Robust Switching Control of DC-DC Boost Converter for EV Charging Stations

The authors are with CODIASE Group at LAPLACE (Université de Toulouse, CNRS, INPT, UPS), Toulouse, France.

email: {ahmad, pitanga, kergus, kader<EMAIL_ADDRESS>This work is funded by ADEME, the French Agency for Ecological Transition, as part of the I-REVE project.

Saif Ahmad
Ryan P. C. de Souza
Pauline Kergus
Zohra Kader
Stéphane Caux

In this work, the problem of switching control design for DC-DC boost converter is considered, in the case of operation under uncertain equilibrium condition arising due to perturbations in the input and load parameters. Assuming that these uncertain parameters are generated via a known linear exo-system, a parameter estimator is designed to update the equilibrium point for the switching controller in real-time. In order to mitigate the noise amplification problem associated with the designed parameter estimator, the estimation error injection term is filtered via a set of first-order filters to obtain the desired level of noise suppression in the final set of estimates. To demonstrate the efficiency of the developed scheme, a realistic application scenario of a DC charging station for electric vehicles is considered, with photovoltaic array as the source and a battery connected at the load side.

DC-DC boost converter, switching control, electric vehicle charging, parameter estimation, measurement noise.

§ INTRODUCTION

Microgrids are small-scale grids that integrate distributed loads and sources, able to operate with or without the contribution of the utility grid, providing significant benefits to the connected users as well as to the utility grid [1]. Comparison between AC and DC microgrids is available in [2], recalling that, with the projected increase in the number of DC powered components and of distributed sources that generate DC power, DC microgrids are a good candidate for future energy systems. Indeed, DC microgrids offer better compatibility with distributed renewable energy resources, higher reliability and efficiency [1]. In particular, they are increasingly popular for Electrical Vehicle (EV) charging [3]. Indeed, due to their capacity to absorb intermittent generation from renewable energy sources, DC microgrids allow to mitigate the burden and negative impacts of EV charging on the electrical grid.

In order to meet the expectation regarding the performance of such DC microgrids, efficient control and energy management are required. As recalled in [4], there are three hierarchical control levels for DC microgrids: primary, secondary and tertiary control. At the highest level, tertiary control, also known as Energy Management System (EMS), schedules the power dispatch between source and storage. The source load is then shared among the different sources through secondary control. This is done by sending references to be tracked by primary control, consisting of the local controllers, enabling proper load sharing for parallel connected converters. At each control level, the performance depends on the accuracy and performance of the following control level, which makes DC-DC converter control of crucial importance.

In this context, a number of linear control strategies have been proposed in literature based on averaged small signal models of DC-DC converters  [5, 6], which ensures desired nominal performance and system stability but only in the neighbourhood of the equilibrium point. However, power converters are generally characterised by highly nonlinear dynamics and continuous parametric variations subject to source and load conditions, which in turn shifts the equilibrium point. This highlights the limitations of closed-loop performance achievable via linear feedback control and the need for more advanced control strategies in this area that account for the switching and parameter-varying nature of converters. Advanced control methods for DC-DC converters in DC microgrids are reviewed in [7] and classified in the following categories: model predictive control, backstepping, sliding mode control (SMC), passivity-based control, observer/estimation based technique and intelligent control.
More recently, principles of switched systems theory has been successfully employed as an alternative to the averaging approach for control of power converters in [8]. One of the main advantages of using switched systems theory is that linearization around the equilibrium is not required, and global stabilization can be ensured. Furthermore, a modulation stage, such as Pulse-Width Modulation (PWM), is not required either.

However, the switching controller in [8] assumes that the input and load parameters remain unchanged for the power converters, which in turn results in a  constant equilibrium point. On the other hand, in most practical scenarios such as DC-microgrids and EV charging stations, power electronic converters are subject to time-varying source and load perturbations [9], resulting in continuous variation of the operating point. This problem has recently been addressed for DC-DC power converters by using augmented state observer [10] and state augmentation via tracking error integral [11]. In the present work, we address the problem of uncertain equilibrium in the context of switching control designed for a DC-DC boost converter (DBC) by estimating the uncertain parameters. The developed solution results in the following advantages: (a). Unlike [10], the designed estimator approximates only the required unknown parameters and not the measurable states. Moreover, noise amplification problem commonly associated with observers/estimators (such as those used in [10]) is addressed by the estimation error filtering approach introduced in [12] which results in better estimation quality and allows the freedom to introduce  desired level of filtering in the obtained estimates. Also, unlike [11], the parameter estimation approach returns physical quantities (input voltage and load current) that are important from an operational perspective (e.g. load sharing via droop control) in an interconnected network of power converters. It is also worth mentioning that the approach used in [11] is designed for a buck converter which is much simpler to handle while the present work considers a boost converter which has nonlinear dynamics. (b).  The resulting estimation error dynamics are devoid of switched terms which in turn facilitates the tuning of estimator, and gives simpler linear matrix inequality (LMI) conditions for stability of the designed estimator.  Furthermore, the robust switched control algorithm is implemented via a hysteresis block which provides a simple and intuitive solution to the infinite switching frequency problem [8] and facilitates practical implementation. In contrast to the previous works which consider constant resistances at the load end and step changes in input voltages, the present work considers a PV array connected at the input side and a Li-ion battery charging at the load side in order to simulate a more practical operating condition.

Remaining sections in this paper are organized as follows. Section <ref> first introduces the switching model for boost converters and highlights the importance of considering variations of operating points in control design of DC-DC converters. Section <ref> then exposes the proposed control and estimation algorithm along with the noise filtering and hysteresis approach to facilitate practical implementation. The efficiency of the proposed approach is then illustrated in Section <ref> through numerical simulations. The paper concludes in Section <ref> with a summary of the results and future research directions.

§.§.§ Notations
$\bm{I_n}$ is an identity matrix of dimension $n\times n$, $\textbf{0}$ is a zero vector or matrix of appropriate dimension, $\textrm{diag}(a_1,\dots,a_n)$ denotes a diagonal matrix with $a_1,\dots,a_n$ as diagonal elements.

§ PROBLEM FORMULATION

Fig. <ref> shows the ideal circuit of a DC-DC boost converter (DBC) where $v_{in}$ is the source voltage, $R_o$ is a load resistance, $i_{Load}$ denotes the current being supplied to another load connected in parallel, $L$ and $C$ are the inductance and capacitance respectively while $i_L$ and $v_o$ are the system states representing inductor current and output voltage respectively. We define $\sigma\in\{0,1\}$ as a switching variable which  corresponds to  the state of the switched device $S$ such that $\sigma=1$ when switch $S$ is on and $\sigma=0$ when the switch is in off state. The dynamics of a DBC can now be defined with the following expression:

[american voltages]
(0,0) to [american voltage source, v<=$v_{in}$,invert] (0,3)

to [L,i=$ i_L $,l=$ L $] (2,3)
to [empty diode] (4,3)
to [short](6,3)

to [R,l=$R_o$] (6,0)
(0,0) to [short] (8,0)
(6,3) to [short](8,3)

(2,3) to [closing switch,l_=$ S $] (2,0)
(4,3) to [C,l=$ C $,v=$ v_o $] (4,0)
(8,3) to [generic,i=$i_{Load}$] (8,0);

