# a more general framework than the $\delta$-primary hyperideals

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

The $\delta$-primary hyperideal is a concept unifing the $n$-ary prime and
$n$-ary primary hyperideals under one frame where $\delta$ is a function which
assigns to each hyperideal $Q$ of $G$ a hyperideal $\delta(Q)$ of the same
hyperring with specific properties. In this paper, for a commutative Krasner
$(m,n)$-hyperring $G$ with scalar identity $1$, we aim to introduce and study
the notion of $(t,n)$-absorbing $\delta$-semiprimary hyperideals which is a
more general structure than $\delta$-primary hyperideals. We say that a proper
hyperideal $Q$ of $G$ is an $(t,n)$-absorbing $\delta$-semiprimary hyperideal
if whenever $k(a_{1}^{tn-t+1})\in Q$ for $a_{1}^{tn-t+1}\in G$, then there
exist $(t-1)n-t+2$ of the $a_{i}^{,}$s whose $k$-product is in $\delta(Q)$.
Furthermore, we extend the concept to weakly $(t,n)$-absorbing
$\delta$-semiprimary hyperideals. Several properties and characterizations of
these classes of hyperideals are determined. In particular, after defining
srongly weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideals, we present
the condition in which a weakly $(t,n)$-absorbing $\delta$-semiprimary
hyperideal is srongly. Moreover, we show that $k(Q^{(tn-t+1)})=0$ where the
weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal $Q$ is not
$(t,n)$-absorbing $\delta$-semiprimary. Also, we investigate the stability of
the concepts under intersection, homomorphism and cartesian product of
hyperrings.

###### Key words and phrases:

$(t,n)$-absorbing $\delta$-semiprimary hyperideal, weakly $(t,n)$-absorbing
$\delta$-semiprimary hyperideal, $\delta$-$(t,n)$-zero.

###### 2010 Mathematics Subject Classification:

16Y99

##  1\. Introduction

For the first time, the idea of 2-absorbing ideals as an extension of prime
ideals was presented by Badawi in [3]. The concept and it,s generalizations
have been widely studied by many researchers. The notion of 2-absorbing
$\delta$-semiprimary ideals in a commutative ring which is an expansion of the
2-absorbing ideals was introduced by Celikel. From [4], a proper ideal $I$ of
a commutative ring $R$ refers to a (weakly) 2-absorbing $\delta$-semiprimary
ideal if $x,y,z\in R$ and ($0\neq xyz\in I$) $xyz\in I$ imply $xy\in\delta(I)$
or $yz\in\delta(I)$ or $xz\in\delta(I)$ where $\delta$ is a function that
assigns to each ideal $I$ an ideal $\delta(I)$ of the same ring.

Study on $n$-ary algebras goes back to Kasner’s lecture [9] at a scientific
meeting in 1904. In 1928, the first paper was written concerning the theory of
$n$-ary groups by Dorente. The $n$-ary structures have been studied in
[5],[7], [10],[11], [12], [13], [14] and [15]. There is a hyperring in which
the addition is a hyperoperation, while the multiplication is an ordinary
binary operation. The hyperring introduced by Krasner is called Krasner
hyperring. The Krasner $(m,n)$-hyperring as a generalization of the Krasner
hyperrings and a subclass of $(m,n)$-hyperrings was introduced by Mirvakili
and Davvaz in [14]. $(G,h,k)$, or simply $G$, is a Krasner $(m,n)$-hyperring
if: (1) $(G,h$) is a canonical $m$-ary hypergroup; (2) $(G,k)$ is a $n$-ary
semigroup; (3) The operation $k$ is distributive with respect to the
hyperoperation $h$ , i.e., for every $g^{i-1}_{1},g^{n}_{i+1},x^{m}_{1}\in G$,
and $1\leq i\leq n$,
$k(g^{i-1}_{1},h(x^{m}_{1}),g^{n}_{i+1})=h(k(g^{i-1}_{1},x_{1},g^{n}_{i+1}),\cdots,k(g^{i-1}_{1},x_{m},g^{n}_{i+1}));$
(4) $0$ is a zero element of the $n$-ary operation $k$, i.e., for every
$g^{n}_{2}\in R$ ,
$k(0,g^{n}_{2})=k(g_{2},0,g^{n}_{3})=\cdots=k(g^{n}_{2},0)=0$. A non-empty
subset $H$ of $G$ is a subhyperring of $G$ if $(H,h,k)$ is a Krasner
$(m,n)$-hyperring. The non-empty subset $I$ of $G$ is a hyperideal if $(I,h)$
is an $m$-ary subhypergroup of $(G,h)$ and
$k(g^{i-1}_{1},I,g_{i+1}^{n})\subseteq I$, for all $g^{n}_{1}\in G$ and $1\leq
i\leq n$. By $g^{j}_{i}$ we mean the sequence $g_{i},g_{i+1},\cdots,g_{j}$ and
it is the empty symbol if $j<i$. Then we have
$h(a_{1},\cdots,a_{i},b_{i+1},\cdots,b_{j},c_{j+1},\cdots,c_{n})=h(a^{i}_{1},b^{j}_{i+1},c^{n}_{j+1})$
and we write $h(a^{i}_{1},b^{(j-i)},c^{n}_{j+1}),$ where
$b_{i+1}=\cdots=b_{j}=b$. We define $h(G^{n}_{1})=\bigcup\\{h(g^{n}_{1})\ |\
g_{i}\in G_{i},1\leq i\leq n\\}$ where $G_{1}^{n}$ are non-empty subsets of
$G$. Recall from [1] that a proper hyperideal $P$ of $G$ is called $n$-ary
prime if $k(G_{1}^{n})\subseteq P$ for hyperideals $G_{1}^{n}$ of $R$ implies
that $G_{i}\subseteq P$ for some $1\leq i\leq n$. If $k(g^{n}_{1})\in P$ for
all $g^{n}_{1}\in G$ and some hyperideal $P$ of $G$ implies that $g_{i}\in P$
for some $1\leq i\leq n$, then $P$ is an $n$-ary prime hyperideal of $G$, by
Lemma 4.5 in [1]. Let $I$ be a hyperideal in $G$ with scalar identity $1$. By
$rad(I)$ we mean the intersection is taken over all $n$-ary prime hyperideals
$P$ which contain $I$. $rad(I)=G$ if the set of all $n$-ary prime hyperideals
containing $I$ is empty . It was shown (Theorem 4.23 in [1]) that if $g\in
rad(I)$, then there exists $s\in\mathbb{N}$ such that $k(g^{(s)},1^{(n-s)})\in
I$ for $s\leq n$, or $k_{(l)}(g^{(s)})\in I$ for $s=l(n-1)+1$. A proper
hyperideal $I$ of $G$ with the scalar identity $1$ is an $n$-ary primary
hyperideal if $k(g^{n}_{1})\in I$ and $g_{i}\notin I$ imply
$k(g_{1}^{i-1},1,g_{i+1}^{n})\in rad(I)$ for some $1\leq i\leq n$. Theorem
4.28 in [1] shows that the radical of an $n$-ary primary hyperideal of $G$ is
an $n$-ary prime hyperideal of $G$. Let $\mathcal{HI}(G)$ denote the set of
all hyperideals of $G$. Recall from [2] that a function $\delta$ from
$\mathcal{HI}(G)$ to $\mathcal{HI}(G)$ is a hyperideal expansion of $G$ if
$I\subseteq\delta(I)$ and if $I\subseteq J$ for any hyperideals $I,J$ of $G$,
then $\delta(I)\subseteq\delta(J)$. For example, the mapping $\delta_{0}$,
$\delta_{1}$ and $\delta_{R}$, defined by $\delta_{0}(I)=I$,
$\delta_{1}(I)=rad(I)$ and $\delta_{R}(I)=R$ for all hyperideals $I$ of $G$,
are hyperideal expansions of $G$. Furthermore, the function $\delta_{q}$,
defined by $\delta_{q}(I/J)=\delta(I)/J$ for any hyperideal $I$ of $G$
containing hyperideal $J$ and an expansion function $\delta$ of $G$, is a
hyperideal expansion of $G/J$. A proper hyperideal $P$ of $G$ is an
$(t,n)$-absorbing hyperideal as in [8] if $k(g_{1}^{tn-t+1})\in P$ for some
$g_{1}^{tn-t+1}\in G$ implies that there exist $(t-1)n-t+2$ of the
$g_{i}^{,}$s whose $k$-product is in $P$. From the paper, a proper $P$ of $G$
is called an $(t,n)$-absorbing primary hyperideal if $k(g_{1}^{tn-t+1})\in P$
for some $g_{1}^{tn-t+1}\in G$ implies that $k(g_{1}^{(t-1)n-t+2})\in P$ or a
$k$-prodect of $(t-1)n-t+2$ of the $g_{i}^{,}$s except $k(g_{1}^{tn-t+2})$ is
in $rad(P)$. The notion of $n$-ary $\delta$-primary hyperideals was introduced
in a Krasner $(m,n)$-hyperring in [2]. This concept unifies the $n$-ary prime
and $n$-ary primary hyperideals under one frame. A proper hyperideal $Q$ of
$G$ is called $\delta$-primary hyperideal if $k(a_{1}^{n})\in Q$ for
$a_{1}^{n}\in G$ implies that $a_{i}\in Q$ or
$k(a_{1}^{i-1},1,a_{i+1}^{n})\in\delta(Q)$ for some $1\leq i\leq n$. Also, a
proper hyperideal $Q$ of $G$ is $(t,n)$-absorbing $\delta$-primary if for
$a_{1}^{tn-t+1}\in G$, $k(a_{1}^{tn-t+1})\in Q$ implies that
$k(a_{1}^{(t-1)n-t+2})\in Q$ or a $k$-product of $(t-1)n-t+2$ of $a_{i}^{,}$ s
except $k(a_{1}^{(t-1)n-t+2})$ is in $\delta(Q)$.

