# Integrating embedded neural networks and self-mixing interferometry for
smart sensors design ††thanks: This project has received financial support
from the CNRS through the MITI interdisciplinary programs

1st Pierre-Emmanuel Novac  Université Côte d’Azur, LEAT
Nice, France
<EMAIL_ADDRESS>2nd Laurent Rodriguez  Université
Côte d’Azur, LEAT
Nice, France
<EMAIL_ADDRESS>3rd Stéphane Barland  INPHYNI, CNRS,
Université Côte d’Azur
Nice, France
<EMAIL_ADDRESS>
###### Abstract

Self-mixing interferometry is a measurement approach in which a laser beam is
re-injected into the emitting laser itself after reflection on a target.
Information about the position of the target can be obtained from monitoring
the voltage across the laser. However, analyzing this signal is difficult. In
previous works, neural networks have been used with great success to process
this data. In this article, we present the first prototype of an integrated
sensor based on self-mixing interferometry with embedded neural networks. It
consists of a semiconductor laser (acting both as light emitter and detector)
equipped with an embedded platform for data processing. The platform includes
an ADC (Analog-to-Digital Converter) and an STM32L476RG microcontroller. The
microcontroller runs the neural network in charge of reconstructing the
displacement of a target from the interferometric signal entering the ADC. We
assess the robustness of the neural network to unwanted signal amplitude
variations and the impact of different network weights quantization choices
required to run the network on the microcontroller. Finally, we provide a
demonstration of target displacement reconstruction fully running on the
embedded platform. Our results pave the way towards robust, low power and
versatile sensors based on self-mixing interferometry and embedded neural
networks.

###### Index Terms:

Self-mixing interferometry, semiconductor lasers, neural networks, embedded
artificial intelligence

##  I Introduction

Self-mixing interferometry is a well-established and versatile measurement
approach in which a laser beam is re-injected into the emitting laser itself
after reflection on a target [1, 2, 3, 4, 5, 6]. Thus, optical interferences
between the emitted and reflected beams take place inside the laser itself,
altering its operation point. In normal operating conditions, a semiconductor
laser emits a monochromatic beam whose intensity (and all other properties)
are fully determined by the laser construction parameters and by the amount of
energy brought into the laser, ie the electrical pumping current. In the case
of self-mixing interferometry, the interference condition between the emitted
and the re-injected beam constitutes an additional parameter which influences
the intensity emitted by the laser. Thus, information about the position of
the target (or many other properties, see eg [7] for an overview) can be
obtained from monitoring the laser operation point, by recording the voltage
across the laser diode as a function of time. Recovering the displacement of
the target per time unit is in principle a simple problem. If the displacement
is less than an optical wavelength a physical model can be fit to the data and
the displacement can be recovered. Alternatively, if the displacement is of
many wavelengths, counting minima and maxima of interference can lead to
reconstruction of the displacement. In practice though, this signal analysis
is often complex and remains error prone due to detection noise, very large
signal bandwidth (even for very simple displacement patterns) or variations of
the reflectivity of non-cooperative targets. Many hardware or software
approaches have been designed to mitigate these long-standing issues [8, 9,
10, 11, 12, 13, 12, 14]. As was shown recently [15] a relatively small
convolutional neural network can recover the displacement of the target from
the voltage across the laser diode with very high precision. This method has
recently been extended to a multichannel measurement scheme, which results in
much improved measurement availability [16]. In [17] neural networks were used
for signal pre-processing, in [18] neural networks were used for fringe
discrimination in the signal and [19] discussed the use of neural networks for
parametric conditions estimation.

Further development of the self-mixing approach as a general purpose sensor
[7] suitable for standalone operation without involvement of specifically
trained personnel therefore requires bringing suitable neural networks to a
widely available microcontroller platform.

Such a device is the STMicroelectronics STM32L476RG, a very popular low-power
32-bit microcontroller which can be used to run the inference of tiny deep
neural networks. To achieve this, a software framework that enables the
deployment of deep neural networks on microcontrollers must be used. Popular
frameworks include TensorFlow Lite for Microcontrollers and STM32Cube.AI, but
we chose to use Qualia-CodeGen[20] for its low memory footprint and its ease
of customization[21]. The neural network is quantized to fixed-point numbers
in order to keep the memory footprint low and to obtain the best performance.

Furthermore, the embedded platform can be designed so that it captures the
signal directly through an ADC (Analog-to-Digital Converter), rather than
requiring an external capture device to acquire the data on a workstation.
Finally, the use of a microcontroller dedicated to this task, with a high
level of control over the data capture, paves the way to real-time operation.
This enables the development of a first prototype of a self-contained device
for the sensing of small displacements through self-mixing interferometry.

Our contributions are as follows:

  * •

design of a tiny deep neural network for small displacement measurements
through self-mixing interferometry,

  * •

evaluation of fixed-point quantization of the deep neural network,

  * •

evaluation of embedded performances after deployment on microcontroller,

  * •

design of a first prototype of a device for small displacement sensing.

The rest of this article is organized as follows. Section II explains the deep
neural network, its quantization and deployment on the microcontroller.
Section III describes the optoelectronics setup and the embedded platform.
Section IV presents results on the prediction performance of the neural
network before and after quantization, as well as results on the embedded
execution. Finally, section V concludes our work and provides future
perspectives.

##  II Proposed method

###  II-A CNN based reconstruction

The task to be realized by the network is represented on Figure 1. On the
basis of segments of the interferometric signal in the course of time, the
network must infer the value of the displacement of the target during the
considered time interval. One of the peculiarities of this signal is that,
since it consists essentially of interference fringes taking place in a
nonlinear medium, the amplitude of the signal is not significant in terms of
the displacement of the target but only in terms of its reflectivity. Thus,
the information which must be extracted from the signal resides in the shape
of the fringes, their more or less peaked aspect, their orientation and their
repetition rate.

Figure 1: Example of the regression task to be realized. Top: the
interferometric signal is measured as a voltage at the edges of a
semiconductor laser diode in the course of time. Bottom, the displacement of
the target during the same time frame. The task is to reconstruct the
displacement of the target (red dots) from the signal during the corresponding
time segment (the alternating light green and white stripes)

The network we start from was originally described in [15] and consists of a
an alternating stack of four convolutional and four maxpooling layers followed
by two fully connected layers for the final regression. In the original work,
the network (featuring $5.7*10^{4}$ parameters) processed segments of 256
points acquired at a sampling rate of 250kS/s. This network was trained and
evaluated on many segments of interferometric signal (data available at [22]
with details in [15]) of comparable amplitude, the whole data set being scaled
to its standard deviation.

In order to run the network on the STM32 platform, we reduced it so that it
operates on non-overlapping windows of 48 samples acquired at a rate of 48kHz.
In addition, a batch normalization layer has been added between each
convolution layer (kernel size of 3) and the following max pooling layer (pool
size of 2). ReLU activation functions are used everywhere except for the final
regression layer which is linear. The resulting network features 21601 tunable
weights.

###  II-B Network training

As we mentioned above Figure 1, the peak to peak amplitude of the
interferometric signal does not contain information about the displacement of
the target and may instead be altered by other factors (reflectivity of the
target or alignment conditions). Thus, the network must be trained to be
robust to variations of the amplitude of the signal. To optimize the network
for the reconstruction task at hand, we start from the data set available in
[22] which consists of 195011 segments of interferometric signal with their
corresponding true displacement value. This data set was downsampled so that
each segment of signal consists of 48 points instead of 256. In addition, the
original data set is normalized by its standard deviation. Here, to simulate
spurious signal amplitude variations, we apply during training a random
multiplication factor comprised between 6 and 60 to each segment of input
signal. The network is trained during 40 epochs with the RMPSprop optimizer
and an initial learning rate of $10^{-3}$ which is decreased progressively
down to $6*10^{-5}$ on learning plateaus, 20% of the data set being kept for
validation.

###  II-C Artificial neural network quantization

After training the artificial neural network (ANN), its weights and
activations are quantized. The inputs are also quantized. This enables a
reduction both in terms of memory and latency. Indeed, going from 32-bit
floating-point numbers to 16-bit fixed-point numbers divides the memory usage
of the weights (in ROM), the activations (in RAM) and the inputs (in RAM) by
2. Furthermore, computation with fixed-point numbers can be better optimized
to be performed faster than floating-point numbers.

The quantization of the neural network follows the method presented in [21].
Post-training quantization with floor rounding mode, a power-of-two scale
factor and a symmetric range is used. The same scale factor is applied to
weights, biases, activations and inputs. For 16-bit quantization, the
empirically chosen format is Q7.9, and for 8-bit quantization the chosen
format is Q2.6. Qx.y means x bits for the integer part (including sign) and y
bits for the fractional part.

###  II-D Deployment on microcontroller

The Qualia-CodeGen tool from the Qualia framework[20] is used in order to
generate a portable C library for inference from the trained Keras model.
Qualia-CodeGen generates an inference function for each layer of the neural
network, along with the layers call chain. In order to speed the inference up,
the CMSIS-NN[23] library is used. CMSIS-NN contains optimizations for
computation of neural network operations such as convolutions, and makes use
of specific instructions when available (e.g. SMLAD to perform two 16-bit
integer multiply-accumulate operations in a single cycle). The generated C
code is then compiled with the GCC compiler, using the -Ofast optimization
level, and uploaded to the microcontroller.

##  III Experimental setup

Figure 2: Experimental setup, LNA: Low-Noise Amplifier, ADC: Analog-to-Digital
Converter, MCU: MicroController Unit

###  III-A Optoelectronics setup

The experimental arrangement is shown schematically on Figure 2. From the
optical point of view it consists of a semiconductor laser driven slightly
above its coherent emission threshold by a stabilized current source emitting
a beam of about 1 $mW$ power at a wavelength $\lambda=1.310~{}\mu m$. The beam
is focused by a lens on a target at a distance of about 15 cm and a tiny part
of it (of the order of $10^{-4}$) is re-injected back into the laser. In this
case the target is a simple speaker whose displacement has been calibrated so
that the voltage at the edges of the speaker can be used as a proxy for the
displacement of the target. The details of this optical system are available
in [15] and we omit them here for brevity. The interferometric signal is
obtained by measuring the voltage at the edges of the laser diode. This
voltage consists of a large (about 1 V) continuous component superimposed with
much smaller fluctuations (sub-mV) which contain the relevant part of the
signal. We amplify the voltage with an AC-coupled (approximately $10^{4}$
gain) amplifier and the resulting signal is sent to the embedded platform.

###  III-B Embedded platform

Figure 3: Embedded platform

Our prototyping embedded platform consists of a Nucleo-L476RG board (white
PCB) mounted with a custom capture interface (green PCB), as seen in Fig. 3.
The Nucleo-L476RG board is designed around an STM32L476RG microcontroller with
a Cortex-M4F core running at 80 MHz, 1 MiB of Flash memory and 128 KiB of
SRAM.

The custom capture interface is originally designed for audio applications
with microphones. However, it can also be used to capture other AC signals,
although the ADC only supports standard audio sampling rates. The
TLV320ADC3101, which contains ADCs, amplification stages and filters, captures
the input signal and sends it to the microcontroller through an I2S interface.
The TLV320ADC3101 is configured with a 10 dB analog gain to the input signal,
a sampling rate of 48 kHz, and a resolution of 16 bits. Only one channel is
used.