Circuit of a DC-DC boost converter.

\begin{equation}\label{eq:system}
\begin{split}
\dot{\bm{x}}(t)=\bm{A_{\sigma}x}(t)+\bm{G}\bm{p}(t)
\end{split}
\end{equation}

where $\bm{x}(t):=[i_L(t),v_o(t)]^T, \ \bm{p}(t)=[p_1(t),p_2(t)]^T:=[v_{in}(t),i_{Load}(t)]^T,$

\begin{equation}
\begin{split}
\bm{G}=&\begin{bmatrix}\frac{1}{L}&0\\ 0&-\frac{1}{C}\end{bmatrix},\bm{A_{\sigma}}=\begin{bmatrix}0&-\frac{(1-\sigma)}{L}\\ \frac{(1-\sigma)}{C}& -\frac{1}{R_oC}\end{bmatrix}.
\end{split}
\end{equation}

It can be shown that the possible equilibria of the instantaneous model (<ref>) are the same as those of the averaged model [10], whose dynamics are given by:

\begin{equation}\label{eq:averagedModel}
\bm{\dot{x}}(t) = \bm{A}(\bar{\sigma})\bm{x}(t) + \bm{Gp}(t),
\end{equation}

where $\bm{A}(\bar{\sigma}) = \bar{\sigma} \bm{A_1} + (1-\bar{\sigma})\bm{A_0}$.

The control objective in this paper is to stabilize the output voltage at a desired equilibrium value $v_o^\star$. A point $\bm{x^*} = [i_L^\star\;\;v_o^\star]^T$ is an equilibrium of (<ref>) if there exists a $\sigma^*\in[0,1]$ such that $\bm{A}(\sigma^*)\bm{x^*} + \bm{Gp} = \textbf{0}$ for a given $\bm{p}(t)$. Here, the value of $\bar{\sigma}^*$ is the steady-state duty cycle. Assuming $\bar{\sigma}^* > 0$, $\bm{x}^*$ can be expressed as $\bm{x}^* = -\bm{A}(\bar{\sigma}^*)^{-1}\bm{Gp}$.

§ ROBUST SWITCHING CONTROL DESIGN

It can easily be checked that (<ref>) is an example of a class of systems known as switched affine systems (SAS). This means that, for each $\sigma \in \{0,1\} $, the dynamics are expressed as an affine function of the state $\bm{x}$. Several works in the literature deal with control design for these systems. In the next subsection, one of these strategies ensuring global stabilization of the equilibrium point is presented.

§.§ Switching Controller

Assuming that $\bm{x^*}$ is perfectly known, the SAS defined by (<ref>) can be asymptotically stabilized to $\bm{x^*}$ if for a given $\alpha$ such that $0\leq\alpha\leq\bar{\alpha}$ (where $\bar{\alpha}>0$ is dependent on the open-loop system matrix $A(\sigma^*)$), there exists $\bm{P}=\bm{P}^T\succ0$ such that

\begin{equation}\label{eq:LMIforP}
\bm{A}^T(\sigma^*)\bm{P}+\bm{PA}(\sigma^*)+2\alpha\bm{P}\leq 0.
\end{equation}

The preceding LMI imposes an implicit decay rate $\alpha$ on the tracking error via appropriate selection of $\bm{P}$. The feasibility of (<ref>) is a convex problem, and efficient solvers are widely available.

It has been shown in [13] that, once $\bm{P}$ is found, the following switching controller stabilizes system (<ref>):

\begin{equation}\label{eq:switchingLaw1}
\sigma(\bm{x}(t)) \in \argmin_{m\in\{0,1\}} (\bm{x}(t)-\bm{x^*})^T \bm{P} (\bm{A_m x}(t) + \bm{G} \bm{p}(t)).
\end{equation}

Since the term $ \bm{G p}(t) $ does not depend on the mode $\sigma$ in DBC, (<ref>) can be rewritten as:

\begin{equation}\label{eq:switchingLaw}
\sigma(\bm{x}(t)) \in \argmin_{m\in\{0,1\}} (\bm{x}(t)-\bm{x^*})^T \bm{P A_m x}(t).
\end{equation}

As shown in [13], control law (<ref>) globally stabilizes system (<ref>). It is worth noting that, unlike more traditional control methods, the switching law presented here does not require linearization around the equilibrium point. Even so, it ensures global stabilization. Another practical advantage is that the controller directly outputs the mode to be selected, thereby bypassing the need for a modulation stage between the controller and the switching device.

However, it can easily be seen that the controller (<ref>) requires the knowledge of $\bm{x}^*$, which, as previously established, is given by $\bm{x}^* = -A(\sigma^*)^{-1}G\bm{p}(t)$. It can be checked that $\bm{x}^*$ depends in fact on both $p_1(t)$ and $p_2(t)$ for the case of DBC:

\begin{equation}\label{eq:xstar}
x^\star = \frac{1}{1-\sigma^*}\left[\begin{array}{c} p_2 \\ p_1 \end{array}\right].
\end{equation}

Note from (<ref>) that $\sigma^*$ can be determined as:

\begin{equation}\label{eq:sigma*}
\sigma^* = 1 - \frac{p_1}{v_o^\star}.
\end{equation}

Therefore, in order to implement (<ref>), the parameter vector $\bm{p}(t)$ must be known. In addition, solving LMI (<ref>) requires the knowledge of $\sigma^*$, which depends on $p_1(t)$. However, $\bm{p}(t)$ itself is uncertain and depends upon the source as well as load conditions which in turn necessitates either the measurement of these additional signals (which incurs additional cost) or estimation in order to ensure accurate tracking of the desired uncertain and possibly time-varying equilibrium point $\bm{x^*}(t)$, where the variation with respect to time is sufficiently slow. A straightforward implementation of (<ref>) requires the equilibrium point to be updated at each time-step using the measurement of $\bm{p}(t)$. If the measurements are unavailable, it becomes necessary to estimate $\bm{p}(t)$, which is the subject of the next subsection.

§.§.§ Remark 1
SAS defined  in (<ref>) and the corresponding switching controller in (<ref>) is not specific to a DBC and can be used to model and control a wider class of DC-DC converters including buck, buck-boost, flyback converters among others [10, 11]. Moreover, the problem of uncertain equilibrium is common to all power converters due to the inherent intermittency of the renewables and uncertain load conditions (time-varying $\bm{p}(t)$) in most applications. Hence, the presented solutions, results and ensuing discussions are more general and can easily be extended for other types of converters.

§.§ Parameter Estimator

Since we assume that now the parameters $\bm{p}$ may change with time, then, as mentioned before, the equilibrium point $\bm{x^*}$ changes accordingly. Let us assume that $p_1$ is bounded in the following way: $v_{in}^- \leq p_1 \leq v_{in}^+ < v_o^\star$, where $v_{in}^-$ and $v_{in}^+$ define the range in which the input voltage may evolve and they depend on its uncertainty. Note that if $v_{in}^+ > v_o^\star$, then the equilibrium point may be unreachable, since the output voltage of a DBC cannot be lower than the input voltage. As $v_{in}$ evolves over time, the steady-state duty cycle $\sigma^*$ varies according to (<ref>).