Now in this paper, we aim to define and study the notion of $(t,n)$-absorbing
$\delta$-semiprimary hyperideals which is more general than the concept of
$\delta$-primary hyperideals in a Krasner $(m,n)$-hyperring. Additionally, we
present an extension of the notion called weakly $(t,n)$-absorbing
$\delta$-semiprimary hyperideals. Among many results in this paper, it is
shown (Theorem 2.30) that if $Q$ is a weakly $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G$ that is not $(t,n)$-absorbing
$\delta$-semiprimary, then $k(Q^{(tn-t+1)})=0$. Let $Q$ be a weakly
$(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$ and $0\neq
k(Q_{1}^{tn-t+1})\subseteq Q$ for some hyperideals $Q_{1}^{tn-t+1}$ of $G$. It
is shown (Theorem 2.25) that if $Q$ is a free $\delta$-$(t,n)$-zero with
respect to $k(Q_{1}^{tn-t+1})$, then $k$-product of $(t-1)n-t+2$ of the
$Q_{i}$ is a subset of $\delta(Q)$. Moreover, the stability of these concepts
are examined under intersection, homomorphism and cartesian product of
hyperrings.

##  2\. (weakly) $(t,n)$-absorbing $\delta$-semiprimay hyperideals

Throughout this section, $G$ is a commutative Krasner $(m,n)$-hyperring with
scalar identity $1$. Initially, we give the deﬁnition of $(t,n)$-absorbing
$\delta$-semiprimay hyperideals of $G$.

######  Definition 2.1.

Let $\delta$ be a hyperideal expansion of $G$ and $t$ be a positive integer. A
proper hyperideal $Q$ of $G$ is called an $(t,n)$-absorbing
$\delta$-semiprimary hyperideal if whenever $k(a_{1}^{tn-t+1})\in Q$ for
$a_{1}^{tn-t+1}\in G$, then there exist $(t-1)n-t+2$ of the $a_{i}^{,}$s whose
$k$-product is in $\delta(Q)$.

######  Example 2.2.

For all $n$-ary prime hyperideal of $G$, we have

$n$-ary prime $\Longrightarrow$ $n$-ary $\delta$-primary $\Longrightarrow$
$(t,n)$-absorbing $\delta$-primary

$\Longrightarrow$ $(t,n)$-absorbing $\delta$-semiprimary

The next example shows that the inverse of $"\Longrightarrow"^{,}$s in Example
2.2, is not true, in general.

######  Example 2.3.

In the Krasner $(2,2)$-hyperring $(G=[0,1],\boxplus,\circ)$ that $"\circ"$ is
the usual multiplication on real numbers and 2-ary hyperoperation $"\boxplus"$
is defined by

$a\boxplus b=\begin{cases}\\{max\\{a,b\\}\\},&\text{if $a\neq b$}\\\
$[0,a]$,&\text{if $a=b,$}\end{cases}$

the hyperideal $Q=[0,0.5]$ is a $(2,2)$-absorbing $\delta_{1}$-semiprimary
hyperideal of $G$ but it is not $2$-ary prime.

######  Theorem 2.4.

Let $Q$ be an $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$ with
$rad(\delta(Q))\subseteq\delta(rad(Q))$. Then $rad(Q)$ is an $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G$.

###### Proof.

Let $k(a_{1}^{tn-t+1})\in rad(Q)$ for $a_{1}^{tn-t+1}\in G$ such that all
products of $(t-1)n-t+2$ of the $a_{i}^{,}$s, other than
$k(a_{1}^{(t-1)n-t+2})$, are not in $\delta(rad(Q))$. By the assumption, we
conclude that none of the $k$-products of the $a_{i}^{,}$s are in
$rad(\delta(Q))$. From $k(a_{1}^{tn-t+1})\in rad(Q)$, it follows that there
exists $s\in\mathbb{N}$ with $k(k(a_{1}^{tn-t+1})^{(s)},1^{(n-s)})\in Q$, for
$s\leq n$ or $k_{(l)}(k(a_{1}^{tn-t+1})^{(s)})\in Q$, for $s>n$ and
$s=l(n-1)+1$. In the first possibility, we get
$k(k(a_{1})^{(s)},k(a_{2})^{(s)},\cdots,k(a_{tn-t+1})^{(s)},1^{(n-s)})\in Q$.
Since $Q$ is an $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$, we
conclude that

$\hskip
56.9055ptk(k(a_{1})^{(s)},k(a_{2})^{(s)},\cdots,k(a_{(t-1)n-t+2})^{(s)},1^{(n-s)})$

$\hskip 71.13188pt=k(k(a_{1}^{(t-1)n-t+2})^{(s)},1^{(n-s)})\in\delta(Q)$
because none of the $k$-products of the $a_{i}^{,}$s are in $rad(\delta(Q))$.
Since $k(a_{1}^{(t-1)n-t+2})\in rad(\delta(Q))$ and
$rad(\delta(Q))\subseteq\delta(rad(Q))$, then we have
$k(a_{1}^{(t-1)n-t+2})\in\delta(rad(Q))$. If
$k_{(l)}(k(a_{1}^{tn-t+1})^{(s)})\in Q$, for $s>n$ and $s=l(n-1)+1$, then we
are done similarly. Thus $rad(Q)$ is an $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G$. ∎

The following result is a direct consequence of the previous theorem.

######  Corollary 2.5.

If $Q$ is an $(t,n)$-absorbing $\delta_{1}$-semiprimary hyperideal of $G$,
then $rad(Q)$ is an $(t,n)$-absorbing hyperideal of $G$.

From [1], the hyperideal generated by an element $g$ in $G$ is defined by
$<g>=k(G,g,1^{(n-2)})=\\{k(r,g,1^{(n-2)})\ |\ r\in G\\}.$ The following
theorem will give us a characterization of $(t,n)$-absorbing
$\delta$-semiprimary hyperideals.

######  Theorem 2.6.

Every proper hyperideal is an $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G$ if and only if every proper principal hyperideal is an
$(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$.

###### Proof.

$\Longrightarrow$ It is obvious.

$\Longleftarrow$ Assume that $Q$ is a proper hyperideal of $G$ and
$k(a_{1}^{tn-t+1})\in Q$ for $a_{1}^{tn-t+1}\in G$. Therefore
$k(a_{1}^{tn-t+1})\in<k(a_{1}^{tn-t+1})>$. Since every proper principal
hyperideal is an $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$,
there exist $(t-1)n-t+2$ of the $a_{i}^{,}$s whose $k$-product is in
$\delta(<k(a_{1}^{tn-t+1})>)\subseteq\delta(Q)$. Hence $Q$ is an
$(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$. ∎

Recall from [2] that a hyperideal expansion $\delta$ of $G$ is called
intersection preserving if it satisfies $\delta(P\cap
Q)=\delta(P)\cap\delta(Q)$, for all hyperideals $P$ and $Q$ of $G$. For
example, hyperideal expansion $\delta_{1}$ of $G$ is intersection preserving.

######  Theorem 2.7.

Let the hyperideal expansion $\delta$ of $G$ be intersection preserving. If
$Q_{1}^{s}$ are $(t,n)$-absorbing $\delta$-semiprimary hyperideals of $G$ Such
that $\delta(Q_{i})=P$ for each $1\leq i\leq s$, then
$Q=\bigcap_{i=1}^{s}Q_{i}$ is an $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G$ with $\delta(Q)=P$.

###### Proof.

Assume that $k(a_{1}^{tn-t+1})\in Q$ for $a_{1}^{tn-t+1}\in G$ such that
$k(a_{1}^{(t-1)n-t+2})\notin\delta(Q)$. Since
$\delta(Q)=\delta(\cap_{i=1}^{s}Q_{i})=\cap_{i=1}^{s}\delta(Q_{i})=P$, then
there exists $1\leq u\leq s$ such that
$k(a_{1}^{(t-1)n-t+2})\notin\delta(Q_{u})$. Since $Q_{u}$ is an
$(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$ and
$k(a_{1}^{tn-t+1})\in Q_{u}$, then there is a $k$-product of $(t-1)n-t+2$ of
the $a_{i}^{,}$s is in $\delta(Q_{u})=P=\delta(Q)$. Thus
$Q=\bigcap_{i=1}^{s}Q_{i}$ is an $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G$ with $\delta(Q)=P$. ∎

Let $(G_{1},h_{1},k_{1})$ and $(G_{2},h_{2},k_{2})$ be two Krasner
$(m,n)$-hyperrings such that $1_{G_{1}}$ and $1_{G_{2}}$ be scalar identities
of $G_{1}$ and $G_{2}$, respectively. Then $(G_{1}\times G_{2},h=h_{1}\times
h_{2},k=k_{1}\times k_{2})$ is a Krasner $(m,n)$-hyperring where

$h_{1}\times h_{2}((a_{1},b_{1}),\cdots,(a_{m},b_{m}))=\\{(a,b)\ |\ a\in
h_{1}(a_{1}^{m}),b\in h_{2}(b_{1}^{m})\\},$

$k_{1}\times
k_{2}((a_{1},b_{1}),\cdots,(a_{n},b_{n}))=(k_{1}(a_{1}^{n}),k_{2}(b_{1}^{n})),$
for all $a_{i}\in G_{1}$ and $b_{i}\in G_{2}$ [6].

######  Theorem 2.8.

Let $\delta_{1}$ and $\delta_{2}$ be two hyperideal expansions of Krasner
$(m,n)$-hyperrings $G_{1}$ and $G_{2}$, respectively, such that
$\delta(Q_{1}\times Q_{2})=\delta_{1}(Q_{1})\times\delta_{2}(Q_{2})$ for
hyperideals $Q_{1}$ and $Q_{2}$ of $G_{1}$ and $G_{2}$, respectively. If
$Q=Q_{1}\times Q_{2}$ is an $(t+1,n)$-absorbing $\delta$-semiprimary
hyperideal of $G=G_{1}\times G_{2}$, then either $Q_{1}$ is an
$(t+1,n)$-absorbing $\delta_{1}$-semiprimary hyperideal of $G_{1}$ and
$\delta_{2}(Q_{2})=G_{2}$ or $Q_{2}$ is an $(t+1,n)$-absorbing
$\delta_{2}$-semiprimary hyperideal of $G_{2}$ and $\delta_{1}(Q_{1})=G_{1}$
or $Q_{i}$ is an $(t,n)$-absorbing $\delta_{i}$-semiprimary hyperideal of
$G_{i}$ for each $i\in\\{1,2\\}$.