The signal is fed through a 3.5 mm jack connector. An external low-noise
amplifier with a gain of approximately $10^{4}$ is placed between the laser
and the capture interface input jack. The amplitude of the signal at the
output of the low-noise amplifier varies depending on the calibration of the
optoelectronics setup, however a good calibration provides around 400 mV peak-
to-peak.

After being captured by the ADC, the signal is divided by $2^{5}$ in software
as an initial guess to better match the amplitude of the signal to the ANN
input layer. No z-score normalization is applied.

For evaluation of the embedded platform using data from the test dataset, the
output of the low-noise amplifier is replaced with the audio output of a
laptop which streams the interferometric signal from the test dataset.

##  IV Results

###  IV-A Robustness to amplitude variations on workstation

As outlined in II, the network must be able to reconstruct the displacement of
the target even in presence of unwanted variations of the signal amplitude. We
assess the performance of the network on workstation by comparing the true and
the inferred displacement on a validation data set (never used during
training) for different magnification factors applied to the input signal. The
results are shown on Figure 4. One good metric for performance is of course
the Pearson correlation coefficient between the true and inferred
displacement, shown on top of Figure 4. The correlation is found to be above
0.8 for scaling factors between 1 and 20, with a maximum of about 0.85 for
amplification factors between 2 and 3. These values can be compared to an
optimal Pearson coefficient of 0.9 with a much larger network in [15].

Beyond correlation, the performance of the network can be assessed by
measuring the linear regression coefficient between the inferred and the true
displacement, as shown on the bottom panel of Figure 4. In this case, the
optimal operation range is found for amplification factors between 2 and 100,
whereas the correlation decreases for much smaller amplifications indicating a
noisier inference, still keeping a good average precision. In the optimal
range, the regression coefficient is found to be slightly smaller than 1,
which would be the most precise inference. This is related to scarcity of
training data for the larger absolute values of displacement which imply a
saturation of the inference in these regions.

The existence of plateaus in both the correlation and the regression
coefficients indicates that the network is robust to about one order of
magnitude of amplitude variations.

Figure 4: Scaling signal influences the inference quality.

In the results that follow, the input data is multiplied by 3 in order to use
the optimal range.

###  IV-B Evaluation of quantization

The effect of quantization of the neural network is evaluated by generating
the C code with Qualia-CodeGen. Three configurations are evaluated: floating-
point numbers with no quantization (float32), fixed-point 16-bit quantization
with format Q7.9 (int16), and fixed-point 8-bit quantization with format Q2.6.
The inference is then performed on a workstation with the test dataset.

Figure 5: Predictions for float32, int16 and int8 models vs. ground truth. The solid lines are optimal linear fits with coefficients 0.74, 0.76 and 0.56 for float32, int16 and int8 respectively.  TABLE I: Pearson Correlation Coefficient for float32, int16 and int8 predictions compared to ground truth Data format | Pearson Correlation Coefficient
---|---
float32 | 0.8551
int16 Q7.9 | 0.8556
int8 Q2.6 | 0.7076

Fig. 5 shows the scatter plot for predictions of the neural network for int8
quantization (green), int16 quantization (orange), and non-quantized float32
(blue) versus the ground truth. Table I provides the Pearson correlation
coefficient with respect to the ground truth for each configuration.

The difference in correlation coefficient between float32 and int16 is only
$+0.0005$, we consider this not to be significant. The scatter plot shows a
slight shift in the distribution towards lower values from float32 to int16.
This can be explained by the floor rounding mode used for quantization. A less
biased result may be obtained by using round to nearest instead.

On the other hand, int8 quantization causes a significant loss in prediction
performance as the correlation coefficient drops by 0.1474 compared to
float32. This highlights the limits of such a simple quantization scheme. In
order to improve the results of 8-bit quantization, the scale factor must be
chosen separately for each layer, and it must also be decoupled for the
inputs, the weights and the activations. Quantization-aware training could
also be considered.

###  IV-C Evaluation of embedded constraints

TABLE II: Latency and memory consumption on STM32L476RG for float32, int16 and
int8 inference, with and without batch normalization, and with and without
CMSIS-NN

Data format | BatchNorm | CMSIS-NN | Latency | ROM | RAM
---|---|---|---|---|---
|  |  | (ms) | (kB) | (kB)
float32 | No | No | 14.8 | 136.672 | 12.120
int16 | No | Yes | 6.2 | 95.840 | 11.336
int16 | No | No | 13.2 | 94.488 | 9.816
float32 | Yes | No | 15.2 | 138.776 | 13.592
int16 | Yes | Yes | 6.8 | 97.104 | 12.072
int16 | Yes | No | 13.9 | 95.752 | 10.552
int8 | Yes | Yes | 7.4 | 78.456 | 10.200
int8 | Yes | No | 14.2 | 74.184 | 8.680

Table II shows latency and memory for different configurations of the neural
network when deployed on the microcontroller. The latency is measured as the
time taken to perform the inference, after collecting a window of data in the
input buffer. For the memory footprint, both ROM and RAM footprints are
provided. They are extracted from the memory allocation in the firmware after
compilation. The ROM contains the program instructions and the neural network
weights. The RAM contains the input buffer and each layer’s output buffer, all
allocated statically. When possible, i.e., on independent layers, buffers
share a common memory space. Note that some memory is also used for other
parts of the firmware, e.g., I2S and serial communications.

In these results, we consider cases with and without batch normalization
layers. For now, the batch normalization layers are not fused with the
preceding convolutional layer. As can be seen, the case with batch
normalization is slower by a few tenths of a millisecond, and has a slightly
higher memory footprint. However, in the future, we can assume that the
inference time with fused batch normalization will be the same as without
batch normalization. Furthermore, this will have the same effect on the memory
footprint.

We also compare the effect of fixed-point quantization, with and without
CMSIS-NN optimizations. As expected, 16-bit quantization significantly reduces
the memory footprint compared to 32-bit floating point by approximately 30%
for the ROM. Without CMSIS-NN, the reduction in inference time is of less than
10%, however using CMSIS-NN allows for a decrease of 55%.

8-bit goes further in terms of memory footprint reduction, by approximately
43% when compared to 32-bit floating point. Unfortunately, the inference time
is not reduced in this case, we suspect that the packing and unpacking of data
may not be as optimized.

All of these configurations fit well within the ROM (MCU limit: 384 kiB) and
RAM (MCU limit: 128 kiB) constraints, however the latency is well above our
real-time constraint of 1 ms. A more optimized neural network and a higher-
performance microcontroller may help reaching this target.

Still, the best performing configuration is using 16-bit quantization with
CMSIS-NN optimization. Furthermore, we already concluded in the previous
section that 16-bit quantization does not significantly deteriorate the
quality of the predictions over 32-bit floating point.

###  IV-D Live inference

(a) Interferometry signal

(b) Predictions and ground truth

Figure 6: Example of live inference on workstation (red) and embedded platform
(gray) to be compared to ground truth (red).

To demonstrate the experimental setup, we selected a displacement signal
composed of a sinusoidal signal for the first half and a more chaotic signal
for the second half (in red in Fig. 6(b)). The associated interferometry
signal that the deep neural network takes as inputs can be seen in Fig. 6(a).
On the workstation, the signal is interpreted as is by the neural network (in
floating point representation) and the predictions are shown in green in Fig.
6(b). On the embedded platform (STM32), the signal goes through the digital-
to-analog converter of the workstation, is then transmitted as an analog
signal through an audio cable to the embedded platform’s capture interface.
The capture interface amplifies the signal, samples it and performs analog-to-
digital conversion, for the microcontroller to handle the inference over the
digitized signal. This introduces variability in the signal, and therefore in
the predictions. Fig. 6(b) shows several of these inferences in gray.

We can see that the predictions do not match the ground truth exactly, but the
reconstruction is still visually similar. In the sinusoidal part, the
reconstruction on the workstation (in green) follows the ground truth (in red)
very closely. The predictions on the microcontroller (in grey) have a similar
shape, but they are scaled down a little bit. We suspect this is due to the
fact that the signal at the input of the neural network on the microcontroller
does not have the exact same amplitude as on the workstation, due to the
multiple transformations and amplification of the signal. For the chaotic
part, the reconstruction looks a bit less precise but the shape stays similar.

Figure 7: Comparison of correlations between the true displacement and the
displacement inferred on workstation and on the STM32 platform. The solid
lines are optimal linear fits with coefficients 0.84 on the workstation and
0.41 on the STM32.

Fig. 7 shows the scatter plot of the predictions vs. the ground truth, both
for the predictions from the workstation and from the embedded platform.
Overall, the predictions from the workstation are of better quality than from
the embedded platform. The workstation (using floating point representation)
provides predictions for this signal with a 0.84 correlation compared to the
ground truth, while for the embedded platform the correlation with the ground
truth is of 0.71. We suspect that the lack of control over the exact amplitude
of the signal and the lack of control of the exact sampling time, as well as
the added noise due to the analog transmission can cause these differences.

##  V Conclusion

In this article, we presented a first prototype of an embedded device for
small displacement measurement through self-mixing interferometry. To achieve
this, a deep neural network compatible with the embedded constraints was
designed and evaluated. Then, the neural network was quantized in order to
reduce the memory footprint and inference latency. Finally, the neural network
was deployed on an STM32L476RG microcontroller and evaluated in terms of
memory footprint and latency. Results showed that the signal reconstruction
could achieve a 0.8556 correlation with the ground truth and that the memory
footprint is kept below 10% of the microcontroller capabilities. However,
real-time constraints are not met yet due to the inference latency being too
high.

Future works will first focus on striving for real-time computation, i.e.,
reduce the latency below 1 ms. This will include optimizations of the neural
network to reduce the amount of computation, and evaluation of more powerful
microcontrollers. If the real-time constraint still cannot be met, a hardware
accelerator could be used instead. Furthermore, the issue related to the
robustness of the network to variations of amplitude must be solved. Apart
from trying to make the network more robust, it may also be possible to
leverage the automatic gain control of the TLV320ADC3101. Finally, the
embedded platform currently does not provide any synchronization mechanism. An
external trigger will be implemented to trigger an interrupt in the
microcontroller and synchronize the start of the measurement.

Beyond that, the present work opens the way to many possible applications of
AI-enabled self-mixing sensors including flow cytometry, microfluidics or
environmental conditions classification.

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# Remarks on $J$-hyperideals and their expansion

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

The aim of this research work is to define and characterize a new class of
hyperideals in a Krasner $(m,n)$-hyperring that we call n-ary $J$-hyperideals.
Also, we study the concept of n-ary $\delta$-$J$-hyperideals as an expansion
of n-ary $J$-hyperideals. Finally, we extend the notion of n-ary
$\delta$-$J$-hyperideals to $(k,n)$-absorbing $\delta$-$J$-hyperideals.

###### Key words and phrases:

$n$-ary $J$-hyperideal, $n$-ary $\delta$-$J$-hyperideal, $(k,n)$-absorbing
$\delta$-$J$-hyperideal.