In order for the control law (<ref>) to stabilize the system in all the range of possible values for $\sigma^*$, one must find a single matrix $\bm{P}$ such that (<ref>) holds for all $\sigma^* \in [\sigma_{min}^*,\sigma_{max}^*]$, where $\sigma_{min}^* = 1-v_{in}^+/v_o^\star$ and $\sigma_{max}^* = 1-v_{in}^-/v_o^\star$. Thanks to the convexity of (<ref>), this is true if the following two LMIs hold:

\begin{equation}\label{eq:LMIforPmod}
\bm{A}^T(\mu)\bm{P}+\bm{PA}(\mu)+2\alpha\bm{P}\leq 0,\quad \mu \in\{\sigma_{min}^*,\sigma_{max}^*\}.
\end{equation}

Now, in order to obtain the estimate of uncertain parameters, it is assumed that the parameter vector $\bm{p}(t)$ is generated by a known linear exo-system of the form

\begin{equation}
\begin{split}
\bm{\dot{\zeta}_p}(t)&=\bm{A_p\zeta_p}(t)\\
\bm{p}(t)&=\bm{C_p\zeta_p}(t),
\end{split}
\label{eq:dist_dynamics}
\end{equation}

where  $\bm{\zeta_p}\in\mathbb{R}^{m}$ is the state-vector defining the exo-system while $\bm{A_p}\in \mathbb{R}^{m\times m}$ and $\bm{C_p}\in \mathbb{R}^{2\times m} $ are the associated matrices. The preceding assumption allows us to consider a wider class of parameter behaviors such as linearly varying with time, sinusoidal etc. or a combination, that arise in practical systems. Furthermore, the common assumption of $\bm{\dot{p}}=\textbf{0}$ considered in [10, 11], is a special case of (<ref>) obtained with $\bm{A_p}=\textbf{0}$ and $\bm{C_p}=\bm{I_2}$, which is also used in the current work. The following switched parameter estimator can now be constructed upon (<ref>) and (<ref>) following the general approach introduced in [14]:

\begin{equation}
\begin{split}
\dot{\hat{\bm{z}}}_{\bm{p}}(t)=&(\bm{A_p}-\bm{\kappa GC_p})\bm{\hat{z}_p}(t)+(\bm{A_p-\kappa GC_p})\bm{\kappa}\bm{x}(t)-\bm{\kappa}\bm{A_{\sigma}}\bm{x}(t)\\
\hat{\bm{\zeta}}_{\bm{p}}(t) =&\hat{\bm{z}}_{\bm{p}}(t)+\bm{\kappa x}(t)\\
\hat{\bm{p}}(t)=&\bm{C_p\hat{\zeta}_p}(t)
\end{split}
\label{eq:pe}
\end{equation}

where $\bm{\hat{z}_p}\in\mathbb{R}^{m}$ is an intermediate variable (introduced to avoid the implementation of derivative $\bm{\dot{x}}(t)$), $ \bm{\hat{\zeta}_p}$ and $\bm{\hat{p}}$ are estimates of $\bm{\zeta_p}$ and $\bm{p}$ respectively, while $\bm{\kappa}\in \mathbb{R}^{m\times2}$ is the observer gain vector that needs to be selected.

From (<ref>) and (<ref>), the error dynamics for the designed estimator is obtained as

\begin{equation}
\bm{\dot{e}}(t)=(\bm{A_p}-\bm{\kappa G C_p})e(t)
\label{eq:estimation_error}
\end{equation}

where $\bm{e}(t):=\bm{\hat{\zeta}_p}-\bm{\zeta_p}$ (and consequently $\bm{e_p}(t):=\bm{p}(t)-\bm{\hat{p}}(t)=\bm{C_pe}(t)$) converges asymptotically if $\bm{\kappa}$ is selected such that $\bm{A_p}-\bm{\kappa G C_p}$ is Hurwitz, which is equivalent to saying that there exists a $\bm{\Bar{P}}=\bm{\Bar{P}}^T\succ\textbf{0}$, such that

\begin{equation}
(\bm{A_p}-\bm{\kappa G C_p})^T\bm{\Bar{P}}+\bm{\Bar{P}}(\bm{A_p}-\bm{\kappa G C_p})\leq \textbf{0}.
\label{eq:LMI_estimator_stability}
\end{equation}
Condition (<ref>) is obtained readily upon selection of a Lyapunov function of the form $V=\bm{e}^T\bm{\bar{P}e}$ and taking its derivative along the trajectories of estimation error dynamics (<ref>), following the approach given in [14]. In this case, it can be shown that the estimation error dynamics is globally asymptotically stable.

§.§.§ Remark 2
For the sake of simplicity, it is assumed in this paper that $R_o$ is a known load resistance. However, uncertainties in this parameter can be addressed by adapting the LMIs in (<ref>) to be solved in a similar way as in [10], for instance.

§.§.§ Remark 3
In a practical scenario, it is not possible to completely characterize the variations in uncertain parameters with a known linear exo-system and hence, it is more appropriate to consider the exo-system dynamics as

\begin{equation}
\begin{split}
\bm{\dot{\zeta}_p}(t)&=\bm{A_p\zeta_p}(t)+\mathcal{H}\\
\bm{p}(t)&=\bm{C_p\zeta_p}(t),
\end{split}
\label{eq:practical_dist_dynamics}
\end{equation}

where $\mathcal{H}\in\mathbb{R}^m$ represents the residual of the exo-system i.e. the model mismatch. The convergence results obtained previously can be extended for the practical scenario considering (<ref>) to show that that the estimation error dynamics is input to state stable (ISS) [12] with respect to $\mathcal{H}$, provided $\|\mathcal{H}\|\leq\mu$ where $\mu$ is a positive constant, which is always the case in practical applications due to physical limitations of real systems and implementation of protection mechanisms. Under a high-gain formulation [12], it is also possible to show that the region in which the estimation error eventually converges under the effect of $\mathcal{H}$ can be made made arbitrary small and the rate of convergence can be made arbitrarily fast by increasing the high-gain parameter [15]. We apply this approach for tuning the estimator parameter $\bm{\kappa}$ in the current work such that all the eigenvalues of $\bm{A_p}-\bm{\kappa G C_p}$ are placed at $``-\lambda"$ which denotes the high-gain parameter in this case. This in turn reduces the number of parameters to be selected and simplifies the tuning process to facilitate practical implementation.

§.§.§ Remark 4
As shown in [15], it is possible to obtain a more accurate estimate of time-varying parameters by approximating their dynamics using a $k^{th}$ order time-polynomial form. However, the noise amplification associated with high-gain construction  is aggravated in this case and additional filtering becomes necessary [15, 12].

