M. Mourigal, Phys. Rev. Lett. 122, 097201 (2019).
  * Conlon and Chalker (2010) P. H. Conlon and J. T. Chalker, Phys. Rev. B 81, 224413 (2010).
  * Sushkov  _et al._ (2005) A. B. Sushkov, O. Tchernyshyov, W. R. II, S. W. Cheong, and H. D. Drew, Phys. Rev. Lett. 94, 137202 (2005).
  * Kant  _et al._ (2009) C. Kant, J. Deisenhofer, T. Rudolf, F. Mayr, F. Schrettle, A. Loidl, V. Gnezdilov, D. Wulferding, P. Lemmens, and V. Tsurkan, Phys. Rev. B 80, 214417 (2009).
  * Nilsen  _et al._ (2015) G. J. Nilsen, Y. Okamoto, T. Masuda, J. Rodriguez-Carvajal, H. Mutka, T. Hansen, and Z. Hiroi, Phys. Rev. B 91, 174435 (2015).
  * Rossi  _et al._ (2019) L. Rossi, A. Bobel, S. Wiedmann, R. Küchler, Y. Motome, K. Penc, N. Shannon, H. Ueda, and B. Bryant, Phys. Rev. Lett. 123, 027205 (2019).
  * Ueda  _et al._ (2005a) H. Ueda, H. Mitamura, T. Goto, and Y. Ueda, Prog. Theo. Phys. Supp. 159, 256 (2005a).
  * Ueda  _et al._ (2005b) H. Ueda, H. A. Katori, H. Mitamura, T. Goto, and H. Takagi, Phys. Rev. Lett. 94, 047202 (2005b).
  * Miyata  _et al._ (2011) A. Miyata, H. Ueda, Y. Ueda, H. Sawabe, and S. Takeyama, Phys. Rev. Lett. 107, 207203 (2011).
  * Tsurkan  _et al._ (2017) V. Tsurkan, S. Zherlitsyn, L. Prodan, V. Felea, P. T. Cong, Y. Skourski, Z. Wang, J. Deisenhofer, H.-A. K. von Nidda, J. Wosnitza, and A. Loidl, Sci. Adv. 3, e1601982 (2017).
  * Tchernyshyov  _et al._ (2002) O. Tchernyshyov, R. Moessner, and S. L. Sondhi, Phys. Rev. Lett. 88, 067203 (2002).
  * Penc  _et al._ (2004) K. Penc, N. Shannon, and H. Shiba, Phys. Rev. Lett. 93, 197203 (2004).
  * Bergman  _et al._ (2006) D. L. Bergman, R. Shindou, G. A. Fiete, and L. Balents, Phys. Rev. B 74, 134409 (2006).
  * Wang and Vishwanath (2008) F. Wang and A. Vishwanath, Phys. Rev. Lett. 100, 077201 (2008).
  * Aoyama and Kawamura (2016) K. Aoyama and H. Kawamura, Phys. Rev. Lett. 116, 257201 (2016).
  * Aoyama and Kawamura (2019) K. Aoyama and H. Kawamura, Phys. Rev. B 99, 144406 (2019).
  * Moessner and Chalker (1998) R. Moessner and J. T. Chalker, Phys. Rev. B 58, 12049 (1998).
  * Samarakoon  _et al._ (2017) A. M. Samarakoon, A. Banerjee, S.-S. Zhang, Y. Kamiya, S. E. Nagler, D. A. Tennant, S.-H. Lee, and C. D. Batista, Phys. Rev. B 96, 134408 (2017).
  * Yan  _et al._ (2018) H. Yan, R. Pohle, and N. Shannon, Phys. Rev. B 98, 140402 (2018).
  * Pohle  _et al._ (2021) R. Pohle, H. Yan, and N. Shannon, Phys. Rev. B 104, 024426 (2021).
  * Lakshmanan (2011) M. Lakshmanan, Philos. Transact. A: Math., Phys. Eng. Sci. 369, 1280 (2011).
  * Verner (2010) J. H. Verner, Numer. Algorithms 53, 383 (2010).
  * Kühner and White (1999) T. D. Kühner and S. R. White, Phys. Rev. B 60, 335 (1999).
  * Nassar  _et al._ (2024) L. S. Nassar, H. Lane, B. Haberl, M. Graves-Brook, B. Winn, S. M. Koohpayeh, and M. Mourigal, Phys. Rev. B 109, 064415 (2024).
  * Szabó and Nilsen (2024) A. Szabó and G. J. Nilsen, Phys. Rev. B 109, 104425 (2024).
  * Nakatsuji  _et al._ (2005) S. Nakatsuji, Y. Nambu, H. Tonomura, O. Sakai, S. Jonas, C. Broholm, H. Tsunetsugu, Y. Qiu, and Y. Maeno, Science 309, 1697 (2005).
  * Stock  _et al._ (2010) C. Stock, S. Jonas, C. Broholm, S. Nakatsuji, Y. Nambu, K. Onuma, Y. Maeno, and J.-H. Chung, Phys. Rev. Lett. 105, 037402 (2010).
  * Valentine  _et al._ (2020) M. E. Valentine, T. Higo, Y. Nambu, D. Chaudhuri, J. Wen, C. Broholm, S. Nakatsuji, and N. Drichko, Phys. Rev. Lett. 125, 197201 (2020).
  * Toth  _et al._ (2012) S. Toth, B. Lake, K. Hradil, T. Guidi, K. C. Rule, M. B. Stone, and A. T. M. N. Islam, Phys. Rev. Lett. 109, 127203 (2012).
  * Wysocki  _et al._ (2011) A. L. Wysocki, K. D. Belashchenko, and V. P. Antropov, Nat. Phys. 7, 485 (2011).
  * Paul  _et al._ (2020) A. Paul, C.-M. Chung, T. Birol, and H. J. Changlani, Phys. Rev. Lett. 124, 167203 (2020).
  * Chen  _et al._ (2010) G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82, 174440 (2010).