###### 2010 Mathematics Subject Classification:

16Y99

##  1\. Introduction

Hyperstructures represent a natural extension of classical algebraic
structures and they were introduced by the French mathematician F. Marty. In
1934, Marty [21] defined the concept of a hypergroup as a generalization of
groups during the $8^{th}$ Congress of the Scandinavian Mathematicians. Many
papers and books concerning hyperstructure theory have appeared in literature
(see [7, 8, 9, 25, 28, 30]). The simplest algebraic hyperstructures which
possess the properties of closure and associativity are called
semihypergroups. $n$-ary semigroups and $n$-ary groups are algebras with one
$n$-ary operation which is associative and invertible in a generalized sense.
The notion of investigations of $n$-ary algebras goes back to Kasner’s lecture
[15] at a scientific meeting in 1904. In 1928, Dorente wrote the first paper
concerning the theory of $n$-ary groups [12]. Later on, Crombez and Timm [5,
6] defined the notion of the $(m,n)$-rings and their quotient structures.
Mirvakili and Davvaz [20] defined $(m,n)$-hyperrings and obtained several
results in this respect. In [10], they introduced a generalization of the
notion of a hypergroup in the sense of Marty and a generalization of an
$n$-ary group, which is called $n$-ary hypergroup. The $n$-ary structures has
been studied in [17, 18, 19, 20, 26]. Mirvakili and Davvaz [23] defined
$(m,n)$-hyperrings and obtained several results in this respect.

One important class of hyperrings was introduced by Krasner, where the
addition is a hyperoperation, while the multiplication is an ordinary binary
operation, which is called Krasner hyperring. In [22], a generalization of the
Krasner hyperrings, which is a subclass of $(m,n)$-hyperrings, was defined by
Mirvakili and Davvaz. It is called Krasner $(m,n)$-hyperring. Ameri and
Norouzi in [1] introduced some important hyperideals such as Jacobson radical,
n-ary prime and primary hyperideals, nilradical, and n-ary multiplicative
subsets of Krasner $(m,n)$-hyperrings. Afterward, the notions of
$(k,n)$-absorbing hyperideals and $(k,n)$-absorbing primary hyperideals were
studied by Hila et. al. [14].

Norouzi et. al. proposed and analysed a new defnition for normal hyperideals
in Krasner $(m,n)$-hyperrings, with respect to that one given in [22] and they
showed that these hyperideals correspond to strongly regular relations [24].
In [26], Ostadhadi-Dehkordi and Davvaz deﬁned the fundamental relation $\eta*$
on $R$ as the smallest equivalence relation on $R$ such that the quotient
$[R:\eta*]$ is an $(m,n)$-ring. Asadi and Ameri introduced and studied direct
limit of a direct system in the category of Krasner $(m,n)$-hyperrigs [4].

Dongsheng defined the notion of $\delta$-primary ideals in a commutative ring
where $\delta$ is a function that assigns to each ideal $I$ an ideal
$\delta(I)$ of the same ring [11]. Moreover, in [13] he and his colleague
investigated 2-absorbing $\delta$-primary ideals which unify 2-absorbing
ideals and 2-absorbing primary ideals. Ozel Ay et al. generalized the notion
of $\delta$-primary on Krasner hyperrings [27]. The concept of
$\delta$-primary hyperideals in Krasner $(m,n)$-hyperrings, which unifies the
prime and primary hyperideals under one frame, was defined in [3]. The notion
of $J$-ideals as a generalization of n-ideals in ordinary rings was studied by
Khashan and Bani-ata in [16].

Now in this paper, first we define the notion of n-ary $J$-hyperideals in a
Krasner $(m,n)$-hyperring which is a generalization of $J$-ideals. We give
several characterizations of n-ary $J$-hyperideals. Afterward, we study the
concept of n-ary $\delta$-$J$-hyperideals as an expansion of n-ary
$J$-hyperideals. Several properties of them are provided. Moreover, we extend
the notion of n-ary $\delta$-$J$-hyperideals to $(k,n)$-absorbing
$\delta$-$J$-hyperideals.

##  2\. Preliminaries

In this section we recall some definitions and results concerning $n$-ary
hyperstructures which we need to develop our paper.
Let $H$ be a nonempty set and $P^{*}(H)$ be the set of all the non-empty
subsets of $H$. Then the mapping $f:H^{n}\longrightarrow P^{*}(H)$ is called
an $n$-ary hyperoperation and the algebraic system $(H,f)$ is called an
$n$-ary hypergroupoid. For non-empty subsets $A_{1},...,A_{n}$ of $H$ we
define

$f(A^{n}_{1})=f(A_{1},...,A_{n})=\bigcup\\{f(x^{n}_{1})\ |\ x_{i}\in
A_{i},i=1,...,n\\}$.
The sequence $x_{i},x_{i+1},...,x_{j}$ will be denoted by $x^{j}_{i}$. For
$j<i$, $x^{j}_{i}$ is the empty symbol. Using this notation,

$f(x_{1},...,x_{i},y_{i+1},...,y_{j},z_{j+1},...,z_{n})$
will be written as $f(x^{i}_{1},y^{j}_{i+1},z^{n}_{j+1})$. The expression will
be written in the form $f(x^{i}_{1},y^{(j-i)},z^{n}_{j+1})$, when
$y_{i+1}=...=y_{j}=y$ .

If for every $1\leq i<j\leq n$ and all $x_{1},x_{2},...,x_{2n-1}\in H$,

$f(x^{i-1}_{1},f(x_{i}^{n+i-1}),x^{2n-1}_{n+i})=f(x^{j-1}_{1},f(x_{j}^{n+j-1}),x_{n+j}^{2n-1}),$
then the n-ary hyperoperation $f$ is called associative. An $n$-ary
hypergroupoid with the associative $n$-ary hyperoperation is called an $n$-ary
semihypergroup.

An $n$-ary hypergroupoid $(H,f)$ in which the equation $b\in
f(a_{1}^{i-1},x_{i},a_{i+1}^{n})$ has a solution $x_{i}\in H$ for every
$a_{1}^{i-1},a_{i+1}^{n},b\in H$ and $1\leq i\leq n$, is called an $n$-ary
quasihypergroup, when $(H,f)$ is an $n$-ary semihypergroup, $(H,f)$ is called
an $n$-ary hypergroup.

An $n$-ary hypergroupoid $(H,f)$ is commutative if for all
$\sigma\in\mathbb{S}_{n}$, the group of all permutations of
$\\{1,2,3,...,n\\}$, and for every $a_{1}^{n}\in H$ we have
$f(a_{1},...,a_{n})=f(a_{\sigma(1)},...,a_{\sigma(n)})$. If $a_{1}^{n}\in H$
then we denote $a_{\sigma(1)}^{\sigma(n)}$ as the
$(a_{\sigma(1)},...,a_{\sigma(n)})$.

If $f$ is an $n$-ary hyperoperation and $t=l(n-1)+1$, then $t$-ary
hyperoperation $f_{(l)}$ is given by

$f_{(l)}(x_{1}^{l(n-1)+1})=f(f(...,f(f(x^{n}_{1}),x_{n+1}^{2n-1}),...),x_{(l-1)(n-1)+1}^{l(n-1)+1})$.

######  Definition 2.1.

([22]). Let $(H,f)$ be an $n$-ary hypergroup and $B$ be a non-empty subset of
$H$. $B$ is called an $n$-ary subhypergroup of $(H,f)$, if
$f(x^{n}_{1})\subseteq B$ for $x^{n}_{1}\in B$, and the equation $b\in
f(b^{i-1}_{1},x_{i},b^{n}_{i+1})$ has a solution $x_{i}\in B$ for every
$b^{i-1}_{1},b^{n}_{i+1},b\in B$ and $1\leq i\leq n$. An element $e\in H$ is
called a scalar neutral element if $x=f(e^{(i-1)},x,e^{(n-i)})$, for every
$1\leq i\leq n$ and for every $x\in H$.

An element $0$ of an $n$-ary semihypergroup $(H,g)$ is called a zero element
if for every $x^{n}_{2}\in H$ we have
$g(0,x^{n}_{2})=g(x_{2},0,x^{n}_{3})=...=g(x^{n}_{2},0)=0$. If $0$ and
$0^{\prime}$are two zero elements, then $0=g(0^{\prime},0^{(n-1)})=0^{\prime}$
and so the zero element is unique.

######  Definition 2.2.

([17]). Let $(H,f)$ be a $n$-ary hypergroup. $(H,f)$ is called a canonical
$n$-ary hypergroup if
(1) there exists a unique $e\in H$, such that for every $x\in
H,f(x,e^{(n-1)})=x$;
(2) for all $x\in H$ there exists a unique $x^{-1}\in H$, such that $e\in
f(x,x^{-1},e^{(n-2)})$;
(3) if $x\in f(x^{n}_{1})$, then for all $i$, we have $x_{i}\in
f(x,x^{-1},...,x^{-1}_{i-1},x^{-1}_{i+1},...,x^{-1}_{n})$.

We say that $e$ is the scalar identity of $(H,f)$ and $x^{-1}$ is the inverse
of $x$. Notice that the inverse of $e$ is $e$.

######  Definition 2.3.

([22])A Krasner $(m,n)$-hyperring is an algebraic hyperstructure $(R,f,g)$, or
simply $R$, which satisfies the following axioms:
(1) $(R,f$) is a canonical $m$-ary hypergroup;
(2) $(R,g)$ is a $n$-ary semigroup;
(3) the $n$-ary operation $g$ is distributive with respect to the $m$-ary
hyperoperation $f$ , i.e., for every $a^{i-1}_{1},a^{n}_{i+1},x^{m}_{1}\in R$,
and $1\leq i\leq n$,

$g(a^{i-1}_{1},f(x^{m}_{1}),a^{n}_{i+1})=f(g(a^{i-1}_{1},x_{1},a^{n}_{i+1}),...,g(a^{i-1}_{1},x_{m},a^{n}_{i+1}))$;
(4) $0$ is a zero element (absorbing element) of the $n$-ary operation $g$,
i.e., for every $x^{n}_{2}\in R$ we have

$g(0,x^{n}_{2})=g(x_{2},0,x^{n}_{3})=...=g(x^{n}_{2},0)=0$.

We assume throughout this paper that all Krasner $(m,n)$-hyperrings are
commutative.

A non-empty subset $S$ of $R$ is called a subhyperring of $R$ if $(S,f,g)$ is
a Krasner $(m,n)$-hyperring. Let $I$ be a non-empty subset of $R$, we say that
$I$ is a hyperideal of $(R,f,g)$ if $(I,f)$ is an $m$-ary subhypergroup of
$(R,f)$ and $g(x^{i-1}_{1},I,x_{i+1}^{n})\subseteq I$, for every $x^{n}_{1}\in
R$ and $1\leq i\leq n$.

######  Definition 2.4.

([1]) For every element $x$ in a Krasner $(m,n)$-hyperring $R$, the hyperideal
generated by $x$ is denoted by $<x>$ and defined as follows:

$<x>=g(R,x,1^{(n-2)})=\\{g(r,x,1^{(n-2)})\ |\ r\in R\\}$

######  Definition 2.5.

([1]) A hyperideal $M$ of a Krasner $(m,n)$-hyperring $R$ is said to be
maximal if for every hyperideal $N$ of $R$, $M\subseteq N\subseteq R$ implies
that $N=M$ or $N=R$.

The Jacobson radical of a Krasner $(m,n)$-hyperring $R$ is the intersection of
all maximal hyperideals of $R$ and it is denoted by $J_{(m,n)}(R)$. If $R$
does not have any maximal hyperideal, we let $J_{(m,n)}(R)=R$.

######  Definition 2.6.

([1]) We say that an element $x\in R$ is invertible if there exists $y\in R$
such that $1_{R}=g(x,y,1_{R}^{(n-2)})$. Also, the subset $U$ of $R$ is
invertible if and only if every element of $U$ is invertible .