###### Proof.

Let $Q=Q_{1}\times Q_{2}$ be an $(t+1,n)$-absorbing $\delta$-semiprimary
hyperideal of $G=G_{1}\times G_{2}$. Assume that $\delta_{1}(Q_{1})\neq G_{1}$
and $\delta_{2}(Q_{2})=G_{2}$. Let us suppose that $k_{1}(a_{1}^{(t+1)n-t})\in
Q_{1}$ for some $a_{1}^{(t+1)n-t}\in G_{1}$ such that all products of $tn-t+1$
of the $a_{i}^{,}$s except $k_{1}(a_{1}^{tn-t+1})$ are not in $\delta(Q_{1})$.
Note that $k((a_{1},0),\cdots,(a_{(t+1)n-t},0))\in Q$ and all products of
$tn-t+1$ of the $(a_{i},0)^{,}$s are not in $\delta(Q)$. Since $Q$ is an
$(t+1,n)$-absorbing $\delta$-semiprimary hyperideal of $G$, we get
$k((a_{1},0),\cdots,(a_{tn-t+1},0))\in\delta(Q)=\delta_{1}(Q_{1})\times\delta_{2}(Q_{2})$
which means $k_{1}(a_{1}^{tn-t+1})\in\delta(Q_{1})$. Thus $Q_{1}$ is an
$(t+1,n)$-absorbing $\delta_{1}$-semiprimary hyperideal of $G_{1}$. Similiar
for the second assertion. For the third assertion, assume
$\delta_{1}(Q_{1})\neq G_{1}$ and $\delta_{2}(Q_{2})\neq G_{2}$. Moreover, let
us suppose that $Q_{1}$ is not an $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G_{1}$ and $k_{1}(a_{1}^{tn-t+1})\in Q_{1}$. We define the
following elements of $G$:
$x_{1}=(a_{1},1_{G_{2}}),x_{2}=(a_{2},1_{G_{2}}),\cdots,x_{tn-t+1}=(a_{tn-t+1},1_{G_{2}}),x_{(t-1)n-t+2}=(1_{G_{1}},0)$.
Therefore we have $k(x_{1}^{(t-1)n-t+2})=(k_{1}(a_{1}^{tn-t+1}),0)\in Q$,
$k(x_{1}^{tn-t+1})=(k_{1}(a_{1}^{tn-t+1}),1_{G_{2}})\notin\delta(Q)$ and
$k(x_{1},\cdots,\hat{x}_{i},\cdots,x_{(t-1)n-t+2})=(k_{1}(a_{1},\cdots,\hat{a}_{i},\cdots,a_{(t-1)n-t+2}),0)\notin\delta(Q)$
for some $1\leq i\leq tn-t+1$, a contradiction. Thus $Q_{1}$ is an
$(t,n)$-absorbing $\delta_{1}$-semiprimary hyperideal of $G_{1}$. Similarly,
we conclude that $Q_{2}$ is an $(t,n)$-absorbing $\delta_{2}$-semiprimary
hyperideal of $G_{2}$ ∎

######  Theorem 2.9.

Let $\delta_{1},\cdots,\delta_{tn-t+1}$ be hyperideal expansions of Krasner
$(m,n)$-hyperrings $G_{1},\cdots,G_{tn-t+1}$ such that
$\delta(Q_{1}\times\cdots\times
Q_{tn-t+1})=\delta_{1}(Q_{1})\times\cdots\times\delta_{tn-t+1}(Q_{tn-t+1})$
for hyperideals $Q_{1},\cdots,Q_{tn-t+1}$ of $G_{1},\cdots,G_{tn-t+1}$,
respectively. If $Q=Q_{1}\times\cdots\times Q_{tn-t+1}$ is an
$(t+1,n)$-absorbing $\delta$-semiprimary hyperideal of $G=G_{1}\times...\times
G_{tn-t+1}$, then either $Q_{u}$ is an $(t+1,n)$-absorbing
$\delta$-semiprimary hyperideal of $G_{u}$ for some $1\leq u\leq tn-t+1$ and
$\delta_{i}(Q_{i})=G_{i}$ for each $1\leq i\leq tn-t+1$ and $i\neq u$ or
$Q_{u}$ and $Q_{v}$ are $(t,n)$-absorbing $\delta_{u,v}$-semiprimary
hyperideals of $G_{u}$ and $G_{v}$, respectively, for some
$u,v\in\\{1,\cdots,tn-t+1\\}$ and $\delta_{i}(Q_{i})=G_{i}$ for all $1\leq
i\leq tn-t+1$ but $i\neq u,v$.

###### Proof.

It can be seen that the idea is true in a similar manner to the proof of
Theorem 2.8. ∎

Now, we want to extend the notion of $(t,n)$-absorbing $\delta$-semiprimary
hyperideals to weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal.
Although different from each other in many aspects, they share quite a number
of similar properties as well.

######  Definition 2.10.

Let $\delta$ be a hyperideal expansion of $G$ and $t$ be a positive integer. A
proper hyperideal $Q$ of $G$ refers to a weakly $(t,n)$-absorbing
$\delta$-semiprimary hyperideal if $a_{1}^{tn-t+1}\in G$ and $0\neq
k(a_{1}^{tn-t+1})\in Q$, then there exist $(t-1)n-t+2$ of the $a_{i}^{,}$s
whose $k$-product is in $\delta(Q)$.

######  Example 2.11.

Suppose that $\mathbb{Z}_{12}$ is the set of all congruence classes of
integers modulo $12$ and $H=\\{1,5,7,11\\}$ is multiplicative subgroup of
units $\mathbb{Z}_{12}$. Construct $G$ as $\mathbb{Z}_{12}/H$. Then we have
$G=\\{\bar{0},\bar{1},\bar{2},\bar{3},\bar{4},\bar{6}\\}$ in which
$\bar{0}=\\{0\\}$, $\bar{1}=\\{1,5,7,11\\}$, $\bar{2}=\bar{10}=\\{2,10\\}$,
$\bar{3}=\bar{9}=\\{3,9\\}$, $\bar{4}=\bar{8}=\\{4,8\\}$, $\bar{6}=\\{6\\}$.
Consider Krasner hyperring $(G,\oplus,\star)$ that for all $\bar{a},\bar{b}\in
G$, $\bar{a}\star\bar{b}=\overline{ab}$ and 2-ary hyperoperation $\oplus$ is
defined as follows:

$\oplus$ | $\bar{0}$ | $\bar{1}$ | $\bar{2}$ | $\bar{3}$ | $\bar{4}$ | $\bar{6}$
---|---|---|---|---|---|---
$\bar{0}$ | $\bar{0}$ | $\bar{1}$ | $\bar{2}$ | $\bar{3}$ | $\bar{4}$ | $\bar{6}$
$\bar{1}$ | $\bar{1}$ | $\bar{0},\bar{2},\bar{\bar{4}},\bar{6}$ | $\bar{1},\bar{3}$ | $\bar{2},\bar{4}$ | $\bar{1},\bar{3}$ | $\bar{1}$
$\bar{2}$ | $\bar{2}$ | $\bar{1},\bar{3}$ | $\bar{0},\bar{4}$ | $\bar{1}$ | $\bar{2},\bar{6}$ | $\bar{4}$
$\bar{3}$ | $\bar{3}$ | $\bar{2},\bar{4}$ | $\bar{1}$ | $\bar{0},\bar{6}$ | $\bar{1}$ | $\bar{3}$
$\bar{4}$ | $\bar{4}$ | $\bar{1},\bar{3}$ | $\bar{2},\bar{6}$ | $\bar{1}$ | $\bar{0},\bar{4}$ | $\bar{2}$
$\bar{6}$ | $\bar{6}$ | $\bar{1}$ | $\bar{4}$ | $\bar{3}$ | $\bar{2}$ | $\bar{0}$

It is easy to see that the hyperideal
$Q=\\{\bar{0},\bar{2},\bar{4},\bar{6}\\}$ of $G$ is a $(2,2)$-absorbing
$\delta_{1}$-semiprimary.

######  Theorem 2.12.

If $Q$ is a (weakly) $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$,
then $Q$ is (weakly) $(v,n)$-absorbing $\delta$-semiprimary for all $v>n$.

###### Proof.

By using an argument similar to that in the proof of Theorem 4.4 in [8], one
can complete the proof. ∎

######  Theorem 2.13.

Let $Q$ be a proper hyperideal of $G$. If $\delta(Q)$ is a (weakly)
$(t,n)$-absorbing hyperideal of $G$, then $Q$ is a (weakly) $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G$.

###### Proof.

Let ($0\neq k(a_{1}^{tn-t+1})\in Q$) $k(a_{1}^{tn-t+1})\in Q$ such that all
products of $(t-1)n-t+2$ of the $a_{i}^{,}$s, other than
$k(a_{1}^{(t-1)n-t+2})$, are not in $\delta(Q)$. Since $\delta(Q)$ is a
(weakly) $(t,n)$-absorbing hyperideal of $G$ and $Q\subseteq\delta(Q)$, we
conclude that $k(a_{1}^{(t-1)n-t+2})\in\delta(Q)$. This shows that $Q$ is a
(weakly) $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$. ∎

######  Theorem 2.14.

Let $Q$ be a proper hyperideal of $G$ such that $\delta(\delta(Q))=\delta(Q)$.
Then $\delta(Q)$ is a (weakly) $(t,n)$-absorbing hyperideal of $G$ if and only
if $\delta(Q)$ is a (weakly) $(t,n)$-absorbing $\delta$-semiprimary hyperideal
of $G$.