§.§ Noise Filtering

In real systems, state measurements are always corrupted by high-frequency noise component $(\nu)$ which gets added during the measurement process. Therefore, the designed estimator and controller receive corrupted state measurements $(\bm{x_m}:=\bm{x}+[1,1]^T\nu)$ instead of actual states. On close inspection of (<ref>), it becomes evident that the estimate $\bm{\hat{p}}(t)$ is directly affected by noise in state measurements i.e. there is no filtering in the designed estimator. Furthermore, a high value value of $\bm{\kappa}$ (obtained by selecting a high value of $\lambda$), which is necessary for tracking fast variations in the source and load parameters, directly amplifies the noise component in these measurements thereby corrupting the obtained estimates and limiting the practical application of the developed scheme particularly in noisy environment. In order to address the noise amplification problem, we modify the estimator dynamics following the approach introduced in [12] such that the estimation error injection term is filtered via a set of first-order filters. The resulting estimator dynamics are obtained in the following expression:

\begin{equation}
\begin{split}
\bm{\dot{\hat{\zeta}}_p}(t)&=\bm{A_p\hat{\zeta}_p}(t)+\bm{\kappa} \bm{z_r}(t) \\
\bm{\dot{\eta}}(t)&=\bm{\theta}\big[-(\bm{A_{\sigma}}+\bm{\theta})\bm{x}(t)-\bm{GC_p\zeta_p}(t)-\bm{\eta}(t)\big]\\
\bm{\dot{z}_i}(t)&=\bm{\theta}\big[\bm{z_{i-1}}(t)-\bm{z_i}(t)\big], \ i=\{2,\dots,r\}\\
\bm{z_1}(t)&=\bm{\eta}(t)+\bm{\theta x}(t)\\
\bm{\hat{p}}(t)&=\bm{C_p\hat{\zeta}_p}(t)\\
\end{split}
\label{eq:estimator_noise}
\end{equation}

where $\bm{\theta}=\textrm{diag}(\lambda_{\theta},\lambda_{\theta})$ is the filter gain vector parameterized in terms of $\lambda_{\theta}$. In order to ensure that the accuracy of the designed estimator remains unaffected, the filter parameter is selected such that $\lambda_{\theta}=\gamma\lambda$ where $\gamma>1$ is a tuning parameter for the filters. It is to be noted that $r$ is a design choice which decides the degree of filtering in the obtained estimates by introducing a relative degree of $r$ between the estimate $\bm{\hat{p}}(t)$ and the noise $(\nu)$ entering via state measurements.

§.§ Hysteresis-based switching

One important drawback of the switching law (<ref>) is that it may lead to a sliding mode [13], which in practical terms means that the switching frequency rises to unacceptably high levels. Note that, according to (<ref>), switching only occurs when both modes minimize the objective function in (<ref>). This can only occur on the surface $s(x) = 0$, where $s(x):=(x - x^\star)^T P D x$ and $D:= A_1 - A_0$.

Here, an hysteresis-based strategy is employed to bound the switching frequency at a finite value. In this manner, switching is not allowed while $|s(x)| < h$, with $h > 0$ a parameter to be determined.

Assuming that the switching frequency is high compared to the system bandwidth (which often holds in practice since the same assumption is made when pulse width modulation (PWM) is used), it can be shown [8] that the steady-state switching frequency $f_s(p)$ is determined as:

\begin{equation}\label{eq:switchingFrequency}
f_s(p) = \frac{1}{2h(p)}\frac{|b_0^T P Dx^\star {x^\star}^TD^T P b_1|}{\left(|b_0^T P Dx^\star| + |b_1^T P Dx^\star|\right)},
\end{equation}

where $b_\sigma := A_\sigma x^\star + Gp $, $\sigma = 0,1$. Since the steady-state switching frequency also depends on the parameters $ p $, then their estimation can also be used to update the hysteresis width $ h(p) $ in order to obtain the user-specified switching frequency $ f_s(p) $ using (<ref>).

§ NUMERICAL SIMULATION

Simulation study is carried out in  Simulink environment of MATLAB with the simulation parameters listed in Table <ref>. The switching frequency of the implemented control scheme was limited to $200$kHz via the hysteresis approach defined in Section <ref>. A high-pass filtered white noise is added to the state variables  going to the estimator designed in (<ref>), in order to replicate the effect of measurement noise $(\nu)$. Nominal circuit, controller and estimator parameters used for the simulation study are also presented in Table <ref>. Estimator parameter $\lambda$ is tuned to be faster than the convergence rate $(\alpha)$ for the switching controller while the filter parameter is selected as $\gamma=2.5$ to ensure appropriate filtering of the noise component without affecting the estimator performance. Sufficient filtering of the estimates is obtained with a unity relative degree between measurement noise and the estimates by selecting $r=1$ in (<ref>). For the simulation study, three different test scenarios are considered which are discussed next.

Circuit    $L=1mH, C=50\mu F, R_o=10\Omega, i_{Load}=0A,v_{in}=350V,v_{in}^+=400V,v_{in}^-=300V,v_o^*=450V$

Controller & Estimator     $\alpha=40,\bm{A_p}=\textbf{0}$, $\bm{C_p}=\bm{I_2}$, $\lambda=4000,\gamma=2.5,r=1$

Simulation     $T_s=1/100f_s$, $f_s=200$kHz, Noise $(\nu)$= High-pass filtered (100kHz cut-off) white noise (power 1e-10)

Parameters used for the simulation study.

Output voltage

Inductor current

Simulation plots for S1.

Output Voltage

Inductor current

$v_{in}$ and $\hat{p}_1$

$i_{Load}$ and $\hat{p}_2$

Simulation plots for S2.

Output voltage

Inductor current

$v_{in}$ and $\hat{p}_1$

$i_{Load}$ and $\hat{p}_2$

PV irradiance

Battery current

Battery voltage

Simulation plots for S3.

§.§.§ [S1] Without parameter estimator $(\bm{\dot{p}}(t)=\textbf{0})$
To illustrate the limitations of the switching controller in (<ref>) without equilibrium point update, a series of step changes in $\bm{p}(t)$ is applied in the following manner: $v_{in}=350V\xrightarrow{0.1s} 300V\xrightarrow{0.2s} 400V \xrightarrow{0.3s} 350V $ and $i_{Load}=0A\xrightarrow{0.15s} 20A\xrightarrow{0.25s} 10A \xrightarrow{0.3s} 0A$, where $i_{Load}$ is applied via a controlled current source block connected in parallel to the nominal resistance $R_o$. Fig. <ref> shows the obtained plots of the inductor current and output voltage over time. Note that, as soon as the first step disturbance is applied at $t=0.1$s, the system stabilizes at an equilibrium point which is different from the desired one. Moreover, at instant $t=0.15$s, the load disturbance is such that the new voltage equilibrium value is lower than the input (300V). Since this cannot be achieved in a DBC, the system stops switching and the voltage converges to 300V. The same thing happens at $t=0.2$s and $t=0.25$s. Then, in $t=0.3$s, with the step disturbances applied on the parameters $\bm{p}$, the nominal conditions are recovered and the switching controller is able to stabilize the system at the desired reference. This behavior is expected since the switching law ensures global stabilization, as noted in Section <ref>.