######  Definition 2.7.

([1])A hyperideal $P$ of a Krasner $(m,n)$-hyperring $R$, such that $P\neq R$,
is called a prime hyperideal if for hyperideals $U_{1},...,U_{n}$ of $R$,
$g(U_{1}^{n})\subseteq P$ implies that $U_{1}\subseteq P$ or $U_{2}\subseteq
P$ or …or $U_{n}\subseteq P$.

######  Lemma 2.8.

(Lemma 4.5 in [1])Let $P\neq R$ be a hyperideal of a Krasner $(m,n)$-hyperring
$R$. Then $P$ is a prime hyperideal if for all $x^{n}_{1}\in R$,
$g(x^{n}_{1})\in P$ implies that $x_{1}\in P$ or … or $x_{n}\in P$.

######  Definition 2.9.

([1]) Let $I$ be a hyperideal in a Krasner $(m,n)$-hyperring $R$ with scalar
identity. The radical (or nilradical) of $I$, denoted by ${\sqrt{I}}^{(m,n)}$
is the hyperideal $\bigcap P$, where the intersection is taken over all prime
hyperideals $P$ which contain $I$. If the set of all prime hyperideals
containing $I$ is empty, then ${\sqrt{I}}^{(m,n)}$ is defined to be $R$.

Ameri and Norouzi showed that if $x\in{\sqrt{I}}^{(m,n)}$ then there exists
$t\in\mathbb{N}$ such that $g(x^{(t)},1_{R}^{(n-t)})\in I$ for $t\leq n$, or
$g_{(l)}(x^{(t)})\in I$ for $t=l(n-1)+1$ [1].

######  Definition 2.10.

([1]) A proper hyperideal $I$ in a Krasner $(m,n)$-hyperring $R$ with the
scalar identity $1_{R}$ is said to be primary if $g(x^{n}_{1})\in I$ and
$x_{i}\notin I$ implies that
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in{\sqrt{I}}^{(m,n)}$ for some $1\leq i\leq
n$.

If $I$ is a primary hyperideal in a Krasner $(m,n)$-hyperring $R$ with the
scalar identity $1_{R}$, then ${\sqrt{I}}^{(m,n)}$ is prime. (Theorem 4.28 in
[1])

######  Definition 2.11.

([1]) Let $S$ be a hyperideal of a Krasner $(m,n)$-hyperring $(R,f,g)$. Then
the set

$R/S=\\{f(x^{i-1}_{1},S,x^{m}_{i+1})\ |\ x^{i-1}_{1},x^{m}_{i+1}\in R\\}$
endowed with m-ary hyperoperation $f$ which for all
$x_{11}^{1m},...,x_{m1}^{mm}\in R$

$f(f(x_{11}^{1(i-1)},S,x^{1m}_{1(i+1)}),...,f(x_{m1}^{m(i-1)},S,x^{mm}_{m(i+1)}))$

$=f(f(x^{m1}_{11}),...,f(x^{m(i-1)}_{1(i-1)}),S,f(x^{m(i+1)}_{1(i+1)}),...,f(x^{mm}_{1m}))$
and with $n$-ary hyperoperation g which for all
$x_{11}^{1m},...,x_{n1}^{nm}\in R$

$g(f(x_{11}^{1(i-1)},S,x^{1m}_{1(i+1)}),...,f(x_{n1}^{n(i-1)},S,x^{nm}_{n(i+1)}))$

$=f(g(x^{n1}_{11}),...,g(x^{n(i-1)}_{1(i-1)}),S,g(x^{n(i+1)}_{1(i+1)}),...,f(x^{nm}_{1m}))$
construct a Krasner $(m,n)$-hyperring, and $(R/S,f,g)$ is called the quotient
Krasner $(m,n)$-hyperring of $R$ by $S$.

######  Definition 2.12.

([22]) Let $(R_{1},f_{1},g_{1})$ and $(R_{2},f_{2},g_{2})$ be two Krasner
$(m,n)$-hyperrings. A mapping $h:R_{1}\longrightarrow R_{2}$ is called a
homomorphism if for all $x^{m}_{1}\in R_{1}$ and $y^{n}_{1}\in R_{1}$ we have

$h(f_{1}(x_{1},...,x_{m}))=f_{2}(h(x_{1}),...,h(x_{m}))$

$h(g_{1}(y_{1},...,y_{n}))=g_{2}(h(y_{1}),...,h(y_{n})).$

######  Definition 2.13.

([3]) Let $R$ be a Krasner $(m,n)$-hyperring. A function $\delta$ is called a
hyperideal expansion of $R$ if it assigns to each hyperideal $I$ of $R$ a
hyperideal $\delta(I)$ of $R$ with the following conditions:

$(i)$ $I\subseteq\delta(I)$.

$(ii)$ if $I\subseteq J$ for any hyperideals $I,J$ of $R$, then
$\delta(I)\subseteq\delta(J)$ .

For example, let $R$ be a Krasner $(m,n)$-hyperring.

1\. Define $\delta_{0}(I)=I$, for each hyperideal $I$ of $R$. Then
$\delta_{0}$ is a hyperideal expansion of $R$.

2\. Define $\delta_{1}(I)=\sqrt{I}^{(m,n)}$, for each hyperideal $I$ of $R$.
Then $\delta_{1}$ is a hyperideal expansion of $R$.

3\. Define $\delta_{R}(I)=R$, for each hyperideal $I$ of $R$. Then
$\delta_{R}$ is a hyperideal expansion of $R$.

4\. Define $\delta_{q}(I/J)=\delta(I)/J$, for each hyperideal $I$ of $R$
containing hyperideal $J$ and expansion function $\delta$ of $R$. Then
$\delta_{q}$ is a hyperideal expansion of $R/J$.

######  Definition 2.14.

([3]) Let $(R_{2},f_{2},g_{2})$ and $(R_{2},f_{2},g_{2})$ be two Krasner
$(m,n)$-hyperrings and $h:R_{1}\longrightarrow R_{2}$ a hyperring
homomorphism. Let $\delta$ and $\gamma$ be hyperideal expansions of $R_{1}$
and $R_{2}$, respectively. Then $h$ is said to be a
$\delta\gamma$-homomorphism if $\delta(h^{-1}(I_{2}))=h^{-1}(\gamma(I_{2}))$
for hyperideal $I_{2}$ of $R_{2}$.

Note that $\gamma(h(I_{1})=h(\delta(I_{1})$ for $\delta\gamma$-epimorphism $h$
and for hyperideal $I_{1}$ of $R_{1}$ with $Ker(h)\subseteq I_{1}$. For
example, let $(R_{1},f_{1},g_{1})$ and $(R_{2},f_{2},g_{2})$ be two Krasner
$(m,n)$-hyperrings. If $\delta_{1}$ of $R_{1}$ and $\gamma_{1}$ of $R_{2}$ be
the hyperideal expansions defined in Example 3.2, in ([3]), then each
homomorphism $h:R_{1}\longrightarrow R_{2}$ is a
$\delta_{1}\gamma_{1}$-homomorphism.

##  3\. $n$-ary $J$-hyperideals

######  Definition 3.1.

A hyperideal proper $Q$ of a Krasner $(m,n)$-hyperring with the scalar
identity $1_{R}$ is said to be n-ary $J$-hyperideal if whenever $x_{1}^{n}\in
R$ with $g(x_{1}^{n})\in Q$ and $x_{i}\notin J_{(m,n)}(R)$ implies that
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in Q$.

######  Example 3.2.

The set $A=\\{0,1,x\\}$ with the following 3-ary hyperoeration$f$ and 3-ary
operation $g$ is a Krasner $(3,3)$-hyperring such that $f$ and $g$ are
commutative.

$f(0,0,0)=0,\ \ \ f(0,0,1)=1,\ \ \ f(0,1,1)=1,\ \ \ f(1,1,1)=1,\ \ \
f(1,1,x)=A$ $f(0,1,x)=A,\ \ \ f(0,0,x)=x,\ \ \ f(0,x,x)=x,\ \ \ f(1,x,x)=A,\ \
\ f(x,x,x)=x$

$\ \ \ g(1,1,1)=1,\ \ \ \ g(1,1,x)=g(1,x,x)=g(x,x,x)=x$

and for $x_{1},x_{2}\in R,g(0,x_{1},x_{2})=0$. In the Krasner
$(3,3)$-hyperring, hyperideals $\\{0\\}$ and $\\{0,x\\}$ are two n-ary
$J$-hyperideals of $A$.

######  Example 3.3.

The set $R=\\{0,1,\alpha,\beta\\}$ whih following 2-hyperoperation $"\oplus"$
is a canonical 2-ary hypergroup.

$\oplus$ |  $0$ $1$ $\alpha$ $\beta$
---|---
0 |  $0$ $1$ $\alpha$ $\beta$
$1$ |  $1$ $A$ $\beta$ $B$
$\alpha$ |  $\alpha$ $\beta$ $0$ $1$
$\beta$ |  $\beta$ $B$ $1$ $A$

In which $A=\\{0,1\\}$ and $B=\\{\alpha,\beta\\}$. Define
$g(a_{1}^{n})=\alpha$ for $a_{1},a_{2},a_{3},a_{4}\in B$, otherwise,
$g(a_{1}^{n})=0$. It follows that $(R,\oplus,g)$ ia a Krasner (2,4)-hyperring.
In the hyperring, $\\{0\\}$ is a 4-ary $J$-hyperideal.

######  Theorem 3.4.

Let $Q$ be an n-ary $J$-hyperideal of a Krasner $(m,n)$-hyperring $R$. Then
$Q\subseteq J_{(m,n)}(R)$.

###### Proof.

Let $Q$ be an n-ary $J$-hyperideal of a Krasner $(m,n)$-hyperring $R$ such
that $Q\nsubseteq J_{(m,n)}(R)$. Suppose that $x\in Q$ but $x\notin
J_{(m,n)}(R)$. Since $Q$ is a n-ary $J$-hyperideal of $R$ and
$g(x,1_{R}^{(n-1)})\in Q$, then we have $g(1_{R}^{(n)})\in Q$ which is a
contradiction. Therefore, $Q\subseteq J_{(m,n)}(R)$. ∎

Next, we characterize the Krasner ${(m,n)}$-hyperring which every proper
hyperideal is an n-ary $J$-hyperideal.

######  Theorem 3.5.

Let $R$ be a Krasner ${(m,n)}$-hyperring. Then $R$ is local if and only if
every proper hyperideal of $R$ is an n-ary $J$-hyperideal.

###### Proof.

$\Longrightarrow$ Let $M$ is the only maximal hyperideal of $R$. Then
$J_{(m,n)}(R)=M$. Suppose that $Q$ is a proper hyperideal of $R$. Let for
$x_{1}^{n}\in R$, $g(x_{1}^{n})\in Q$ such that $x_{i}\notin M$. Therefore
$x_{i}$ is invertible. Then we have

$\displaystyle g(x_{i}^{-1},g(x_{1}^{n}),1_{R}^{(n-2)})$
$\displaystyle=g(g(x_{i},x_{i}^{-1},1_{R}^{(n-2)}),g(x_{1}^{i-1},1_{R},x_{i+1}^{n}),1_{R}^{(n-2)})$
$\displaystyle=g(x_{1}^{i-1},1_{R},x_{i+1}^{n})$ $\displaystyle\subseteq Q$

Hence, $Q$ is a an n-ary $J$-hyperideal of $R$.