###### Proof.

$\Longrightarrow$ Assume that $\delta(Q)$ is a (weakly) $(t,n)$-absorbing
hyperideal of $G$. Since $\delta(\delta(Q))=\delta(Q)$, we are done by Theorem
2.13.

$\Longleftarrow$ Let $\delta(Q)$ be a (weakly) $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G$. Suppose that ($0\neq
k(a_{1}^{tn-n+1})\in\delta(Q)$) $k(a_{1}^{tn-n+1})\in\delta(Q)$. Since
$\delta(Q)$ is a (weakly) $(t,n)$-absorbing $\delta$-semiprimary hyperideal of
$G$, then there exist $(t-1)n-t+2$ of the $a_{i}^{,}$s whose $k$-product is in
$\delta(\delta(Q))$. Since $\delta(\delta(Q))=\delta(Q)$, then the $k$-product
of the $(t-1)n-t+2$ of the $a_{i}^{,}$s is in $\delta(Q)$ which means
$\delta(Q)$ is a (weakly) $(t,n)$-absorbing hyperideal of $G$. ∎

######  Theorem 2.15.

Let $Q$ be a (weakly) $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$
and $P$ be a proper hyperideal of $G$ such that $P\subseteq Q$. If
$\delta(Q)=\delta(P)$, then $P$ is a (weakly) $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G$.

###### Proof.

Assume that ($0\neq k(a_{1}^{tn-t+1})\in P$) $k(a_{1}^{tn-t+1})\in P$ for
$a_{1}^{tn-t+1}\in G$. By the assumption, we get ($0\neq k(a_{1}^{tn-t+1})\in
Q$) $k(a_{1}^{tn-t+1})\in Q$ which implies there exist $(t-1)n-t+2$ of the
$a_{i}^{,}$s whose $k$-product is in $\delta(Q)$ because $Q$ is a (weakly)
$(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$. From
$\delta(Q)=\delta(P)$, it follows that the $k$-product of $(t-1)n-t+2$ of the
$a_{i}^{,}$s is in $\delta(P)$ which means $P$ is a (weakly) $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G$. ∎

######  Definition 2.16.

Let $Q$ be a proper hyperideal of $G$. $Q$ refers to a strongly (weakly)
$(t,n)$-absorbing $\delta$-semiprimary hyperideal if ($0\neq
k(Q_{1}^{tn-t+1})\subseteq Q$) $k(Q_{1}^{tn-t+1})\subseteq Q$ for some
hyperideals $Q_{1}^{tn-t+1}$ of $G$, then there exist $(t-1)n-t+2$ of
$Q_{i}^{,}$s whose $k$-product is a subset of $\delta(Q)$.

######  Definition 2.17.

Assume that $G$ is a commutative Krasner $(m,2)$-hyperring and $Q$ is a weakly
$(2,2)$-absorbing $\delta$-semiprimary hyperideal of $G$. Then $(x,y,z)$ is
said to be an $\delta$-$(2,2)$-zero of $Q$ for some $x,y,z\in G$ if
$k(x,y,z)=0$, $k(x,y)\notin\delta(Q)$, $k(y,z)\notin\delta(Q)$ and
$k(x,z)\notin\delta(Q)$.

######  Theorem 2.18.

Let $G$ be a commutative Krasner $(m,2)$-hyperring, $Q$ a weakly
$(2,2)$-absorbing $\delta$-semiprimary hyperideal of $G$ and
$k(Q_{1},x,y)\subseteq Q$ for some $x,y\in G$ and a hyperideal $Q_{1}$ of $G$.
If $(q,x,y)$ is not a $\delta$-$(2,2)$-zero of $Q$ for all $q\in Q_{1}$ and
$k(x,y)\notin\delta(Q)$, then $k(Q_{1},x)\subseteq\delta(Q)$ or
$k(Q_{1},y)\subseteq\delta(Q)$.

###### Proof.

Let $k(Q_{1},x,y)\subseteq Q$ for some $x,y\in G$ and a hyperideal $Q_{1}$ of
$G$ but $k(x,y)\notin\delta(Q)$, $k(Q_{1},x)\nsubseteq\delta(Q)$ and
$k(Q_{1},y)\nsubseteq\delta(Q)$. Then we have $k(q_{1},x)\nsubseteq\delta(Q)$
and $k(q_{2},y)\nsubseteq\delta(Q)$ for some $q_{1},q_{2}\in Q_{1}$. Since
$(q_{1},x,y)$ is not a $\delta$-$(2,2)$-zero of $Q$ and $k(q_{1},x,y)\in Q$,
we get $k(q_{1},y)\in\delta(Q)$. Similarly, we have $k(q_{2},x)\in\delta(Q)$.
Note that
$k(h(q_{1},q_{2},0^{(m-2)}),x,y)=h(k(q_{1},x,y),k(q_{2},x,y),0^{(m-2)})\subseteq
Q$. Then we obtain
$k(h(q_{1},q_{2},0^{(m-2)}),x)=h(k(q_{1},x),k(q_{2},x),0^{(m-2)})\subseteq\delta(Q)$
or
$k(h(q_{1},q_{2},0^{(m-2)}),y)=h(k(q_{1},y),k(q_{2},y),0^{(m-2)})\subseteq\delta(Q)$.
This follows that $k(q_{1},x)\in h(-k(q_{2},x),0^{(m-1)})\subseteq\delta(Q)$
or $k(q_{2},y)\in h(-k(q_{1},y),0^{(m-1)})\subseteq\delta(Q)$ which both of
them are a contradiction. Consequently, $k(Q_{1},x)\subseteq\delta(Q)$ or
$k(Q_{1},y)\subseteq\delta(Q)$. ∎

######  Theorem 2.19.

Let $G$ be a commutative Krasner $(m,2)$-hyperring, $Q$ a weakly
$(2,2)$-absorbing $\delta$-semiprimary hyperideal of $G$ and
$k(Q_{1},Q_{2},x)\subseteq Q$ for some $x\in G$ and two hyperideals
$Q_{1},Q_{2}$ of $G$. If $(q_{1},q_{2},x)$ is not a $\delta$-$(2,2)$-zero of
$Q$ for all $q_{1}\in Q_{1}$ and $q_{2}\in Q_{2}$, then
$k(Q_{1},x)\subseteq\delta(Q)$ or $k(Q_{2},x)\subseteq\delta(Q)$ or
$k(Q_{1},Q_{2})\subseteq\delta(Q)$.

###### Proof.

Let $k(Q_{1},Q_{2},x)\subseteq Q$, $k(Q_{1},x)\nsubseteq\delta(Q)$,
$k(Q_{2},x)\nsubseteq\delta(Q)$ and $k(Q_{1},Q_{2})\nsubseteq\delta(Q)$. Then
we get $k(q,x)\notin\delta(Q)$ and $k(q_{1},Q_{2})\nsubseteq\delta(Q)$ for
some $q,q_{1}\in Q_{1}$. By Theorem 2.18, we conclude that
$k(q,Q_{2})\subseteq\delta(Q)$ because $k(q,Q_{2},x)\subseteq Q$,
$k(q,x)\notin\delta(Q)$ and $k(Q_{2},x)\nsubseteq\delta(Q)$. Also, from
Theorem 2.18, we obtain $k(q_{1},x)\in\delta(Q)$. Note that
$k(h(q,q_{1},0^{(m-2)}),Q_{2},x)=h(k(q,Q_{2},x),k(q_{1},Q_{2},x),0^{(m-2)})\subseteq
Q$. Then we have
$k(h(q,q_{1},0^{(m-2)}),Q_{2})=h(k(q,Q_{2}),k(q_{1},Q_{2}),0^{(m-2)})\subseteq\delta(Q)$
which means $k(q_{1},Q_{2})\subseteq
h(-k(q,Q_{2}),0^{(m-1)})\subseteq\delta(Q)$ or
$k(h(q,q_{1},0^{(m-2)}),x)=h(k(q,x),k(q_{1},x),0^{(m-2)})\subseteq\delta(Q)$
which implies $k(q,x)\in h(-k(q_{1},x),0^{(m-1)})\subseteq\delta(Q)$. This is
a contradiction. Hence $k(Q_{1},x)\subseteq\delta(Q)$ or
$k(Q_{2},x)\subseteq\delta(Q)$ or $k(Q_{1},Q_{2})\subseteq\delta(Q)$. ∎

######  Definition 2.20.

Suppose that $G$ is a commutative Krasner $(m,2)$-hyperring and $Q_{1}^{3},Q$
be some proper hyperideals of $G$ such that $Q$ is a weakly $(2,2)$-absorbing
$\delta$-semiprimary hyperideal of $G$. $Q$ is said to be a free
$\delta$-$(2,2)$-zero with respect to $k(Q_{1}^{3})$ if $(q_{1},q_{2},q_{3})$
is not a $\delta$-$(2,2)$-zero of $Q$ for every $q_{1}\in Q_{1}$, $q_{2}\in
Q_{2}$ and $q_{3}\in Q_{3}$.

######  Theorem 2.21.

Let $G$ be a commutative Krasner $(m,2)$-hyperring, $Q$ a weakly
$(2,2)$-absorbing $\delta$-semiprimary hyperideal of $G$ and $0\neq
k(Q_{1},Q_{2},Q_{3})\subseteq Q$ for some hyperideals $Q_{1}^{3}$ of $G$. If
$Q$ is a free $\delta$-$(2,2)$-zero with respect to $k(Q_{1}^{3})$, then
$k(Q_{1}^{2})\subseteq\delta(Q)$ or $k(Q_{2}^{3})\subseteq\delta(Q)$ or
$k(Q_{1},Q_{3})\subseteq\delta(Q)$.