§.§.§ [S2] Idea Scenario $(\bm{\dot{p}}(t)=\textbf{0})$ with parameter estimator
Parametric perturbations in S1 are reapplied on the closed-loop system, but now with the estimator in (<ref>) providing the required information to update $\bm{x^*}$ at each time-step. Simulation plots obtained in Fig. <ref> clearly indicate that $v_o(t)$ tracks the desired reference of $v_o^*=450V$ without any steady-state error as the designed estimator is able to accurately reconstruct the uncertain parameters using the state measurements and the mode $(\sigma)$ information which is subsequently used to update the equilibrium point $i_L^*$. Furthermore, the closed-loop system is able to recover from a simultaneous input voltage and load perturbations applied at $t=0.3s$. Also, since the assumption $\bm{\dot{p}}=\textbf{0}$ is satisfied, the estimates $\bm{\hat{p}}$ track the actual parameters asymptotically resulting in zero static error, which is evident from Fig. <ref> and <ref>. It is also worth noting that the estimates $\bm{\hat{p}}$ are not affected much by the measurement noise due to the additional filtering introduced in the designed estimator.

§.§.§ [S3] Practical scenario $(\bm{\dot{p}}(t)\neq\textbf{0})$ with parameter estimator for EV charging station with PV input
In this scenario, a PV array is connected at the input side of the DBC with an input capacitor $(C_{in}=10\mu F)$ connected in parallel across the PV terminals and actual irradiation data (captured on-site at LAPLACE) with contracted time-scale is supplied to the PV array block.  On the load side, a Li-ion battery (50Ah, 300V) is connected via another DC-DC converter in order to ensure that the desired charging condition demanded by the EV user is satisfied. The charging starts at $t=0.1s$ with ramp type current having slope 0.1A/ms, which increases up to 10A till $t=0.2s$ as shown in Fig. <ref> where charging current is considered as negative. Furthermore, the charging mode for the battery switches to constant voltage from constant current at $t=0.3s$, after which the charging current starts decreasing until it becomes zero close to full charge, however, the changes in current and voltage in different charging modes are too subtle to be notices on a short time-scale shown in Fig. <ref>. Note that the voltage  across the battery terminals and charging current is different from the output voltage across the converter terminals and reflected load current $i_{Load}$ as the battery is interfaced via another controlled DC-DC converter which ensures that the appropriate charging conditions are met. The PV irradiance, input voltage,  charging current for the battery and voltage across its terminals are all shown in Fig. <ref>. Simulation results obtained in Fig. <ref> to <ref> illustrate the effectiveness of the designed control algorithm in tracking the desired reference despite the continuous fluctuations in the input and load parameters $(\bm{\dot{p}}(t)\neq\textbf{0})$.

§ CONCLUSION

A robust switching controller was proposed in this work for a DC-DC boost converter operating under input voltage and load perturbations. To address the problem of uncertain equilibrium point in the context of switching controllers, a parameter estimator was constructed under the assumption that the uncertain parameters are generated via a known linear exo-system. A sketch of the stability proof under ideal and practical conditions was provided for the estimator. Furthermore, the noise amplification and infinite switching frequency problems were addressed to facilitate practical implementation on real systems. Simulation results obtained for the ideal and practical test scenarios illustrate the effectiveness of the developed scheme. The designed estimator allows the flexibility to consider time varying parameters characterized by linear dynamics which covers a wider class of perturbations. For future works, it will be interesting to explore the effectiveness of designed estimator for different types of parametric variations and how higher-order time-polynomial approximations for such variations affect the estimation and closed-loop performance in the context of switching controller. Furthermore, closed-loop performance of the designed estimator with other types of switching controller, such as those designed for local stabilization,  can also be investigated.

[1]
D. Kumar, F. Zare, and A. Ghosh, “Dc microgrid technology: System
architectures, ac grid interfaces, grounding schemes, power quality,
communication networks, applications, and standardizations aspects,”
IEEE Access, vol. 5, pp. 12 230–12 256, 2017.

[2]
J. J. Justo, F. Mwasilu, J. Lee, and J.-W. Jung, “AC-microgrids versus
DC-microgrids with distributed energy resources: A review,” Renewable
and sustainable energy reviews, vol. 24, pp. 387–405, 2013.

[3]
S. Kaur, T. Kaur, R. Khanna, and P. Singh, “A state of the art of DC
microgrids for electric vehicle charging,” in 2017 4th Int. Conf.
Signal Proc., Comp. Control (ISPCC).1em plus 0.5em minus 0.4emIEEE, 2017, pp. 381–386.

[4]
S. Dahale, A. Das, N. M. Pindoriya, and S. Rajendran, “An overview of DC-DC
converter topologies and controls in DC microgrid,” in 2017 7th Int.
Conf. Power Systems (ICPS).1em plus 0.5em minus 0.4emIEEE,
2017, pp. 410–415.

[5]
T. Kobaku, S. C. Patwardhan, and V. Agarwal, “Experimental evaluation of
internal model control scheme on a dc–dc boost converter exhibiting
nonminimum phase behavior,” IEEE Trans. Power Electron., vol. 32,
no. 11, pp. 8880–8891, 2017.

[6]
T. Kobaku, R. Jeyasenthil, S. Sahoo, R. Ramchand, and T. Dragicevic,
“Quantitative feedback design-based robust pid control of voltage mode
controlled dc-dc boost converter,” IEEE Trans. Circ. Sys. II: Express
Briefs, vol. 68, no. 1, pp. 286–290, 2020.

[7]
Q. Xu, N. Vafamand, L. Chen, T. Dragičević, L. Xie, and F. Blaabjerg,
“Review on advanced control technologies for bidirectional DC/DC converters
in DC microgrids,” IEEE J. Emer. Selec. Topics Power Electron.,
vol. 9, no. 2, pp. 1205–1221, 2020.

[8]
R. P. De Souza, Z. Kader, and S. Caux, “Switching control applied to
interconnected boost converters: A comparison with hysteresis current
control,” in 2022 International Symp. Power Electronics, Electrical
Drives, Automation and Motion (SPEEDAM).1em plus 0.5em minus
0.4emIEEE, 2022, pp. 541–546.

[9]
M. Bayati, M. Abedi, G. B. Gharehpetian, and M. Farahmandrad,
“Sinusoidal-ripple current control in battery charger of electric
vehicles,” IEEE Trans. Vehic. Techn., vol. 69, no. 7, pp. 7201–7210,

[10]
G. Beneux, P. Riedinger, J. Daafouz, and L. Grimaud, “Adaptive stabilization
of switched affine systems with unknown equilibrium points: Application to
power converters,” Automatica, vol. 99, pp. 82–91, 2019.

[11]
A. Ndoye, R. Delpoux, J.-F. Trégouët, and X. Lin-Shi, “Switching
control design for lti system with uncertain equilibrium: Application to
parallel interconnection of dc/dc converters,” Automatica, vol. 145,
p. 110522, 2022.

[12]
D. Astolfi, L. Zaccarian, and M. Jungers, “On the use of low-pass filters in
high-gain observers,” Systems & Control Letters, vol. 148, p.
104856, 2021.

[13]
P. Bolzern and W. Spinelli, “Quadratic stabilization of a switched affine
system about a nonequilibrium point,” in Proc. 2004 American Control
Conf., Boston, MA, USA, 2004, pp. 3890–3895.

[14]
W.-H. Chen, “Disturbance observer based control for nonlinear systems,”
IEEE/ASME Trans. Mech., vol. 9, no. 4, pp. 706–710, 2004.

[15]
S. Ahmad and A. Ali, “On active disturbance rejection control in presence of
measurement noise,” IEEE Trans. Ind. Electron., vol. 69, no. 11, pp.
11 600–11 610, 2021.