$\Longleftarrow$ Let every proper hyperideal of $R$ is an n-ary
$J$-hyperideal. Assume that the hyperideal $M$ of $R$ is maximal. Let $x\in
M$. By the hypothesis, the principal hyperideal $\langle x\rangle$ is an n-ary
$J$-hyperideal of $R$. Since $g(x,1_{R}^{(n-1)})\in\langle x\rangle$, then we
get $x\in J_{(m,n)}(R)$ or $g(1_{R}^{(n)})\in\langle x\rangle$. Since the
second case is a contradiction, then $x\in J_{(m,n)}(R)$ which implies
$J_{(m,n)}(R)=M$. Consequently, $R$ is a local Krasner $(m,n)$-hyperring. ∎

######  Theorem 3.6.

Let $\\{Q_{i}\\}_{i\in\Delta}$ be a nonmepty set of n-ary $J$-hyperideals of a
Krasner $(m,n)$-hyperring $R$. Then $\bigcap_{i\in\Delta}Q_{i}$ is an n-ary
$J$-hyperideal of $R$.

###### Proof.

Let $g(x_{1}^{n})\in\bigcap_{i\in\Delta}Q_{i}$ for some $x_{1}^{n}\in R$ such
that $x_{i}\notin J_{(m,n)}(R)$. Then $g(x_{1}^{n})\in Q_{i}$ for every
$i\in\Delta$. Since $Q_{i}$ is an n-ary $J$-hyperideal of $R$, we have
$g(x_{1}^{i-1},1_{R},x_{i-1}^{n})\in Q_{i}$. Then
$g(x_{1}^{i-1},1_{R},x_{i-1}^{n})\in\bigcap_{i\in\Delta}Q_{i}$. ∎

######  Theorem 3.7.

Let $Q$ be a proper hyperideal of a Krasner $(m,n)$-hyperring $R$. Then the
following statements are equivalent:

(1) $Q$ is an n-ary $J$-hyperideal of $R$.

(2) $Q=U_{x}$ where $U_{x}=\\{y\in R\ |\ g(x,y,1_{R}^{(n-2)})\in Q\\}$ for
every $x\notin J_{(m,n)}(R)$.

(3) $g(I_{1}^{n})\subseteq Q$ for some hyperideals $I_{1}^{n}$ of $R$ and
$I_{i}\nsubseteq J_{(m,n)}(R)$ imply that
$g(I_{1}^{i-1},1_{R},I_{i+1}^{n})\subseteq Q$.

###### Proof.

$(1)\Longrightarrow(2)$ Let $Q$ be an n-ary $J$-hyperideal of $R$. We have
$Q\subseteq U_{x}$ for every $x\in R$. Suppose that $y\in U_{x}$ such that
$x\notin J_{(m,n)}(R)$. This means $g(x,y,1_{R}^{(n-2)})\in Q$. Since $Q$ is
an n-ary $J$-hyperideal of $R$ and $x\notin J_{(m,n)}(R)$, then
$y=g(y,1_{R}^{(n-2)})\in Q$. Hence, we get $Q=U_{x}$.

$(2)\Longrightarrow(3)$ Let $g(I_{1}^{n})\subseteq Q$ for some hyperideals
$I_{1}^{n}$ of $R$ such that $I_{i}\nsubseteq J_{(m,n)}(R)$. Take $x_{i}\in
I_{i}$ such that $x_{i}\notin J_{(m,n)}(R)$. Hence,
$g(I_{1}^{i-1},x_{i},I_{I+1}^{n})\subseteq Q$ which means
$g(I_{1}^{i-1},1_{R},I_{i+1}^{n})\subseteq U_{x_{i}}$. Since $Q=U_{x_{i}}$ for
every $x_{i}\notin J(R)$, then $g(I_{1}^{i-1},1_{R},I_{i+1}^{n})\subseteq Q$.

$(iii)\Longrightarrow(i)$ Let $g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$
with $x_{i}\notin J_{(m,n)}(R)$. We have $g(\langle x_{1}\rangle,...,\langle
x_{n}\rangle)=g(\langle g(x_{1}^{n})\rangle,1_{R}^{(n-1)})\subseteq Q$ but
$\langle x_{i}\rangle\nsubseteq J_{(m,n)}(R)$. Then we get $g(\langle
x_{1}\rangle,...,\langle x_{i-1}\rangle,1_{R},\langle
x_{i+1}\rangle,...,\langle x_{n}\rangle)=g(\langle
g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\rangle,1^{(n-1)})\in Q$ which implies
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in Q$. Therefore, $Q$ is an n-ary
$J$-hyperideal of $R$. ∎

######  Theorem 3.8.

Let $Q$ be a proper hyperideal of a Krasner $(m,n)$-hyperring $R$. Then $Q$ is
an n-ary $J$-hyperideal of $R$ if and only if $U_{x}\subseteq J_{(m,n)}(R)$
where $U_{x}=\\{y\in R\ |\ g(x,y,1_{R}^{(n-2)}\in Q\\}$ for every $x\notin Q$.

###### Proof.

$\Longrightarrow$ Let $y\in U_{x}$ such that $x\notin Q$. So,
$g(x,y,1^{(n-2)})\in Q$. Then we have $y\in J(R)$ as $Q$ is an n-ary
$J$-hyperideal of $R$ and $x=g(x,1^{(n-2)})\notin Q$ .

$\Longleftarrow$ Let $g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$ such that
$x_{i}\notin J_{(m,n)}(R)$. If $g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\notin Q$,
then $x_{i}\in U_{g(x_{1}^{i-1},1_{R},x_{i+1}^{n})}\subseteq J_{(m,n)}(R)$
which is a contradiction. Then we conclude that
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in Q$. Thus, $Q$ is an n-ary $J$-hyperideal
of $R$. ∎

######  Theorem 3.9.

Let $Q$ be a hyperideal of a Krasner $(m,n)$-hyperring $R$ and $S$ be a
nonempty subset of $R$ such that $S\nsubseteq Q$. If $Q$ is an n-ary
$J$-hyperideal of $R$, then $U_{S}=\\{x\in R\ |\ g(x,S,1_{R}^{(n-2)})\in Q\\}$
is an n-ary $J$-hyperideal of $R$.

###### Proof.

Let $U_{S}=R$. Then $1_{R}\in U_{S}$ which implies $S\subseteq Q$ a
contradiction. Hence, $U_{S}$ is a proper hyperideal of $R$. Suppose that
$g(x_{1}^{n})\in U_{S}$ for some $x_{1}^{n}\in R$ such that $x_{i}\notin
J_{(m,n)}(R)$. This means $g(g(x_{1}^{n}),S,1_{R}^{(n-2)})=\bigcup_{s\in
S}g(g(x_{1}^{n}),s,1^{(n-2)})\subseteq Q$ which implies for each $s\in S$,
$g(g(x_{1}^{n}),s,1_{R}^{(n-2)})=g(g(x_{1}^{i-1},s,x_{i+1}^{n}),x_{i},1_{R}^{(n-2)})\in
Q$. Then
$g(x_{1}^{i-1},s,x_{i+1}^{n})=g(g(x_{1}^{i-1},s,x_{i+1}^{n}),1^{(n-1)})\in Q$
for all $s\in S$ as $Q$ is an n-ary $J$-hyperideal of $R$. This means
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in U_{S}$. Thus, $U_{S}$ is an n-ary
$J$-hyperideal of $R$. ∎

######  Theorem 3.10.

Let $Q$ be an n-ary $J$-hyperideal of a Krasner $(m,n)$-hyperring $R$ such
that there is no n-ary $J$-hyperideal which contains $Q$ properly. Then $Q$ is
an n-ary prime hyperideal of $R$.

###### Proof.

Assume that $Q$ is an n-ary $J$-hyperideal of a Krasner $(m,n)$-hyperring $R$
such that there is no n-ary $J$-hyperideal which contains $Q$ properly.
Suppose that $g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$ such that for every
$1\leq i\leq n-1$, $x_{i}\notin Q$. Put $S=\\{x_{1},...,x_{n-1}\\}$. By
Theorem 3.9, $(Q:S)$ is an n-ary $J$-hyperideal of $R$. Since $Q\subseteq
U_{S}=\\{x\in R\ |\ g(x,S,1^{(n-2)})\in Q\\}$, we conclude that
$x_{n}\in(Q:S)=Q$, by the hypothesis. Thus, $Q$ is a prime hyperideal. ∎

In Theorem 3.10, if $Q=J_{(m,n)}(R)$, then the inverse of the theorem is true.

######  Theorem 3.11.

Let $J_{(m,n)}(R)$ be an n-ary prime of $R$. Then $J_{(m,n)}(R)$ is an n-ary
$J$-hyperideal of $R$ such that there is no $J$-hyperideal which contains
$J_{(m,n)}(R)$ properly.

###### Proof.

Put $Q=J_{(m,n)}(R)$ such that $J_{(m,n)}(R)$ is an n-ary prime of $R$. Let
$g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$ with $x_{i}\notin J(R)$. Since
$Q$ is an n-ary prime hyperideal of $R$, then there exists $1\leq j\leq i-1$
or $i+1\leq j\leq n$ such that $x_{j}\in Q=J_{(m,n)}(R)$ which means the
hyperideal $J(R)$ of $R$ is an n-ary $J$-hyperideal. By Theorem 3.4, there is
no $J$-prime hyperideal which contains $Q$ properly. ∎

##  4\. $n$-ary $\delta$-$J$-hyperideals

In this section, we define and study the concept of $n$-ary
$\delta$-$J$-hyperideals as an expansion of $n$-ary $J$-hyperideals.

######  Definition 4.1.

Let $\delta$ be a hyperideal expansion of a Krasner $(m,n)$-hyperring $R$. A
proper hyperideal $Q$ of $R$ is called n-ary $\delta$-$J$-hyperideal if for
$x_{1}^{n}\in R$, $g(x_{1}^{n})\in Q$ implies that $x_{i}\in J_{(m,n)}(R)$ or
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q).$

######  Example 4.2.

The hyperideal $I=\\{0\mathbb{Z}^{\star}_{12},4\mathbb{Z}^{\star}_{12}\\}$ in
$\mathbb{Z}_{12}/\mathbb{Z}_{12}^{\star}$ of Example 4.1 in [14] is a
$\delta_{1}$-$J$-hyperideal.

######  Theorem 4.3.

Let $Q$ be a proper hyperideal of a Krasner $(m,n)$-hyperring $R$. If
$\delta(Q)$ is an n-ary $J$-hyperideal of $R$, then $Q$ is an n-ary
$\delta$-$J$-hyperideal of $R$.

###### Proof.

Suppose that $\delta(Q)$ is an n-ary $J$-hyperideal of $R$. Let
$g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$ such that $x_{i}\notin
J_{(m,n)}(R)$. Since $\delta(Q)$ is an n-ary $J$-hyperideal of $R$ and
$Q\subseteq\delta(Q)$, then $g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q)$
which implies $Q$ is a $\delta$-$J$-hyperideal of $R$. ∎

The next Theorem shows that the inverse of Theorem 4.3 is true
$\delta=\delta_{1}$.

######  Theorem 4.4.

Let $Q$ be a proper hyperideal of a Krasner $(m,n)$-hyperring $R$. If $Q$ is
an n-ary $\delta_{1}$-$J$-hyperideal of $R$, then $\sqrt{Q}^{(m,n)}$ is an
n-ary $J$-hyperideal of $R$.