###### Proof.

Suppose that $k(Q_{1}^{3})\subseteq Q$ but $k(Q_{1}^{2})\nsubseteq\delta(Q)$
or $k(Q_{2}^{3})\nsubseteq\delta(Q)$ or $k(Q_{1},Q_{3})\nsubseteq\delta(Q)$.
This implies that $k(q,Q_{2})\nsubseteq\delta(Q)$ and
$k(q_{1},Q_{3})\nsubseteq\delta(Q)$ for some $q,q_{1}\in Q_{1}$. By Theorem
2.19, we get $k(q,Q_{3})\subseteq\delta(Q)$ because $k(q,Q_{2}^{3})\subseteq
Q$, $k(Q_{2}^{3})\nsubseteq\delta(Q)$ and $k(q,Q_{2})\nsubseteq\delta(Q)$.
Also, from Theorem 2.19, we obtain $k(q_{1},Q_{2})\nsubseteq\delta(Q)$ as
$k(q_{1},Q_{2}^{3})\subseteq Q$, $k(Q_{2}^{3})\nsubseteq\delta(Q)$ and
$k(q_{1},Q_{2})\nsubseteq\delta(Q)$. Since $k(h(q,q_{1}),Q_{2}^{3})\subseteq
Q$ and $k(Q_{2}^{3})\nsubseteq\delta(Q)$, we have
$k(h(q,q_{1},0^{(m-2)}),Q_{2})=h(k(q,Q_{2}),k(q_{1},Q_{2}),0^{(m-2)})\subseteq\delta(Q)$
or
$k(h(q,q_{1},0^{(m-2)}),Q_{3})=h(k(q,Q_{3}),k(q_{1},Q_{3}),0^{(m-2)})\subseteq\delta(Q)$.
In the first case, we conclude that $k(q,Q_{2})\in
h(-k(q_{1},Q_{2}),0^{(m-1)})\subseteq\delta(Q)$, a controdiction. Moreover,
the second case leads to a contradiction because $k(q_{1},Q_{3})\in
h(-k(q,Q_{3}),0^{(m-1)})\subseteq\delta(Q)$. Thus
$k(Q_{1}^{2})\subseteq\delta(Q)$ or $k(Q_{2}^{3})\subseteq\delta(Q)$ or
$k(Q_{1},Q_{3})\subseteq\delta(Q)$. ∎

######  Definition 2.22.

Assume that $Q$ is a weakly $(k,n)$-absorbing $\delta$-semiprimary hyperideal
of $G$. Then $(a_{1},\cdots,a_{tn-t+1})$ is called $\delta$-$(t,n)$-zero of
$Q$ if $k(a_{1}^{tn-t+1})=0$ and none $k$-product of the terms $(t-1)n-t+2$ of
$a_{i}^{,}$s is in $\delta(Q)$.

######  Theorem 2.23.

Assume that $Q$ is a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal
of $G$ and
$k(a_{1},\cdots,\hat{a}_{i_{1}},\cdots,\hat{a}_{i_{2}},\cdots,\hat{a}_{i_{s}},\cdots,a_{tn-t+1},Q_{1}^{s})\subseteq
Q$ for some $a_{1}^{tn-t+1}\in G$ and some hyperideals $Q_{1},\cdots Q_{s}$ of
$G$ such that $1\leq i_{1},\cdots,i_{s}\leq tn-t+1$ and $1\leq
s\leq(t-1)n-t+2$. If
$(a_{1},\cdots,\hat{a}_{i_{1}},\cdots,\hat{a}_{i_{2}},\cdots,\hat{a}_{i_{s}},\cdots,a_{tn-t+1},q_{1}^{s})$
is not a $\delta$-$(t,n)$-zero of $Q$ for all $q_{i}\in Q_{i}$, then
$k$-product of $(t-1)n-t+2$ of
$a_{1},\cdots,\hat{a}_{i_{1}},\cdots,\hat{a}_{i_{2}},\cdots,\hat{a}_{i_{s}},\cdots,a_{tn-t+1},Q_{1}^{s}$
including at least one of the $Q_{i}^{,}$s is in $\delta(Q)$.

###### Proof.

We prove it with induction on $s$. Let us consider $s=1$. In this case we show
that $k$-product of $(t-1)n-t+2$ of
$a_{1},\cdots,\hat{a}_{i_{1}},\cdots,a_{tn-t+1},Q_{1}$ including $Q_{1}$ is in
$\delta(Q)$. Assume that all products of $(t-1)n-t+2$ of
$a_{1},\cdots,\hat{a}_{i_{1}},\cdots,a_{tn-t+1},Q_{1}$ are not in $\delta(Q)$.
We consider $k(a_{2}^{(t-1)n-t+2},Q_{1})\notin\delta(Q)$. Since
$(a_{2}^{tn-t+1},q_{1})$ is not a $\delta$-$(t,n)$-zero of $Q$ for all
$q_{1}\in Q_{1}$, then we conclude that $k$-product of the $(t-1)n-t+2$ of
$a_{i}^{,}$s with $q_{1}$ is in $\delta(Q)$. By a similar argument given in
the proof of Theorem 2.18, we have
$k(a_{3}^{tn-t+1},h(a_{1},q_{1},0^{(m-2)}))=h(k(a_{3}^{tn-t+1},a_{1}),k(a_{3}^{tn-t+1},q_{1}),0^{(m-2)})\subseteq\delta(Q)$
which implies $k(a_{3}^{tn-t+1},a_{1})\in
h(-k(a_{3}^{tn-t+1},q_{1}),0^{(m-1)})\subseteq\delta(Q)$, a contradiction.
This implies that $k$-product of $(t-1)n-t+2$ of
$a_{1},\cdots,\hat{a}_{i_{1}},\cdots,a_{tn-t+1},Q_{1}$ including $Q_{1}$ is in
$\delta(Q)$. Now, we suppose that the claim holds for all positive integers
which are less than $s$. Let
$k(a_{1},\cdots,\hat{a}_{i_{1}},\cdots,\hat{a}_{i_{2}},\cdots,\hat{a}_{i_{s}},\cdots,a_{tn-t+1},Q_{1}^{s})\subseteq
Q$ but all products of $(t-1)n-t+2$ of
$a_{1},\cdots,\hat{a}_{i_{1}},\cdots,\hat{a}_{i_{2}},\cdots,\hat{a}_{i_{s}},\cdots,a_{tn-t+1},Q_{1}^{s}$
including at least one of the $Q_{i}^{,}$s are not in $\delta(Q)$. We may
assume that $k(a_{s+1}^{tn-t+1},Q_{1}^{s})\notin\delta(Q)$. Note that
$(a_{s+1}^{tn-t+1},q_{1}^{s})$ is not a $\delta$-$(t,n)$-zero of $Q$ for all
$q_{1}^{s}\in Q$. We get
$k(a_{s+1}^{tn-t+1},h(a_{1},q_{1},0^{(m-2)}),\cdots,h(a_{s},q_{s},0^{(m-2)}))\subseteq\delta(Q)$
by induction hypothesis and Theorem 2.19. Then we conclude that

$\hskip
19.91684ptk(a_{s+1}^{tn-t+1},h(a_{1},q_{1},0^{(m-2)}),\cdots,h(a_{1},\hat{q}_{1},0^{(m-2)})_{i_{1}},\cdots,h(a_{2},\hat{q}_{2},0^{(m-2)})_{i_{2}},$

$\hskip
28.45274pt\cdots,h(a_{n-1},\hat{q}_{n-1},0^{(m-2)})_{i_{n-1}},\cdots,h(a_{s},q_{s},0^{(m-2)}))\subseteq\delta(Q)$
or

$\hskip
19.91684ptk(a_{s+1},\cdots,\hat{a}_{i_{s+1}},\cdots,\hat{a}_{i_{s+2}},\cdots,\hat{a}_{i_{s+n-1}},\cdots,a_{tn-t+1},h(a_{1},q_{1},0^{(m-2)}),$

$\hskip 28.45274pt\cdots,h(a_{s},q_{s},0^{(m-2)}))\subseteq\delta(Q)$
for some $i\in\\{1,\cdots,s\\}$. This implies that
$k(a_{s+1},\cdots,a_{tn-t+1},\cdots,a_{n},\cdots,a_{s})\in\delta(Q)$ or
$k(a_{s+n},\cdots,a_{tn-t+1},\cdots,a_{1}^{s})\in\delta(Q)$, a contradiction.
Then we conclude that $k$-product of $(t-1)n-t+2$ of
$a_{1},\cdots,\hat{a}_{i_{1}},\cdots,\hat{a}_{i_{2}},\cdots,\hat{a}_{i_{s}},\cdots,a_{tn-t+1},Q_{1}^{s}$
including at least one of the $Q_{i}^{,}$s is in $\delta(Q)$. ∎

######  Definition 2.24.

Suppose that $Q_{1}^{n},Q$ be some proper hyperideals of $G$ such that $Q$ is
a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$ and
$k(Q_{1}^{tn-t+1})\subseteq Q$. $Q$ is called a free $\delta$-$(t,n)$-zero
with respect to $k(Q_{1}^{tn-t+1})$ if $(q_{1},\cdots,q_{tn-t+1})$ is not a
$\delta$-$(t,n)$-zero of $Q$ for every $q_{i}\in Q_{i}$ with $1\leq i\leq
tn-t+1$.

######  Theorem 2.25.

Assume that $Q$ is a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal
of $G$ and $0\neq k(Q_{1}^{tn-t+1})\subseteq Q$ for some hyperideals
$Q_{1}^{tn-t+1}$ of $G$. If $Q$ is a free $\delta$-$(t,n)$-zero with respect
to $k(Q_{1}^{tn-t+1})$, then $k$-product of $(t-1)n-t+2$ of the $Q_{i}$ is a
subset of $\delta(Q)$.