# extensions of $n$-ary prime hyperideals via an $n$-ary multiplicative subset
in a Krasner $(m,n)$-hyperring

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

Let $R$ be a Krasner $(m,n)$-hyperring and $S$ be an n-ary multiplicative
subset of $R$. The purpose of this paper is to introduce the notion of n-ary
$S$-prime hyperideals as a new expansion of n-ary prime hyperideals. A
hyperideal $I$ of $R$ disjoint with $S$ is said to be an n-ary $S$-prime
hyperideal if there exists $s\in S$ such that whenever $g(x_{1}^{n})\in I$ for
all $x_{1}^{n}\in R$, then $g(s,x_{i},1^{(n-2)})\in I$ for some $1\leq i\leq
n$. Several properties and characterizations concerning n-ary $S$-prime
hyperideals are presented. The stability of this new concept with respect to
various hyperring-theoretic constructions are studied. Furthermore, we extend
this concept to n-ary $S$-primary hyperideals. We obtained some specific
results explaining the structure.

###### 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

Prime and primary ideals which are quite important in commutative rings have
been studied by many authors. In 2019, Hamed and Malek [16] introduced the
notion of $S$-prime ideal which is a generalization of prime ideals. Suppose
that $R$ is a commutative ring with identity and $S\subseteq R$ a
multiplicative subset. A proper ideal $P$ of $R$ disjoint from $S$ is called
an $S$-prime of $R$ if there exists an $s\in S$ such that for all $x,y\in R$
if $xy\in P$, then $sx\in P$ or $sy\in P$. In [24] , Massaoud defined and
investigated the concept of $S$-primary ideals of a commutative ring in a way
that generalizes essentially all the results concerning primary ideals. A
proper ideal $Q$ of $R$ disjoint from $S$ is called an $S$-primary of $R$ if
there exists an $s\in S$ such that for all $x,y\in R$ if $xy\in P$, then
$sx\in P$ or $sy\in\sqrt{Q}$. Furthermore, some results on S-primary ideals of
a commutative ring were studied by Visweswaran in [31].

Hyperstructures represent a natural extension of classical algebraic
structures and they were defined by the French mathematician F. Marty. In
1934, Marty [23] defined the concept of a hypergroup as a generalization of
groups during the $8^{th}$ Congress of the Scandinavian Mathematicians. A
comprehensive review of the theory of hyperstructures can be found in [9, 10,
11, 28, 32, 34]. 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 [18] at a
scientific meeting in 1904. In 1928, Dorente wrote the first paper concerning
the theory of $n$-ary groups [14]. Later on, Crombez and Timm [7, 8] defined
and described 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 [12], they introduced and illustrated 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 [19, 20, 21, 22, 29]. Mirvakili and
Davvaz [26] defined $(m,n)$-hyperrings and obtained several results in this
respect.

It was Krasner, who introduced one important class of hyperrings, where the
addition is a hyperoperation, while the multiplication is an ordinary binary
operation, which is called Krasner hyperring. In [25], 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. [17]. Norouzi et. al. proposed and analysed a new
defnition for normal hyperideals in Krasner $(m,n)$-hyperrings, with respect
to that one given in [25] and they showed that these hyperideals correspond to
strongly regular relations [27]. Asadi and Ameri introduced and studied direct
limit of a direct system in the category of Krasner $(m,n)$-hyperrigs [6].
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 [13]. Moreover, in [15] 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 [30]. 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].
In this paper, we aim to complete this circle of ideas. Motivated by the
research works on $S$-prime ideals and $S$-primary ideals of commutative
rings, we define and investigate the notions of n-ary $S$-prime and n-ary
$S$-primary hyperideals in a commutative Krasner $(m,n)$-hyperring.

##  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.

([25]). 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.

[19] 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.

[25] 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] 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 2.6.

[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 2.7.

(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 2.8.

([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 2.9.

[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 2.10.

[1] A non-empty subset $S$ of a Krasner $(m,n)$-hyperring $R$ is called an
n-ary multiplicative, if $g(s_{1}^{n})\in S$ for $s_{1},...,s_{n}\in S$.

######  Definition 2.11.

[25] 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

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

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

##  3\. $n$-ary S-prime hyperideals

We start this section by introducing the concept of n-ary $S$-prime
hyperideals of Krasner $(m,n)$-hyperring $R$ where $S$ is an n-ary
multiplicative subset of $R$. The following definition constitutes the
$S$-version of n-ary prime hyperideals.

######  Definition 3.1.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I$ be a hyperideal of $R$ with $I\cap S=\varnothing$. $I$ refers to an
n-ary $S$-prime hyperideal if there exists an $s\in S$ such that for all
$x_{1}^{n}\in R$ with $g(x_{1}^{n})\in I$, we get $g(s,x_{i},1^{(n-2)})\in I$
for some $1\leq i\leq n$.

######  Example 3.2.

The set $R=\\{0,1,2\\}$ 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.

$f(0,0,0)=0,\ \ \ f(0,0,1)=1,\ \ \ f(0,1,1)=1,\ \ \ f(1,1,1)=1,\ \ \
f(1,1,2)=R,$ $f(0,1,2)=R,\ \ \ f(0,0,2)=2,\ \ \ f(0,2,2)=2,\ \ \ f(1,2,2)=R,\
\ \ f(2,2,2)=2,$

$\ \ \ g(1,1,1)=1,\ \ \ \ g(1,1,2)=g(1,2,2)=g(2,2,2)=2,$
and for $x_{1},x_{2}\in R,\ g(0,x_{1},x_{2})=0$.

Consider 3-ary multiplicative subset $S=\\{1,2\\}$ of Krasner
$(3,3)$-hyperring $(R,f,g)$. Then hyperideal $P=\\{0,2\\}$ is a 3-ary
$S$-prime hyperideal of $R$.

The following example shows that an n-ary $S$-prime hyperideal may not be an
n-ary prime hyperideal of $R$.

######  Example 3.3.

The set $R=\\{0,1,2,3\\}$ with following 2-hyperoperation $"\oplus"$ is a
canonical 2-ary hypergroup.

$\oplus$ |  $0$ $1$ $2$ $3$
---|---
0 |  $0$ $1$ $2$ $3$
$1$ |  $1$ $A$ $3$ $B$
$2$ |  $2$ $3$ $0$ $1$
$3$ |  $3$ $B$ $1$ $A$

In which $A=\\{0,1\\}$ and $B=\\{2,3\\}$. Define a 4-ary operation $g$ on $R$
as follows: If $a_{1},a_{2},a_{3},a_{4}\in B$ then $g(a_{1}^{4})=2$ and
otherwise $g(a_{1}^{4})=0$. It follows that $(R,\oplus,g)$ is a Krasner
(2,4)-hyperring. $S=\\{2,3\\}$ is a 4-ary multiplicative subset of $R$. In the
hyperring, $I=\\{0\\}$ is a 4-ary $S$-prime hyperideal of $R$ but it is not
prime, because $g(1,2,2,3)=0\in I$ while $1,2,3\notin I$.

Our first theorem gives a characterization of n-ary $S$-prime hyperideals.