###### Proof.

Let for $x_{1}^{n}\in R$, $g(x_{1}^{n})\in\sqrt{Q}^{(m,n)}$ such that
$x_{i}\notin J_{(m,n)}(R)$. By $g(x_{1}^{n})\in\sqrt{Q}^{(m,n)}$, it follows
that there exists $t\in\mathbb{N}$ such that if $t\leq n$, then
$g(g(x_{1}^{n})^{(t)},1_{R}^{(n-t)})\in Q$. Hence by associativity we get

$g(x_{i}^{(t)},g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},1_{R}^{(n-2t)})$

$=g(x_{i}^{(t)},g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},g(1_{R}^{(n)}),1_{R}^{(n-2t-1)})$

$=g(g(x_{i}^{(t)},1_{R}^{(n-t)}),g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},1_{R}^{(n-t-1)})$

$\subseteq Q$.
Since $Q$ is an n-ary $\delta_{1}$-$J$-hyperideal of $R$, then

$g(x_{i}^{(t)},1^{(n-t)})\in J_{(m,n)}(R)$
or

$g(g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},1_{R}^{(n-t)})\in\delta_{1}(Q)=\sqrt{Q}^{(m,n)}$.
If $g(x_{i}^{(t)},1^{(n-t)})\in J_{(m,n)}(R)$, then
$x_{i}\in\sqrt{J_{(m,n)}(R)}^{(m,n)}=J_{(m,n)}(R)$ which is a contradiction.
Then we have

$g(g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},1_{R}^{(n-t)})\in\sqrt{Q}^{(m,n)}$
which means $g(x_{i}^{i-1},1_{R},x_{i+1}^{n})\in\sqrt{Q}^{(m,n)}$. Thus we
conclude that $\sqrt{Q}^{(m,n)}$ is an n-ary $J$-hyperideal of $R$. If
$t=l(n-1)+1$, then by using a similar argument, one can easily complete the
proof. ∎

######  Theorem 4.5.

Let $Q$ be a proper hyperideal of a Krasner $(m,n)$-hyperring $R$ and let
$\delta$ and $\gamma$ be two hyperideal expansions of $R$. If $\delta(Q)$ is
an n-ary $\gamma$-$J$-hyperideal of $R$, then $Q$ is an n-ary
$\gamma\circ\delta$-$J$-hyperideal of $R$.

###### Proof.

Suppose that $\delta(Q)$ is an n-ary $\gamma$-$J$-hyperideal of $R$. Let
$g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$ such that $x_{i}\notin
J_{(m,n)}(R)$. We get $g(x_{1}^{n})\in\delta(Q)$ as $Q\subseteq\delta(Q)$.
Since $\delta(Q)$ is an n-ary $\gamma$-$J$-hyperideal of $R$ and $x_{i}\notin
J_{(m,n)}(R)$, then $g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\gamma(\delta(Q))$
which means $g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\gamma\circ\delta(Q)$. Thus,
$Q$ is an n-ary $\gamma\circ\delta$-$J$-hyperideal of $R$. ∎

######  Theorem 4.6.

Let $Q_{1},Q_{2}$ and $Q_{3}$ be three proper hyperideals of a Krasner
$(m,n)$-hyperring $R$ such that $Q_{1}\subseteq Q_{2}\subseteq Q_{3}$. If
$Q_{3}$ is an n-ary $\delta$-$J$-hyperideal of $R$ and
$\delta(Q_{1})=\delta(Q_{3})$, then $Q_{2}$ is an n-ary
$\delta$-$J$-hyperideal of $R$.

###### Proof.

Let $g(x_{1}^{n})\in Q_{2}$ for some $x_{1}^{n}\in R$ such that $x_{i}\notin
J_{(m,n)}(R)$. Since $Q_{2}\subseteq Q_{3}$ and $Q_{3}$ is an n-ary
$\delta$-$J$-hyperideal of $R$, then we have
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q_{3})$. Then we conclude that
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q_{1})$, by the hypothesis. Since
$Q_{1}\subseteq Q_{2}$, then $\delta(Q_{1})\subseteq\delta(Q_{2})$. This
implies that $g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q_{2})$ as needed. ∎

######  Theorem 4.7.

Let $Q$ be a $\delta$-$J$-hyperideal of a Krasner $(m,n)$-hyperring $R$ such
that $\sqrt{\delta(Q)}^{(m,n)}\subseteq\delta(\sqrt{Q}^{(m,n)})$. Then
$\sqrt{Q}^{(m,n)}$ is a $\delta$-$J$-hyperideal of $R$.

###### Proof.

Let for $x_{1}^{n}\in R$, $g(x_{1}^{n})\in\sqrt{Q}^{(m,n)}$ such that
$x_{i}\notin J_{(m,n)}(R)$. By $g(x_{1}^{n})\in\sqrt{Q}^{(m,n)}$, it follows
that there exists $t\in\mathbb{N}$ such that if $t\leq n$, then
$g(g(x_{1}^{n})^{(t)},1_{R}^{(n-t)})\in Q$. Hence by associativity we get

$g(x_{i}^{(t)},g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},1_{R}^{(n-2t)})$

$=g(x_{i}^{(t)},g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},g(1_{R}^{(n)}),1_{R}^{(n-2t-1)})$

$=g(g(x_{i}^{(t)},1_{R}^{(n-t)}),g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},1_{R}^{(n-t-1)})$

$\subseteq Q$.
Since $Q$ is an n-ary $\delta$-$J$-hyperideal of $R$, then

$g(x_{i}^{(t)},1^{(n-t)})\in J_{(m,n)}(R)$ or
$g(g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},1_{R}^{(n-t)})\in\delta(I)$.
If $g(x_{i}^{(t)},1^{(n-t)})\in J_{(m,n)}(R)$, then
$x_{i}\in\sqrt{J_{(m,n)}(R)}^{(m,n)}=J_{(m,n)}(R)$ which is a contradiction.
Then we have

$g(g(x_{i}^{i-1},1_{R},x_{i+1}^{n})^{(t)},1_{R}^{(n-t)})\in\delta(Q)$
which means $g(x_{i}^{i-1},1_{R},x_{i+1}^{n})\in\sqrt{\delta(Q)}^{(m,n)}$. By
the assumption, we obtain

$g(x_{i}^{i-1},1_{R},x_{i+1}^{n})\in\delta(\sqrt{Q}^{(m,n)})$.
Thus we conclude that $\sqrt{Q}^{(m,n)}$ is a $\delta$-$J$-hyperideal of $R$.
If $t=l(n-1)+1$, then by using a similar argument, one can easily complete the
proof. ∎

We say that $\delta$ has the property of intersection preserving if it
satisfies $\delta(I\cap J)=\delta(I)\cap\delta(J)$, for all hyperideals $I,J$
of $R$. For example, the hyperideal expansion $\delta_{1}$ of a Krasner
$(m,n)$-hyperring $R$ has the property of intersection preserving.

######  Theorem 4.8.

Suppose that $Q_{1}^{n}$ are n-ary $\delta$-$J$-hyperideal of a Krasner
$(m,n)$-hyperring $R$ and the hyperideal expansion $\delta$ of $R$ has the
property of intersection preserving. Then $Q=\bigcap_{i=1}^{n}Q_{i}$ is an
n-ary $\delta$-$J$-hyperideal of $R$.

###### Proof.

Let $g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$ such that
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\notin\delta(Q)$. Since the hyperideal
expansion $\delta$ of $R$ has the property of intersection preserving, then
there exists $1\leq j\leq n$ such that
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\notin\delta(Q_{j})$. Thus, we get $x_{i}\in
J_{(m,n)}(R)$ as $Q_{j}$ ia an n-ary $\delta$-$J$-hyperideal of $R$.
Consequently, $Q=\bigcap_{i=1}^{n}Q_{i}$ is an n-ary $\delta$-$J$-hyperideal
of $R$. ∎

######  Theorem 4.9.

Let $Q$ be a proper hyperideal of a Krasner $(m,n)$-hyperring $R$. Then the
following are equivalent:

(1) $Q$ is an n-ary $\delta$-$J$-hyperideal of $R$.

(2) If $Q_{1}^{n-1}$ are some hyperideals of $R$ and $x\in R$ such that
$g(Q_{1}^{n-1},x)\subseteq Q$, then $x\in J_{(m,n)}(R)$ or
$g(Q_{1}^{n-1},1_{R})\subseteq\delta(Q)$.

(3) If $Q_{1}^{n}$ are some hyperideals of $R$ and $g(Q_{1}^{n})\subseteq Q$,
then $Q_{i}\subseteq J_{(m,n)}(R)$ or
$g(Q_{1}^{i-1},1_{R},Q_{i+1}^{n})\subseteq\delta(Q).$

###### Proof.

$(1)\Longrightarrow(2)$ Let $Q$ be an n-ary $\delta$-$J$-hyperideal of $R$.
Assume that $g(Q_{1}^{n-1},x)\subseteq Q$ for some hyperideals $Q_{1}^{n-1}$
of $R$ such that $x\notin J_{(m,n)}(R)$. Therefore, for each $q_{i}\in Q_{i}$
with $1\leq i\leq n-1$ we have $g(q_{1}^{n-1},x)\in Q$. Since $Q$ is an n-ary
$\delta$-$J$-hyperideal of $R$ and $x\notin J_{(m,n)}(R)$, then
$g(q_{1}^{n-1},1_{R})\in\delta(Q)$ which means $g(Q_{1}^{n-1},1_{R})\subseteq
Q$.

$(2)\Longrightarrow(3)$ Let $g(Q_{1}^{n})\subseteq Q$ for some hyperideals
$Q_{1}^{n}$ of $R$ such that $Q_{i}\nsubseteq J_{(m,n)}(R)$. Take $x\in Q_{i}$
but $x\notin J_{(m,n)}(R)$. Since $g(Q_{1}^{i-1},x,Q_{i+1}^{n})\subseteq Q$
and $x\notin J_{(m,n)}$, then
$g(Q_{1}^{i-1},1_{R},Q_{i+1}^{n})\subseteq\delta(Q)$, by the hypothesis.

$(3)\Longrightarrow(1)$ Let $g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$ but
$x_{i}\notin J_{(m,n)}(R)$. Therefore $g(\langle x_{1}\rangle,...,\langle
x_{n}\rangle)\subseteq Q$ but $\langle x_{i}\rangle\nsubseteq J_{(m,n)}$. Thus
we have

$g(\langle x_{1}\rangle,...,\langle x_{i-1}\rangle,1_{R},\langle
x_{i+1}\rangle,...,\langle x_{n}\rangle)\subseteq\delta(Q)$.
This means $g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q)$. ∎

######  Theorem 4.10.

Let $Q$ be a proper hyperideal of a Krasner $(m,n)$-hyperring $R$. Then the
following are equivalent:

(1) $Q$ is an n-ary $\delta$-$J$-hyperideal of $R$.

(2) $Q\subseteq J_{(m,n)}(R)$ and if $g(x_{1}^{n})\in Q$ for some
$x_{1}^{n}\in R$, then either $x_{i}$ is in the intersection of all maximal
hyperideals of $R$ containing $Q$ or
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q)$.

###### Proof.

$(1)\Longrightarrow(2)$ Suppose that $Q$ is an n-ary $\delta$-$J$-hyperideal
of $R$. Let $Q\nsubseteq J_{(m,n)}(R)$. Take $x\in Q$ such that $x\notin
J_{(m,n)}(R)$. Since $g(x,1^{(n-1)})\in Q$, then $g(1^{(n)})\in Q$, a
contradiction. Hence, $Q\subseteq J_{(m,n)}(R)$. Since $J_{(m,n)}(R)$ is in
the intersection of all maximal hyperideals of $R$ containing $Q$, then the
second assertion follows.