###### Proof.

This can be proved by Theorem 2.23, in a very similar manner to the way in
which Theorem 2.21 was proved. ∎

######  Theorem 2.26.

Let $G$ be a commutative Krasner $(m,2)$-hyperring and $Q$ be a weakly
$(2,2)$-absorbing $\delta$-semiprimary hyperideal of $G$. If $(x,y,z)$ is an
$\delta$-$(2,2)$-zero of $Q$ for some $x,y,z\in G$, then

  * (i)

$k(x,y,Q)=k(y,z,Q)=k(x,z,Q)=0$

  * (ii)

$k(x,Q^{(2)})=k(y,Q^{(2)})=k(z,Q^{(2)})=0$

###### Proof.

(i) Let $Q$ be a weakly $(2,2)$-absorbing $\delta$-semiprimary hyperideal of
$G$ and $(x,y,z)$ be an $\delta$-$(2,2)$-zero of $Q$. Let us assume that
$k(x,y,Q)\neq 0$. This means that $k(x,y,q)\neq 0$ for some $q\in Q$. So we
have $0\neq k(x,h(z,q,0^{(m-2)}),y)=h(k(x,z,y),k(x,q,y),0^{(m-2)})\subseteq
Q$. Since $Q$ is weakly $(2,2)$-absorbing $\delta$-semiprimary and
$k(x,y)\notin\delta(Q)$, we get
$k(x,h(z,q,0^{(m-2)}))=h(k(x,z),k(x,q),0^{(m-2)})\subseteq\delta(Q)$ or
$k(h(z,q,0^{(m-2)}),y)=h(k(z,y),k(q,y),0^{(m-2)})\subseteq\delta(Q)$. In the
first case, we have $k(x,z)\in h(-k(x,q),0^{(m-1)})\subseteq\delta(Q)$ which
is a contradiction. The second case leads to a contradiction because
$k(z,y)\in h(-k(q,y),0^{(m-1)})\subseteq\delta(Q)$. Thus $k(x,y,Q)=0$.
Similiar for the other cases.

(ii) Let $k(x,Q^{(2)})\neq 0$. This implies that $k(x,q_{1}^{2})\neq 0$ for
some $q_{1},q_{2}\in Q$. Therefore

$\hskip 76.82234pt0\neq k(x,h(y,q_{1},0^{(m-2)}),h(z,q_{2},0^{(m-2)}))$

$\hskip
85.35826pt=h(k(x,y,z),k(x,y,q_{2}),k(x,q_{1},z),k(x,q_{1}^{2}),0^{(m-4)})$

$\hskip 85.35826pt\subseteq Q$.
Since $Q$ is a weakly $(2,2)$-absorbing $\delta$-semiprimary hyperideal of
$G$, we obtain the following cases:

  * Case 1.

$k(x,h(y,q_{1},0^{(m-2)}))\subseteq\delta(Q)$ which implies
$h(k(x,y),k(x,q_{1}),0^{(m-2)})\subseteq\delta(Q)$. Then we have $k(x,y)\in
h(-k(x,q_{1}),0^{(m-1)})\subseteq\delta(Q)$, a contradiction.

  * Case 2.

$k(x,h(z,q_{2},0^{(m-2)}))\subseteq\delta(Q)$ which means
$h(k(x,z),k(x,q_{2}),0^{(m-2)})\subseteq\delta(Q)$. This follows that
$k(x,z)\in h(-k(x,q_{2}),0^{(m-1)})\subseteq\delta(Q)$, a contradiction.

  * Case 3.

$k(h(y,q_{1},0^{(m-2)}),h(z,q_{2},0^{(m-2)}))\subseteq\delta(Q)$ and so

$h(k(y,z),k(q_{1},z),k(y,q_{2}),k(q_{1}^{2}),0^{(m-4)})\subseteq\delta(Q)$.
This implies that $k(y,z)\in
h(-k(q_{1},z),-k(y,q_{2}),-k(q_{1}^{2}),0^{(m-2)})\subseteq\delta(Q)$ which is
a contradiction.

Therefore $k(x,Q^{(2)})=0$. Similiar for the other cases. ∎

######  Theorem 2.27.

Let $G$ be a commutative Krasner $(m,2)$-hyperring and $Q$ be a weakly
$(2,2)$-absorbing $\delta$-semiprimary hyperideal of $G$ but is not
$(2,2)$-absorbing $\delta$-semiprimary. Then $k(Q^{(3)})=0$.

###### Proof.

Let $Q$ be a weakly $(2,2)$-absorbing $\delta$-semiprimary hyperideal of $G$
but is not $(2,2)$-absorbing $\delta$-semiprimary. This implies that we have
an $\delta$-$(2,2)$-zero of $Q$ for some $x,y,z\in G$. Let us assume that
$k(Q^{(3)})\neq 0$. Then $k(q_{1}^{3})\neq 0$ for some $q_{1}^{3}\in Q$.
Therefore we have

$k(h(x,q_{1},0^{(m-2)}),h(y,q_{2},0^{(m-2)}),h(z,q_{3},0^{(m-2)}))$

$\hskip
5.69046pt=h(h(h(k(x,y,z),k(q_{1},y,z),0^{(m-2)}),h(k(x,y,q_{3}),k(q_{1},y,q_{3}),0^{(m-2)})),$

$\hskip
22.76228pth(h(k(q_{2},x,z),h(q_{1}^{2},z),0^{(m-2)})),h(h(k(q_{1}^{3}),k(x,q_{2}^{3})),0).$
From $k(q_{1}^{3})\neq 0$, it follows that

$\hskip 28.45274pt0\neq
k(h(x,q_{1},0^{(m-2)}),h(y,q_{2},0^{(m-2)}),h(z,q_{3},0^{(m-2)}))\subseteq Q$
by Theorem 2.26. Since $Q$ is weakly $(2,2)$-absorbing $\delta$-semiprimary,
we have $k(h(x,q_{1},0^{(m-2)}),h(y,q_{2},0^{(m-2)}))\subseteq\delta(Q)$ or
$k(h(x,q_{1},0^{(m-2)}),h(z,q_{3},0^{(m-2)}))\subseteq\delta(Q)$ or
$k(h(x,q_{1},0^{(m-2)}),h(z,q_{3},0^{(m-2)}))\subseteq\delta(Q)$. In the first
possibilty, we obtain
$h(k(x,y),k(x,q_{2}),k(q_{1},y),k(q_{1}^{2}),0^{(m-4)})\subseteq\delta(Q)$
which means $k(x,y)\in
h(-k(x,q_{2}),-k(q_{1},y),-k(q_{1}^{2}),0^{(m-3)})\subseteq\delta(Q)$ which is
a contradiction. Moreover, the other possibilities lead to a contradiction.
Thus $k(Q^{(3)})=0$. ∎

######  Definition 2.28.

Assume that $Q$ is a weakly $(k,n)$-absorbing $\delta$-semiprimary hyperideal
of $G$. Then $(a_{1},\cdots,a_{tn-t+1})$ is called $\delta$-$(t,n)$-zero of
$Q$ if $k(a_{1}^{tn-t+1})=0$ and none $k$-product of the terms $(t-1)n-t+2$ of
$a_{i}^{,}$s is in $\delta(Q)$.

######  Theorem 2.29.

If $Q$ is a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$
and $(a_{1},\cdots,a_{tn-t+1})$ is a $\delta$-$(t,n)$-zero of $Q$, then for
$1\leq i_{1},\cdots,i_{s}\leq tn-t+1$ and $1\leq s\leq(t-1)n-t+2$,

$k(a_{1},\cdots,\hat{a}_{i_{1}},\cdots,\hat{a}_{i_{2}},\cdots,\hat{a}_{i_{s}},\cdots,a_{tn-t+1},Q^{(s)})=0.$

###### Proof.

We use the induction on $s$. Assume that $s=1$. Let us suppose that
$k(a_{1},\cdots,\hat{a}_{i_{1}},\cdots,a_{tn-t+1},Q)\neq 0$. We may assume
that $k(a_{2}^{tn-t+1},Q)\neq 0$. Therefore $k(a_{2}^{tn-t+1},q)\neq 0$ for
some $q\in Q$. Hence every $k$-product of the $(t-1)n-t+2$ of $a_{i}^{,}$s
including $q$ is in $\delta(Q)$. By the same argument given in Theorem 2.26,
we have
$k(a_{3}^{tn-t+1},h(a_{1},q,0^{(m-2)}))=h(k(a_{3}^{tn-t+1},a_{1}),k(a_{3}^{tn-t+1},q),0^{(m-2)})\subseteq\delta(Q)$
which implies $k(a_{3}^{tn-t+1},a_{1})\in
h(-k(a_{3}^{tn-t+1},q),0^{(m-1)})\subseteq\delta(Q)$, a contradiction. This
means that $k(a_{1},\cdots,\hat{a}_{i_{1}},\cdots,a_{tn-t+1},Q)=0$. Now, let
us suppose that
$k(a_{1},\cdots,\hat{a}_{i_{1}},\cdots,\hat{a}_{i_{2}},\cdots,\hat{a}_{i_{s}},\cdots,a_{tn-t+1},Q^{(s)})\neq
0$. We may assume that $k(a_{s+1}^{tn-t+1},Q^{(s)})\neq 0$. Hence $0\neq
k(a_{s+1}^{tn-t+1},q_{1}^{s})\in Q$ for some $q_{1}^{s}\in Q$. It follows that
$0\neq
k(a_{s+1}^{tn-t+1},h(a_{1},q_{1},0^{(m-2)}),\cdots,h(a_{s},q_{s},0^{(m-2)}))\subseteq
Q$ by Theorem 2.26 and induction hypothesis. Then we conclude that

$\hskip
19.91684ptk(a_{s+1}^{tn-t+1},h(a_{1},q_{1},0^{(m-2)}),\cdots,h(a_{1},\hat{q}_{1},0^{(m-2)})_{i_{1}},\cdots,h(a_{2},\hat{q}_{2},0^{(m-2)})_{i_{2}},$

$\hskip
28.45274pt\cdots,h(a_{n-1},\hat{q}_{n-1},0^{(m-2)})_{i_{n-1}},\cdots,h(a_{s},q_{s},0^{(m-2)}))\subseteq\delta(Q)$
or

$\hskip
19.91684ptk(a_{s+1},\cdots,\hat{a}_{i_{s+1}},\cdots,\hat{a}_{i_{s+2}},\cdots,\hat{a}_{i_{s+n-1}},\cdots,a_{tn-t+1},h(a_{1},q_{1},0^{(m-2)}),$

$\hskip 28.45274pt\cdots,h(a_{s},q_{s},0^{(m-2)}))\subseteq\delta(Q)$
for some $i\in\\{1,\cdots,s\\}$. This implies that
$k(a_{s+1},\cdots,a_{tn-t+1},\cdots,a_{n},\cdots,a_{s})\in\delta(Q)$ or
$k(a_{s+n},\cdots,a_{tn-t+1},\cdots,a_{1}^{s})\in\delta(Q)$, a contradiction.
Thus we conclude that
$k(a_{1},\cdots,\hat{a}_{i_{1}},\cdots,\hat{a}_{i_{2}},\cdots,\hat{a}_{i_{s}},\cdots,a_{tn-t+1},Q^{(s)})=0$.
∎

######  Theorem 2.30.

Let $Q$ be a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$
but is not $(t,n)$-absorbing $\delta$-semiprimary. Then $k(Q^{(tn-t+1)})=0$.