######  Theorem 3.4.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I$ be a hyperideal of $R$ with $I\cap S=\varnothing$. Then $I$ is n-ary
$S$-prime if and only if $(I:s)=\\{r\in R\ |\ g(r,s,1^{(n-2)})\in I\\}$ is an
n-ary prime hyperideal of $R$ for some $s\in S$.

###### Proof.

$\Longrightarrow$ Let $I$ be is an n-ary $S$-prime hyperideal of $R$. Then
there exists $s\in S$ such that for all $x_{1}^{n}\in R$ with $g(x_{1}^{n})\in
I$, we get $g(s,x_{i},1^{(n-2)})\in I$ for some $1\leq i\leq n$. Suppose that
$g(y_{1}^{n})\in(I:s)$ for $y_{1}^{n}\in R$. Then
$g(s,g(y_{1}^{n}),1^{(n-2)})=g(g(s,y_{1},1^{(n-2)}),y_{2}^{n})\in I$ which
means $g(s,g(s,y_{1},1^{(n-2)}),1^{(n-2)})=g(s^{2},y_{1},1^{(n-3)})\in I$ or
$g(s,y_{i},1^{(n-2)})\in I$ for some $2\leq i\leq n$. Since $I\cap
S=\varnothing$, then we conclude that $g(s,y_{1},1^{(n-2)})\in I$ or
$g(s,y_{i},1^{(n-2)})\in I$ for some $2\leq i\leq n$. So,
$g(s,y_{i},1^{(n-2)})\in I$ for some $1\leq i\leq n$ which implies
$y_{i}\in(I:s)$ for some $1\leq i\leq n$. Consequently, $(I:s)$ is an n-ary
prime hyperideal of $R$.
$\Longleftarrow$ Let $(I:s)$ be an n-ary prime hyperideal of $R$ for some
$s\in S$. Suppose that $g(x_{1}^{n})\in I$ for $x_{1}^{n}\in R$. Since
$I\subseteq(I:s)$, then $g(x_{1}^{n})\in(I:s)$. Since $(I:s)$ is an n-ary
prime hyperideal of $R$, then $x_{i}\in(I:s)$ for some $1\leq i\leq n$. This
implies that $g(s,x_{i},1^{(n-2)})\in I)$ for some $1\leq i\leq n$ which means
$I$ is an n-ary $S$-prime hyperideal of $R$. ∎

######  Theorem 3.5.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $R\subseteq G$ be an extension of $R$. If $I$ is an n-ary $S$-prime
hyperideal of $G$, then $I\cap R$ is an n-ary $S$-prime hyperideal of $R$.

###### Proof.

Let $I$ be an n-ary $S$-prime hyperideal of $G$. Then there exist $s\in S$
such that for all $x_{1}^{n}\in R$ with $g(x_{1}^{n})\in I$, we get
$g(s,x_{i},1^{(n-2)})\in I$ for some $1\leq i\leq n$. Let $g(x_{1}^{n})\in
I\cap R$ for $x_{1}^{n}\in R$. Since $g(x_{1}^{n})\in I$, then
$g(s,x_{i},1^{(n-2)})\in I$ for some $1\leq i\leq n$ which means
$g(s,x_{i},1^{(n-2)})\in I\cap R$. Thus, $I\cap R$ is an n-ary $S$-prime
hyperideal of $R$. ∎

######  Theorem 3.6.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I$ be a hyperideal of $R$. If $I\subseteq\cup_{i=1}^{n}I_{i}$ for some
n-ary $S$-prime hyperideals $I_{1}^{n}$ of $R$, then there exists $s\in S$
such that $g(s,I,1^{(n-2)})\subseteq P_{i}$ for some $1\leq i\leq n$.

###### Proof.

Let $I\subseteq\cup_{i=1}^{n}I_{i}$ for some n-ary $S$-prime hyperideals
$I_{1}^{n}$ of $R$. For each $1\leq i\leq n$, we get $s_{i}\in S$ such that
$(I_{i}:s_{i})$ is an n-ary prime hyperideal of $R$, by Theorem 3.4. Since
$I\subseteq\cup_{i=1}^{n}I_{i}\subseteq\cup_{i=1}^{n}(I_{i}:s_{i})$, we have
$I\subseteq(I_{i}:s_{i})$ for some $1\leq i\leq n$, by Theorem 5.1 in [4].
Thus $g(s_{i},I,1^{(n-2)})\subseteq I_{i}$. ∎

######  Theorem 3.7.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I$ be a hyperideal of $R$ with $I\cap S=\varnothing$. Then $I$ is n-ary
$S$-prime if and only if there exists $s\in S$, for all hyperideals
$I_{1}^{n}$ of $R$, if $g(I_{1}^{n})\subseteq I$, then
$g(s,I_{i},1^{(n-1)})\subseteq I$ for some $1\leq i\leq n$.

###### Proof.

$\Longrightarrow$ Let $I$ is an n-ary $S$-prime hyperideal of $R$. Then there
exists $s\in S$ such that for all $x_{1}^{n}\in R$ with $g(x_{1}^{n})\in I$ we
have $g(s,x_{i},1^{(n-2)})\in I$ for some $1\leq i\leq n$. Let for some
hyperideals $I_{1}^{n}$ of $R$ with $g(I_{1}^{n})\in I$ we have
$g(s,I_{i},1^{(n-2)})\nsubseteq I$ for all $1\leq i\leq n$. This means
$g(s,a_{i},1^{(n-2)})\notin I$ for some $a_{i}\in I_{i}$ and $1\leq i\leq n$
which is a contradiction, since $I$ is an n-ary $S$-prime hyperideal of $R$
and $g(a_{1}^{n})\in g(I_{1}^{n})\subseteq I$.
$\Longleftarrow$ Let $g(x_{1}^{n})\in I$ for $x_{1}^{n}\in R$. Then $g(\langle
x_{1}\rangle,\cdots,\langle x_{n}\rangle)\subseteq I$. Hence we have
$g(s,\langle x_{i}\rangle,1^{(n-2)})\subseteq I$ for some $1\leq i\leq n$
which implies $g(s,x_{i},1^{(n-2)})\in I$. Thus, $I$ is an n-ary $S$-prime
hyperideal of $R$. ∎

In view of Theorem 3.7, we have the following result.

######  Corollary 3.8.

Let $I$ be a proper hyperideal of a Krasner $(m,n)$-hyperring $R$. Then $I$ is
an n-ary prime hyperideal if and only if for all hyperideals $I_{1}^{n}$ of
$R$, If $g(I_{1}^{n})\subseteq I$, then $I_{i}\subseteq I$ for some $1\leq
i\leq n$.

###### Proof.

Consider $S=\\{1\\}$. Then we are done, by Theorem 3.7. ∎

######  Theorem 3.9.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I$ be an n-ary $S$-prime hyperideal of $R$. If $J$ be a hyperideal of $R$
with $J\subseteq I$, then $g(s,\sqrt{J}^{(m,n)},1^{(n-2)})\subseteq I$ for
some $s\in S$.