$(2)\Longrightarrow(1)$ Let $g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$ such
that $x_{i}\notin J_{(m,n)}(R)$. The intersection of all maximal hyperideals
of $R$ containing $Q$ is in $J_{(m,n)}(R)$ as $Q\subseteq J_{(m,n)}(R)$. This
means $x_{i}$ is not in the intersection of all maximal hyperideals of $R$
containing $Q$. Then $g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q)$.
Consequently, $Q$ is an n-ary $\delta$-$J$-hyperideal of $R$. ∎

######  Theorem 4.11.

Let $R$ be a Krasner $(m,n)$-hyperring. Then the following are equivalent:

(1) $R$ is a local Krasner $(m,n)$-hyperring such that $J_{(m,n)}(R)$ is the
only maximal hyperideal of $R$.

(2) Every proper principal hyperideal is an n-ary $\delta$-$J$-hyperideal of
$R$.

(3) Every proper hyperideal is an n-ary $\delta$-$J$-hyperideal of $R$.

###### Proof.

$(1)\Longrightarrow(2)$ Let $R$ be a local Krasner $(m,n)$-hyperring such that
$J_{(m,n)}(R)$ is the only maximal hyperideal of $R$. Consider the principal
hyperideal $Q=\langle a\rangle$ for some element $a$ of $R$ which is not
invertible. Suppose that $g(x_{1}^{n})\in Q$ such that $x_{i}\notin
J_{(m,n)}(R)$. This means $x_{i}$ is invertible member of $R$. Then we have
$g(g(x_{1}^{n}),x_{i}^{-1},1_{R}^{(n-2)})=g(g(x_{1}^{i-1},1_{R},x_{i+1}^{n}),x_{i},x_{i}^{-1},1_{R}^{(n-3)})=g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\subseteq
Q\subseteq\delta(Q)$. Thus, $Q$ is an n-ary $\delta$-$J$-hyperideal of $R$.

$(2)\Longrightarrow(3)$ Suppose that $P$ is a proper hyperideal of $R$ and
$g(x_{1}^{n})\in P$ for some $x_{1}^{n}\in R$ such that $x_{i}\notin
J_{(m,n)}(R)$. Consider the principal hyperideal $Q=\langle
g(x_{1}^{n})\rangle$. By the assumption, $Q$ is an n-ary
$\delta$-$J$-hyperideal of $R$. Since $g(x_{1}^{n})\in Q$ and $x_{i}\notin
J_{(m,n)}(R)$, then $g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q)$. Since
$Q\subseteq P$, then $\delta(Q)\subseteq\delta(P)$ and so
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(P)$. This completes the proof.

$(3)\Longrightarrow(1)$ Suppose that $M$ is a maximal hyperideal of $R$. By
the hypothesis, $M$ is an n-ary $\delta$-$J$-hyperideal of $R$. By theorem
4.10, we have $M\subseteq J_{(m,n)}(R)$. Since $J_{(m,n)}(R)\subseteq M$, then
$M=J_{(m,n)}(R)$. This implies that $J_{(m,n)}(R)$ is the only maximal
hyperideal of $R$. Thus, $R$ is a local Krasner $(m,n)$-hyperring. ∎

Recall that a proper hyperideal $I$ of $R$ is said to be $\delta$-primary if
for all $x_{1}^{n}\in R$, $g(x_{1}^{n})\in I$ implies that $x_{i}\in I$ or
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(I)$ for some $1\leq i\leq n$ [3].

######  Theorem 4.12.

Let $Q$ be a $\delta$-primary hyperideal of $R$. Then $Q$ is an n-ary
$\delta$-$J$-hyperideal of $R$ if and only if $Q\subseteq J_{(m,n)}(R)$.

###### Proof.

$\Longrightarrow$ It follows by 4.10.

$\Longleftarrow$ Let $Q\subseteq J_{(m,n)}(R)$. Suppose that $g(x_{1}^{n})\in
Q$ for some $x_{1}^{n}\in R$ such that $x_{i}\notin J_{(m,n)}(R)$. Then
$x_{i}\notin Q$. Thus, we get $g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q)$
as $Q$ is a $\delta$-primary hyperideal of $R$. This implies that $Q$ is an
n-ary $\delta$-$J$-hyperideal of $R$. ∎

######  Theorem 4.13.

Let $Q$ be a maximal hyperideal of $R$. Then $Q$ is an n-ary
$\delta$-$J$-hyperideal of $R$ if and only if $Q=J_{(m,n)}(R)$.

###### Proof.

$\Longrightarrow$ Let $Q$ be an n-ary $\delta$-$J$-hyperideal of $R$. Since
$Q$ is a maximal hyperideal of $R$, then $J_{(m,n)}(R)\subseteq Q$. Let
$g(x_{1}^{n})\in Q$ for some $x_{1}^{n}\in R$ such that $x_{i}\notin Q$. Then
$x_{i}\notin J_{(m,n)}(R)$. Since $Q$ is an n-ary $\delta$-$J$-hyperideal of
$R$ and $x_{i}\notin J_{(m,n)}(R)$, we conclude that
$g(x_{1}^{i-1},1_{R},x_{i+1}^{n})\in\delta(Q)$. This implies that $Q$ is a
$\delta$-primary hyperideal of $R$. Therefore, we get $Q\subseteq
J_{(m,n)}(R)$, by Theorem 4.12. Hence $Q=J_{(m,n)}(R)$.

$\Longleftarrow$ It is obvious. ∎

######  Theorem 4.14.

Let $(R_{1},f_{1},g_{1})$ and $(R_{2},f_{2},g_{2})$ be two Krasner
$(m,n)$-hyperrings and $h:R_{1}\longrightarrow R_{2}$ be a
$\delta\gamma$-homomorphism such that $\delta$ and $\gamma$ be hyperideal
expansions of $R_{1}$ and $R_{1}$, respectively. Then the following statements
hold :

$(1)$ If $h$ is a monomorphism and $I_{2}$ is an n-ary
$\gamma$-$J_{2}$-hyperideal of $R_{2}$, then $h^{-1}(I_{2})$ is an n-ary
$\delta$-$J_{1}$-hyperideal of $R_{1}$.

$(2)$ Let $h$ be an epimorphism and $I_{1}$ be a hyperideal of $R$ such that
$Ker(h)\subseteq I_{1}$. If $I_{1}$ is an n-ary $\delta$-$J_{1}$-hyperideal of
$R_{1}$, then $h(I_{1})$ is an n-ary $\gamma$-$J_{2}$-hyperideal of $R_{2}$.

###### Proof.

$(1)$ Let for $x_{1}^{n}\in R_{1}$, $g_{1}(x_{1}^{n})\in h^{-1}(I_{2})$. Then
we get $h(g_{1}(x_{1}^{n}))=g_{2}(h(x_{1}^{n}))\in I_{2}$. Since $I_{2}$ is a
$\gamma$-$J_{2}$-hyperideal of $R_{2}$, it implies that either $h(x_{i})\in
J_{(m,n)}(R_{2})$ which follows $x_{i}\in J_{(m,n)}(R_{1})$ as $h$ is a
monomorphism, or

$g_{2}(h(x_{1}),...,h(x_{i-1}),1_{R_{2}},h(x_{i+1}),...,h(x_{n}))$

$=h(g_{1}(x_{1}^{i-1},1_{R_{1}},x_{i+1}^{n}))$

$\in\gamma(I_{2})$,
which follows $g_{1}(x_{1}^{i-1},1_{R_{1}},x_{i+1}^{n})\in
h^{-1}(\gamma(I_{2}))=\delta(h^{-1}(I_{2})$. Thus $h^{-1}(I_{2})$ is a
$\delta$-$J_{1}$-hyperideal of $R_{1}$.

$(2)$ Let for $y_{1}^{n}\in R_{2}$, $g_{2}(y_{1}^{n})\in h(I_{1})$ such that
$y_{i}\notin J_{(m,n)}(R_{2})$. Since $h$ is an epimorphism, then there exist
$x_{1}^{n}\in R_{1}$ such that $h(x_{1})=y_{1},...,h(x_{n})=y_{n}$. Hence

$h(g_{1}(x_{1}^{n}))=g_{2}(h(x_{1}),...,h(x_{n}))=g_{2}(y_{1}^{n})\in
h(I_{1})$.
Since $Ker(h)\subseteq I_{1}$, then we get $g_{1}(x_{1}^{n})\in I_{1}$. Since
$y_{i}\notin J_{(m,n)}(R_{2})$, then $x_{i}\notin J_{(m,n)}(R_{1})$. Since
$I_{1}$ is a $\delta$-$J_{1}$-hyperideal of $R_{1}$ and $x_{i}\notin
J_{(m,n)}(R_{1})$, it implies that
$g_{1}(x_{1}^{i-1},1_{R_{1}},x_{i+1}^{n})\in\delta(I_{1})$ which implies

$h(g_{1}(x_{1}^{i-1},1_{R_{1}},x_{i+1}^{n}))=g_{2}(h(x_{1}),...,h(x_{i-1}),1_{R_{2}},h(x_{i+1}),...,h(x_{n}))$

$\hskip 96.73918pt=g_{2}(y_{1}^{i-1},1_{R_{2}},y_{i+1}^{n})$

$\hskip 96.73918pt\in h(\delta(I_{1}))$

$\hskip 96.73918pt=\gamma(h(I_{1}))$
Thus $h(I_{1})$ is a $\gamma$-$J_{2}$-hyperideal of $R_{2}$. ∎

######  Corollary 4.15.

Let $Q$ and $J$ be hyperideals of a Krasner $(m,n)$-hyperring $R$ such that
$I\subseteq Q$. If $Q$ is an n-ary $\delta$-$J$-hyperideal of $R$, then $Q/I$
is an n-ary $\delta_{q}$ -$J$-hyperideal of $R/I$.

###### Proof.

Consider the map $\pi:R\longrightarrow R/I$, defined by $r\longrightarrow
f(r,I,0^{(m-2)})$. The map is a homomorphism of Krasner $(m,n)$-hyperrings, by
Theorem 3.2 in [1]. Now, by using Theorem 4.14 (2), the claim can be proved. ∎

##  5\. $(k,n)$-absorbing $\delta$-$J$-hyperideals

In this section, we extend the notion of n-ary $\delta$-$J$-hyperideals to
$(k,n)$-absorbing $\delta$-$J$-hyperideals.

######  Definition 5.1.

Let $\delta$ be a hyperideal expansion of a Krasner $(m,n)$-hyperring $R$ and
$k$ be a positive integer. A proper hyperideal $Q$ of $R$ is called
$(k,n)$-absorbing $\delta$-$J$-hyperideal if for $x_{1}^{kn-k+1}\in R$,
$g(x_{1}^{kn-k+1})\in Q$ implies that $g(x_{1}^{(k-1)n-k+2})\in J_{(m,n)}(R)$
or a $g$-product of $(k-1)n-k+2$ of $x_{i}^{,}$ s except
$g(x_{1}^{(k-1)n-k+2})$ is in $\delta(Q)$.

######  Example 5.2.

Suppose that $H=[0,1]$ and define $a\boxplus b=\\{max\\{a,b\\}\\}$, for $a\neq
b$ and $a\boxplus b=[0,a]$, for $a=b$. Let $\cdot$ is the usual multiplication
on real numbers. In the Krasner $(2,3)$-hyperring $H$, the hyperideal
$T=[0,0.5]$ is a $(2,2)$-absorbing $\delta_{1}$-$J$-hyperideal of $R$.