###### Proof.

Assume that $Q$ is a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal
of $G$ but is not $(t,n)$-absorbing $\delta$-semiprimary. Then there exists a
$\delta$-$(t,n)$-zero $(a_{1},\cdots,a_{tn-t+1})$ of $Q$. Now, the claim
follows by using Theorem 2.29, in a very similar manner to the way in which
Theorem 2.27 was proved. ∎

As an instant consequence of the previous theorem, we have the following
explicit results.

######  Corollary 2.31.

Let $Q$ be a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$
but is not $(t,n)$-absorbing $\delta$-semiprimary. Then $Q\subseteq rad(0)$.

######  Corollary 2.32.

Assume that the commutative Krasner $(m,n)$-hyperring $G$ has no non-zero
nilpotent elements. If $Q$ is a weakly $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G$, then $Q$ is an $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G$.

The next theorem provides us how to determine weakly $(t,n)$-absorbing
$\delta$-semiprimary hyperideal to be $(t,n)$-absorbing $\delta$-semiprimary.

######  Theorem 2.33.

Let $Q$ be a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$
such that $\delta(Q)=\delta(0)$. Then $Q$ is not $(t,n)$-absorbing
$\delta$-semiprimary if and only if there exists a $\delta$-$(t,n)$-zero of
$0$.

###### Proof.

$\Longrightarrow$ Assume that $Q$ is not an $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G$. This implies that
$k(a_{1}^{tn-t+1})=0$ and none $k$-product of the terms $(t-1)n-t+2$ of
$a_{i}^{,}$s is in $\delta(Q)$ for some $a_{1}^{tn-t+1}\in G$. From
$\delta(Q)=\delta(0)$, it follows that $(a_{1}^{tn-t+1})$ is a
$\delta$-$(t,n)$-zero of $0$.

$\Longleftarrow$ Straightforward ∎

Let $(G_{1},h_{1},k_{1})$ and $(G_{2},h_{2},k_{2})$ be two commutative Krasner
$(m,n)$-hyperrings. Recall from [14] that a mapping $f:G_{1}\longrightarrow
G_{2}$ is called a homomorphism if we have
$f(h_{1}(a_{1}^{m}))=h_{2}(f(a_{1}),\cdots,f(a_{m}))$ and
$f(k_{1}(b_{1}^{n}))=k_{2}(f(b_{1}),...,f(b_{n}))$ for all $a^{m}_{1}\in
G_{1}$ and $b^{n}_{1}\in G_{1}$. Let $\delta$ and $\delta^{\prime}$ be
hyperideal expansions of $G_{1}$ and $G_{2}$, respectively. Recall from [2]
that $f:G_{1}\longrightarrow G_{2}$ is called a
$\delta\delta^{\prime}$-homomorphism if
$\delta(f^{-1}(Q_{2}))=f^{-1}(\delta^{\prime}(Q_{2}))$ for hyperideal $Q_{2}$
of $G_{2}$. Note that $\delta^{\prime}(h(Q_{1})=h(\delta(Q_{1})$ for
$\delta\delta^{\prime}$-epimorphism $f$ and for hyperideal $Q_{1}$ of $G_{1}$
with $Ker(f)\subseteq Q_{1}$.

######  Theorem 2.34.

Let $(G_{1},h_{1},k_{1})$ and $(G_{2},h_{2},k_{2})$ be two Krasner
$(m,n)$-hyperrings and $f:G_{1}\longrightarrow G_{2}$ be a
$\delta\delta^{\prime}$-homomorphism. Then the followings hold:

  * (i)

If $Q_{2}$ is an $(t,n)$-absorbing $\delta^{\prime}$-semiprimary hyperideal of
$G_{2}$, then $f^{-1}(Q_{2})$ is an $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G_{1}$.

  * (ii)

If $Q_{2}$ is a weakly $(t,n)$-absorbing $\delta^{\prime}$-semiprimary
hyperideal of $G_{2}$ and $Kerf$ is a weakly $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G_{1}$, then $f^{-1}(Q_{2})$ is a weakly
$(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G_{1}$.

  * (iii)

Let $f$ be an epimorphism and $Q_{1}$ be a proper hyperideal of $G_{1}$
containing $Kerf$. If $Q_{1}$ is a (weakly) $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G_{1}$, then $f(Q_{1})$ is a (weakly)
$\delta^{\prime}$-semiprimary hyperideal of $G_{2}$.

###### Proof.

$(i)$ Let $k_{1}(a_{1}^{tn-t+1})\in f^{-1}(Q_{2})$ for $a_{1}^{kn-k+1}\in
G_{1}$. Then we get
$f(k_{1}(a_{1}^{tn-t+1}))=k_{2}(f(a_{1}),...,f(a_{tn-t+1}))\in Q_{2}$. Since
$Q_{2}$ is an $(t,n)$-absorbing $\delta^{\prime}$-semiprimary hyperideal of
$G_{2}$, then there exist $(t-1)n-t+2$ of $f(a_{i})^{,}$s whose
$k_{2}$-product is an element in $\delta^{\prime}(Q_{2})$. It follows that the
image $f$ of $(t-1)n-t+2$ of $a_{i}^{,}$ whose $k_{2}$-product is in
$\delta^{\prime}(Q_{2})$ which means there exist $(t-1)n-t+2$ of $a_{i}^{,}$
whose $k_{1}$-product is in
$f^{-1}(\delta^{\prime}(Q_{2}))=\delta(f^{-1}(Q_{2}))$. Thus $f^{-1}(Q_{2})$
is an $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G_{1}$.

$(ii)$ Assume that $k_{1}(a_{1}^{tn-t+1})\in f^{-1}(Q_{2})$ for
$a_{1}^{kn-k+1}\in G_{1}$. Therefore
$f(k_{1}(a_{1}^{tn-t+1}))=k_{2}(f(a_{1}),...,f(a_{tn-t+1}))\in Q_{2}$. If
$0\neq f(k_{1}(a_{1}^{tn-t+1}))$, then it can be proved by using an argument
similar to that in the proof of the part (i). Let us assume that
$f(k_{1}(a_{1}^{tn-t+1}))=0$. Then we obtain $k_{1}(a_{1}^{tn-t+1})\in Kerf$.
Since $Kerf$ is a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal of
$G_{1}$, then there exist $(t-1)n-t+2$ of $a_{i}^{,}$s whose $k_{1}$-product
is an element in $\delta(Kerf)$. From
$\delta(Kerf)=\delta(f^{-1}(0))\subseteq\delta(f^{-1}(Q_{2}))$, it follows
that $f^{-1}(Q_{2})$ is a weakly $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G_{1}$.

$(iii)$ Let ($0\neq k_{2}(b_{1}^{tn-t+1})\in f(Q_{1})$)
$k_{2}(b_{1}^{tn-t+1})\in f(Q_{1})$ for some $b_{1}^{tn-t+1}\in G_{2}$. Since
$f$ be an epimorphism, then there exist $a_{i}\in G_{1}$ for each $1\leq i\leq
tn-t+1$ such that $f(a_{i})=b_{i}$. Hence
$k_{2}(b_{1}^{tn-t+1})=k_{2}(f(a_{1}),\cdots,f(a_{tn-t+1}))=f(k_{1}(a_{1}^{tn-t+1}))\in
f(Q_{1})$. Since $Q_{1}$ containing $Kerf$, we conclude that ($0\neq
k_{1}(a_{1}^{tn-t+1})\in Q_{1}$) $k_{1}(a_{1}^{tn-t+1})\in Q_{1}$. As $Q_{1}$
is a (weakly) $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G_{1}$,
then there exist $(t-1)n-t+2$ of $a_{i}^{,}$s whose $k_{1}$-product is in
$\delta(Q_{1})$. Now, since $f$ is a homomorphism and
$f(\delta(Q_{1}))=\delta^{\prime}(f(Q_{1}))$, the proof is completed. ∎

Let $P$ be a hyperideal of $(G,h,k)$. Then the set
$G/P=\\{h(g^{i-1}_{1},P,g^{m}_{i+1})\ |\ g^{i-1}_{1},g^{m}_{i+1}\in G\\}$ with
$h$ and $k$ which are defind by

$\hskip
56.9055pth(h(g_{11}^{1(i-1)},P,g^{1m}_{1(i+1)}),...,h(g_{m1}^{m(i-1)},P,g^{mm}_{m(i+1)}))$

$\hskip
71.13188pt=h(h(g^{m1}_{11}),...,h(g^{m(i-1)}_{1(i-1)}),P,h(g^{m(i+1)}_{1(i+1)}),...,h(g^{mm}_{1m}))$
and

$\hskip
56.9055ptk(h(g_{11}^{1(i-1)},P,g^{1m}_{1(i+1)}),...,h(g_{n1}^{n(i-1)},P,g^{nm}_{n(i+1)}))$

$\hskip
71.13188pt=h(k(g^{n1}_{11}),...,k(g^{n(i-1)}_{1(i-1)}),P,k(g^{n(i+1)}_{1(i+1)}),...,k(g^{nm}_{1m}))$
for all $g_{11}^{1m},...,g_{m1}^{mm}\in G$ and $g_{11}^{1m},...,g_{n1}^{nm}\in
G$, construct a Krasner $(m,n)$-hyperring [1].

######  Theorem 2.35.

Let $P$ and $Q$ be two proper hyperideals of $G$ with $P\subseteq Q$. If $Q$
is an $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$, then $Q/P$ is
an $(t,n)$-absorbing $\delta_{q}$-semiprimary hyperideal of $G/P$.

###### Proof.

By considering the natural homomorphism $\pi:G\longrightarrow G/P$, defined by
$\pi(a)=f(a,P,0^{(m-2)})$ and using Theorem 2.34, we are done. ∎

######  Theorem 2.36.

Let $Q$ be an $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$. If
$G^{\prime}$ is a subhyperring of $G$ such that $G^{\prime}\nsubseteq Q$, then
$Q\cap G^{\prime}$ is an $(t,n)$-absorbing $\delta$-semiprimary hyperideal of
$G^{\prime}$.

###### Proof.

It follows by Theorem 2.34. ∎

######  Theorem 2.37.

Let $\delta_{1}$ and $\delta_{2}$ be two hyperideal expansions of Krasner
$(m,n)$-hyperrings $G_{1}$ and $G_{2}$, respectively, such that
$\delta(Q_{1}\times Q_{2})=\delta_{1}(Q_{1})\times\delta_{2}(Q_{2})$ for
hyperideals $Q_{1}$ and $Q_{2}$ of $G_{1}$ and $G_{2}$, respectively. If
$Q=Q_{1}\times G_{2}$ is a weakly $(t,n)$-absorbing $\delta$-semiprimary
hyperideal of $G_{1}\times G_{2}$, then it is an $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G_{1}\times G_{2}$.

###### Proof.

Assume that $Q_{1}\times G_{2}$ is a weakly $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G_{1}\times G_{2}$. Since
$k(Q^{(tn-t+1)})\neq 0$, we conclude that $Q=Q_{1}\times G_{2}$ is an
$(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G_{1}\times G_{2}$ by
Theorem 2.30. ∎

We say that $\delta$ has $(\mathfrak{P})$ property if it satisfies the
condition: $\delta(Q)=G$ if and only if $Q=G$ for all hyperideals $Q$ of $G$.

######  Theorem 2.38.

Let $\delta_{1},\cdots,\delta_{tn-t+1}$ be hyperideal expansions of Krasner
$(m,n)$-hyperrings $G_{1},\cdots,G_{tn-t+1}$ such that each $\delta_{i}$ has
$(\mathfrak{P})$ property and $\delta(Q_{1}\times\cdots\times
Q_{tn-t+1})=\delta_{1}(Q_{1})\times\cdots\times\delta_{tn-t+1}(Q_{tn-t+1})$
for hyperideals $Q_{1},\cdots,Q_{tn-t+1}$ of $G_{1},\cdots,G_{tn-t+1}$,
respectively. If $Q=Q_{1}\times\cdots\times Q_{tn-t+1}$ is a weakly
$(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G=G_{1}\times...\times
G_{tn-t+1}$, then $Q$ is an $(t,n)$-absorbing $\delta$-semiprimary hyperideal
of $G=G_{1}\times...\times G_{tn-t+1}$.

###### Proof.

Let $Q$ is a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$.
Let us consider the following elements of $G$:
$x_{i}=(1_{G_{1}}.\cdots,1_{G_{i-1}},a_{i},1_{G_{i+1}},\cdots,1_{G_{tn-t+1}})$
for all $1\leq i\leq tn-t+1$. Then we have $0\neq k(x_{1}^{tn-t+1})\in Q$.
Since $Q=Q_{1}\times\cdots\times Q_{tn-t+1}$ is a weakly $(t,n)$-absorbing
$\delta$-semiprimary hyperideal of $G=G_{1}\times...\times G_{tn-t+1}$, then
there exists $(t-1)n-t+2$ of the $x_{i}^{,}$s whose $k$-product is in
$\delta(Q)=\delta_{1}(Q_{1})\times\cdots\times\delta_{tn-t+1}(Q_{tn-t+1})$.
This implies that there exists some $1\leq j\leq tn-t+1$ such that
$1_{G_{j}}\in\delta_{j}(Q_{j})$ which means $\delta_{j}(Q_{j})=G_{j}$. Since
$\delta_{j}$ has $(\mathfrak{P})$ property, then $Q_{j}=G_{j}$. Hence we
conclude that $k(Q^{(tn-t+1)})\neq 0$ which implies $Q$ is an
$(t,n)$-absorbing $\delta$-semiprimary hyperideal of $G$ by Theorem 2.30. ∎

######  Theorem 2.39.

Let $\delta_{1},\cdots,\delta_{tn-t+1}$ be hyperideal expansions of Krasner
$(m,n)$-hyperrings $G_{1},\cdots,G_{tn-t+1}$ such that each $\delta_{i}$ has
$(\mathfrak{P})$ property and $\delta(Q_{1}\times\cdots\times
Q_{tn-t+1})=\delta_{1}(Q_{1})\times\cdots\times\delta_{tn-t+1}(Q_{tn-t+1})$
for hyperideals $Q_{1},\cdots,Q_{tn-t+1}$ of $G_{1},\cdots,G_{tn-t+1}$,
respectively. If $Q=Q_{1}\times\cdots\times Q_{tn-t+1}$ is a weakly
$(t+1,n)$-absorbing $\delta$-semiprimary hyperideal of $G=G_{1}\times...\times
G_{tn-t+1}$, then either there exists $1\leq u\leq tn-t+1$ such that $Q_{u}$
is an $(t+1,n)$-absorbing $\delta$-semiprimary hyperideal of $G_{u}$ and
$Q_{i}=G_{i}$ for each $1\leq i\leq tn-t+1$ and $i\neq u$ or $Q_{u}$ and
$Q_{v}$ are $(t,n)$-absorbing $\delta_{u,v}$-semiprimary hyperideals of
$G_{u}$ and $G_{v}$, respectively, for some $u,v\in\\{1,\cdots,tn-t+1\\}$ and
$Q_{i}=G_{i}$ for all $1\leq i\leq tn-t+1$ but $i\neq u,v$.

###### Proof.

Let $Q=Q_{1}\times\cdots\times Q_{tn-t+1}$ be a weakly $(t+1,n)$-absorbing
$\delta$-semiprimary hyperideal of $G=G_{1}\times...\times G_{tn-t+1}$.
Therefore we conclude that $Q$ is an $(t+1,n)$-absorbing $\delta$-semiprimary
hyperideal of $G$ by Theorem 2.38. Now, by using Theorem 2.9, we are done. ∎

##  3\. conclusion

In this paper, our purpose was to study the structure of $(t,n)$-absorbing
$\delta$-semiprimary hyperideals which is more general than $\delta$-primary
hyperideals. Additionally, we generalized the notion to weakly
$(t,n)$-absorbing $\delta$-semiprimary hyperideals. We gave many special
results illustrating the structures. Indeed, this paper makes a major
contribution to classify hyperideals in Krasner $(m,n)$-hyperrings.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

## References

  * [1] R. Ameri, M. Norouzi, Prime and primary hyperideals in Krasner $(m,n)$-hyperrings, European Journal Of Combinatorics, (2013) 379-390.
  * [2] M. Anbarloei, Unifing the prime and primary hyperideals under one frame in a Krasner $(m,n)$-hyperring, Communications in Algebra, (2021) DOI: 10.1080/00927872.2021.1897988.
  * [3] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75(2007) 417-429.
  * [4] E.Y. Celikel, 2-absorbing $\delta$-semiprimary ideals of commutative rings, KYUNGPOOK Math. J., 61 (2021) 711-725.
  * [5] B. Davvaz, T. Vougiouklis, n-ary hypergroups, Iran. J. Sci. Technol., 30 (A2) (2006) 165-174.
  * [6] B. Davvaz, Fuzzy Krasner $(m,n)$-hyperrings, Computers and Mathematics with Applications, 59 (2010) 3879-3891.
  * [7] W. Dorente, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Math. Z., 29 (1928) 1-19.
  * [8] K. Hila, K. Naka, B. Davvaz, On $(k,n)$-absorbing hyperideals in Krasner $(m,n)$-hyperrings, Quarterly Journal of Mathematics, 69 (2018) 1035-1046.
  * [9] E. Kasner, An extension of the group concept (reported by L.G. Weld), Bull. Amer. Math. Soc., 10 (1904) 290-291.
  * [10] V. Leoreanu, Canonical n-ary hypergroups, Ital. J. Pure Appl. Math., 24(2008).
  * [11] V. Leoreanu-Fotea, B. Davvaz, n-hypergroups and binary relations, European J. Combin., 29 (2008) 1027-1218.
  * [12] V. Leoreanu-Fotea, B. Davvaz, Roughness in n-ary hypergroups, Inform. Sci., 178 (2008) 4114-4124.
  * [13] X. Ma, J. Zhan, B. Davvaz, Applications of rough soft sets to Krasner $(m,n)$-hyperrings and corresponding decision making methods, Filomat, 32 (2018) 6599-6614.
  * [14] S. Mirvakili, B. Davvaz, Relations on Krasner $(m,n)$-hyperrings, European J. Combin., 31(2010) 790-802.
  * [15] S. Ostadhadi-Dehkordi, B. Davvaz, A Note on Isomorphism Theorems of Krasner $(m,n)$\- hyperrings, Arabian Journal of Mathematics, 5 (2016) 103-115.
  * [16] M.M. Zahedi, R. Ameri, On the prime, primary and maximal subhypermodules, Ital. J. Pure Appl. Math., 5 (1999) 61-80.