######  Theorem 5.3.

If $Q$ is $\delta$-$J$-hyperideal of a Krasner $(m,n)$-hyperring $R$, then $Q$
is $(2,n)$-absorbing $\delta$-$J$-hyperideal.

###### Proof.

Let for $x_{1}^{2n-1}\in R$, $g(x_{1}^{2n-1})\in Q$. Since $Q$ is a
$\delta$-$J$-hyperideal of $R$, then $g(x_{1}^{n})\in J_{(m,n)}(R)$ or
$g(x_{n+1}^{2n-1})\in\delta(Q)$. This implies that for $1\leq i\leq n$,
$g(x_{i},x_{n+1}^{2n-1})\in\delta(Q)$, since $x_{1}^{n}\in R$ and $\delta(Q)$
is a hyperideal of $R$. Consequently, hyperideal $Q$ is $(2,n)$-absorbing
$\delta$-primary. ∎

######  Theorem 5.4.

If $Q$ is $(k,n)$-absorbing $\delta$-$J$-hyperideal of a Krasner
$(m,n)$-hyperring $R$, then $Q$ is $(s,n)$-absorbing $\delta$-$J$-hyperideal
for $s>n$.

###### Proof.

Let for $x_{1}^{(k+1)n-(k+1)+1}\in R$, $g(x_{1}^{(k+1)n-(k+1)+1})\in Q$. Put
$g(x_{1}^{n+2})=x$. Since hyperideal $Q$ of $R$ is $(k,n)$-absorbing
$\delta$-$J$-hyperideal, $g(x,x_{n+3}^{(k+1)n-(k+1)+1})\in J_{(m,n)}(R)$ or a
$g$-product of $kn-k+1$ of the $x_{i}^{,}$s except
$g(x,x_{n+3}^{(k+1)n-(k+1)+1})$ is in $\delta(I)$. This implies that for
$1\leq i\leq n+2$, $g(x_{i},x_{n+3}^{(k+1)n-(k+1)+1})\in\delta(I)$ which means
hyperideal $Q$ is $(k+1,n)$-absorbing $\delta$-$J$-hyperideal. Thus hyperideal
$Q$ of $R$ is $(s,n)$-absorbing $\delta$-$J$-hyperideal for $s>n$. ∎

######  Theorem 5.5.

If $Q$ is a $(k,n)$-absorbing $J$-hyperideal of a Krasner $(m,n)$-hyperring
$R$, then $\sqrt{Q}^{(m,n)}$ is a $(k,n)$-absorbing $\delta$-$J$-hyperideal.

###### Proof.

Let for $x_{1}^{kn-k+1}\in R$, $g(x_{1}^{kn-k+1})\in\sqrt{Q}^{(m,n)}$. We
suppose that none of the $g$-products of $(k-1)n-k+2$ of the $x_{i}^{,}$s
other than $g(x_{1}^{(k-1)n-k+2})$ are in $\delta(\sqrt{Q}^{(m,n)})$. Since
$g(x_{1}^{kn-k+1})\in\sqrt{Q}^{(m,n)}$ then for some $t\in\mathbb{N}$ we have
for $t\leq n$, $g(g(x_{1}^{kn-k+1})^{(t)},1_{R}^{(n-t)})\in Q$ or for $t>n$
with $t=l(n-1)+1$, $g_{(l)}(g(x_{1}^{kn-k+1})^{(t)})\in Q$. In the former
case, since all $g$-products of the $x_{i}^{,}$s other than
$g(x_{1}^{(k-1)n-k+2})$ are not in $\delta(\sqrt{Q}^{(m,n)})$, then they are
not in $Q$. Since $Q$ is a $(k,n)$-absorbing $J$-hyperideal of $R$, then we
have $g(g(x_{1}^{(k-1)n-k+2)})^{(l)},1_{R}^{(n-t)})\in J_{(m,n)}(R)$ which
means $g(x_{1}^{(k-1)n-k+2})\in\sqrt{J_{(m,n)}(R)}^{(m,n)}=J_{(m,n)}(R)$. By
using similar argument for the second case, the claim is completed. ∎

######  Theorem 5.6.

If $\delta(Q)$ is $(2,n)$-absorbing $J$-hyperideal for hyperideal $Q$ of a
Krasner $(m,n)$-hyperring $R$, then $Q$ is a $(3,n)$-absorbing
$\delta$-$J$-hyperideal of $R$.

###### Proof.

Let for $x_{1}^{3n-2}\in R$, $g(x_{1}^{3n-2})\in Q$ but $g(x_{1}^{2n-1})\notin
J_{(m,n)}(R)$. By $g(x_{1}^{3n-2})\in Q$ it follows that
$g(g(x_{1},x_{2n}^{3n-2}),x_{2}^{2n-1})\in Q\subseteq\delta(Q)$. Since
$\delta(Q)$ is a $(2,n)$-absorbing $J$-hyperideal and $g(x_{2}^{2n-1})\notin
J_{(m,n)}(R)$, then we have $g(x_{1}^{n},x_{2n}^{3n-2})\in\delta(Q)$ or
$g(x_{1},x_{n+1}^{2n-1},x_{2n}^{3n-2})\in\delta(Q)$. Hence $Q$ is a
$(3,n)$-absorbing $\delta$-$J$-hyperideal of $R$. ∎

######  Theorem 5.7.

If $\delta(Q)$ is a $(k+1,n)$-absorbing $\delta$-$J$-hyperideal for the
hyperideal $Q$ of a Krasner $(m,n)$-hyperring $R$, then $Q$ is
$(k+1,n)$-absorbing $\delta$-$J$-hyperideal.

###### Proof.

Let for $x_{1}^{(k+1)n-(k+1)+1}\in R$, $g(x_{1}^{(k+1)n-(k+1)+1})\in Q$ but
$g(x_{1}^{kn-k+1})\notin J_{(m,n)}(R)$. Then we get
$g(x_{1}^{(k+1)n-(k+1)+1})=g(x_{1}^{kn-k},g(x_{kn-k+1}^{(k+1)n-(k+1)+1}))\in
Q\subseteq\delta(Q)$. Since hyperideal $\delta(I)$ is $(k+1,n)$-absorbing
$\delta$-$J$-hyperideal and $g(x_{1}^{kn-k})\notin J_{(m,n)}(R)$, we get for
$1\leq i\leq n$,
$g(x_{1}^{i-1},x_{i+1}^{kn-k},g(x_{kn-k+1}^{(k+1)n-(k+1)+1}))\in\delta(I)$.
Consequently, hyperideal $I$ is a $(k+1,n)$-absorbing $\delta$-$J$-hyperideal.
∎

######  Theorem 5.8.

Let $(R_{1},f_{1},g_{1})$ and $(R_{2},f_{2},g_{2})$ be two Krasner
$(m,n)$-hyperrings and $h:R_{1}\longrightarrow R_{2}$ be a
$\delta\gamma$-homomorphism such that $\delta$ and $\gamma$ are two hyperideal
expansions of Krasner $(m,n)$-hyperring $R_{1}$ and $R_{2}$, respectively.
Then the following statements hold :

$(1)$ Let $h$ be a monomorphism. If $Q_{2}$ is a $(k,n)$-absorbing
$\gamma$-$J_{2}$-hyperideal of $R_{2}$, then $h^{-1}(Q_{2})$ is a
$(k,n)$-absorbing $\delta$-$J_{1}$-hyperideal of $R_{1}$.

$(2)$ If $h$ is an epimorphism and $Q_{1}$ is a $(k,n)$-absorbing
$\delta$-$J_{1}$-hyperideal of $R_{1}$ such that $Ker(h)\subseteq Q_{1}$, then
$h(Q_{1})$ is a $\gamma$-$J_{2}$-hyperideal of $R_{2}$.

###### Proof.

$(1)$ Let for $x_{1}^{kn-k+1}\in R_{1}$, $g_{1}(x_{1}^{kn-k+1})\in
h^{-1}(Q_{2})$. It means
$h(g_{1}(x_{1}^{kn-k+1}))=g_{2}(h(x_{1}),...,h(x_{kn-k+1}))\in Q_{2}$. Since
$Q_{2}$ is a $(k,n)$-absorbing $\gamma$-$J_{2}$-hyperideal of $R_{2}$, we get
$g_{2}(h(x_{1}),...,h(x_{(k-1)n-k+2}))=h(g_{1}(x_{1}^{(k-1)n-k+2})\in
J_{(m,n)}(R_{2})$ which means $g_{1}(x_{1}^{(k-1)n-k+2})\in J_{(m,n)}(R_{1})$,
as $h$ is a monomorphism, or

$g_{2}(h(x_{1}),...,h(x_{i-1}),h(x_{i+1}),...,h(x_{kn-k+1)})=h(g_{1}(x_{1}^{i-1},x_{i+1}^{kn-k+1}))\in\gamma(Q_{2})$
which means $g_{1}(x_{1}^{i-1},x_{i+1}^{kn-k+1}))\in h^{-1}(\gamma(Q_{2})$ for
$1\leq i\leq n$. Since $h$ is a $\delta\gamma$-homomorphism then
$g_{1}(x_{1}^{i-1},x_{i+1}^{kn-k+1}))\in\delta(h^{-1}(Q_{2}))$ for $1\leq
i\leq n$. Therefore we conclude that $h^{-1}(Q_{2})$ is a $(k,n)$-absorbing
$\delta$-$J_{1}$-hyperideal of $R_{1}$.

$(2)$ Let for $y_{1}^{kn-k+1}\in R_{2}$, $g_{2}(y_{1}^{kn-k+1})\in h(Q_{1})$
such that $g_{2}(y_{1}^{(k-1)n-k+2})\notin J_{(m,n)}(R_{2})$. Then there are
$x_{1}^{(k-1)n-k+2}\in R_{1}$ such that $h(x_{i})=y_{i}$ for $1\leq
i\leq(k-1)n-k+2$. Hence,
$h(g_{1}(x_{1}^{kn-k+1})=g_{2}(h(x_{1}),...,h(x_{kn-k+1})\in h(Q_{1})$. Since
$Q_{1}$ containing $Ker(h)$ then $g_{1}(x_{1}^{kn-k+1})\in Q_{1}$. Since
$Q_{1}$ is a $(k,n)$-absorbing $\delta$-$J_{1}$-hyperideal of $R_{1}$ and
$g_{1}(x_{1}^{(k-1)n-k+2})\notin J_{(m,n)}(R_{1})$ , then
$g_{1}(x_{1}^{i-1},x_{i+1}^{kn-k+1})\in\delta(Q_{1})$ which means

$h(g_{1}(x_{1}^{i-1},x_{i+1}^{kn-k+1}))=g_{2}(h(x_{1}),...,h(x_{i-1}),h(x_{i+1}),...,h(x_{kn-k+1}))$

$\hskip 93.89418pt=g_{2}(y_{1}^{i-1},y_{i+1}^{kn-k+1})\in h(\delta(Q_{1}))$
for $1\leq i\leq(k-1)n-k+2$. Since $h$ is a $\delta\gamma$-epimorphism then we
have $g_{2}(y_{1}^{i-1},y_{i+1}^{kn-k+1})\in\gamma(h((Q_{1}))$. Consequently,
$h(Q_{1})$ is a $\gamma$-$J_{2}$-hyperideal of $R_{2}$. ∎

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