###### Proof.

Let $a\in\sqrt{J}^{(m,n)}$. Then there exists $t\in\mathbb{N}$ such that
$g(a^{(t)},1^{(n-t)})\in J$ for $t\leq n$ or $g_{(l)}(a^{(t)})\in I$ for
$t=l(n-1)+1$. If $g(a^{(t)},1^{(n-t)})\in J\subseteq I$, then $g(\langle
a\rangle^{(t)},1^{(n-t)})\subseteq I$. By Theorem 3.7, we get $g(s,\langle
a\rangle,1^{(n-2)})\subseteq I$ for some $s\in S$ which implies
$g(s,a,1^{(n-2)})\in I$. Consequently,
$g(s,\sqrt{J}^{(m,n)},1^{(n-2)})\subseteq I$. If $t=l(n-1)+1$, then by using a
similar argument, one can easily complete the proof. ∎

######  Theorem 3.10.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I_{1}^{n}$ be n-ary $S$-prime hyperideals of $R$. Then there exists $s\in
S$ such that
$g(s,\sqrt{\cap_{i=1}^{n}I_{i}},1^{(n-2)})\subseteq\cap_{i=1}^{n}I_{i}$.

###### Proof.

Let $I_{1}^{n}$ be n-ary $S$-prime hyperideals of $R$. Then for each $1\leq
i\leq n$ we have $s_{i}\in S$ such that for all $x_{1}^{n}$ of $R$, if
$g(x_{1}^{n})\in I_{i}$, then $g(s_{i},x_{j},1^{(n-2)})\in I_{i}$ for some
$1\leq j\leq n$. By Theorem 3.9, we get
$g(s_{i},\sqrt{I_{i}}^{(m,n)},1^{(n-2)})\subseteq I_{i}$ for each $1\leq i\leq
n$. Put $s=g(s_{1}^{n})$. Hence we obtain
$g(s,\sqrt{\cap_{i=1}^{n}I_{i}},1^{(n-2)})=g(s,\cap_{i=1}^{n}\sqrt{I_{i}}^{(m,n)},1^{(n-2)})\subseteq\cap_{i=1}^{n}I_{i}$.
∎

######  Theorem 3.11.

Let $(R_{1},f_{1},g_{1}),(R_{2},f_{2},g_{2})$ be Krasner $(m,n)$-hyperrings
and $h:R_{1}\longrightarrow R_{2}$ be a homomorphism such that $0\notin h(S)$.
If $I_{2}$ is an n-ary $h(S)$-prime hyperideal of $R_{2}$, then
$h^{-1}(I_{2})$ is an n-ary $S$-prime hyperideal of $R_{1}$.

###### Proof.

Suppose that $I_{2}$ is an n-ary $h(S)$-prime hyperideal of $R_{2}$. Then
there exists $s\in S$ such that for all $y_{1}^{n}\in R_{2}$ with
$g_{2}(y_{1}^{n})\in I_{2}$ we have $g_{2}(h(s),y_{i},1^{(n-2)})\in I_{2}$ for
some $1\leq i\leq n$. Put $I_{1}=h^{-1}(I_{2})$. It is easy to see that
$I_{1}\cap S=\varnothing$. Let $g_{1}(x_{1}^{n})\in I_{1}$ for $x_{1}^{n}\in
R_{1}$. Then $h(g_{1}(x_{1}^{n}))=g_{2}(h(x_{1}),...,h(x_{n}))\in I_{2}$. So,
we have $g_{2}(h(s),h(x_{i}),1^{(n-2)})=h(g_{1}(s,x_{i},1^{(n-2)})\in I_{2}$
for some $1\leq i\leq n$ which implies $g_{1}(s,x_{i},1^{(n-2)})\in
h^{-1}(I_{2})=I_{1}$. Consequently, $h^{-1}(I_{2})$ is an n-ary $S$-prime
hyperideal of $R_{1}$. ∎

The concept of Krasner $(m,n)$-hyperring of fractions was introduced in [5].

######  Theorem 3.12.

Let $R$ be a Krasner $(m,n)$-hyperring and $S$ be an $n$-ary multiplicative
subset of $R$ with $1\ \in S$. If $I$ is an $n$-ary $S$-prime hyperideal of
$R$ with $I\cap S=\varnothing$, then $S^{-1}I$ is an $n$-ary prime hyperideal
of $S^{-1}R$.

###### Proof.

Let $G(\frac{a_{1}}{s_{1}},...,\frac{a_{n}}{s_{n}})\in S^{-1}I$ for
$\frac{a_{1}}{s_{1}},...,\frac{a_{n}}{s_{n}}\in S^{-1}R$. Then we have
$\frac{g(a_{1}^{n})}{g(s_{1}^{n})}\in S^{-1}I$. It implies that there exists
$t\in S$ such that $g(t,g(a_{1}^{n}),1^{(n-2)})\in I$. Since $I$ is an $n$-ary
$S$-prime hyperideal of $R$ and $I\cap S=\varnothing$, then there exists $s\in
S$ such that $g(s,a_{i},1^{(n-2)})\in I$ or
$g(s,g(t,a_{i},1^{(n-2)}),1^{(n-2)})=g(s,t,a_{i},1^{(n-3)})\in I$ for some
$1\leq i\leq n$. Hence we conclude that
$G(\frac{a_{i}}{s_{i}},\frac{1}{1}^{(n-1)})=\frac{g(a_{i},1^{(n-1)})}{g(s_{i},1^{(n-1)}))}=\frac{g(s,a_{i},1^{(n-2)})}{g(s,s_{i},1^{(n-2)})}\in
S^{-1}I$ or
$G(\frac{a_{i}}{s_{i}},\frac{1}{1}^{(n-1)})=\frac{g(a_{i},1^{(n-1)})}{g(s_{i},1^{(n-1)})}=\frac{g(s,t,a_{i},1^{(n-3)})}{g(s,t,s_{i},1^{(n-2)})}\in
S^{-1}I$ for some $1\leq i\leq n$ . Thus $S^{-1}I$ is an $n$-ary prime
hyperideal of $S^{-1}R$. ∎

######  Theorem 3.13.

Let $R$ be a Krasner $(m,n)$-hyperring, $S$ be an $n$-ary multiplicative
subset of $R$ with $1\in S$ and $I$ be a hyperideal of $R$ with $I\cap
S=\varnothing$ . If $S^{-1}I$ is an $n$-ary prime hyperideal of $S^{-1}R$ and
$S^{-1}I\cap R=(I:s)$ for some $s\in S$, then $I$ is an $n$-ary $S$-prime
hyperideal of $R$.

###### Proof.

Let $S^{-1}I$ be an $n$-ary prime hyperideal of $S^{-1}R$ and $S^{-1}I\cap
R=(I:s)$ for some $s\in S$. Assume that $g(a_{1}^{n})\in I$ for some
$a_{1}^{n}\in R$. Then we get $G(\frac{a_{1}}{1},\cdots,\frac{a_{n}}{1})\in
S^{-1}I$. Since $S^{-1}I$ is an $n$-ary prime hyperideal of $S^{-1}R$, we
obtain $\frac{a_{i}}{1}\in S^{-1}I$ for some $1\leq i\leq n$ which implies
$g(t,a_{i},1^{(n-2)})\in I$ for some $t\in S$. Hence
$a_{i}=\frac{g(t,a_{i},1^{(n-2)})}{g(t,1^{(n-1)})}\in S^{-1}I$. This means
$a_{i}\in(I:s)$ for somr $s\in S$. Therefore we have $g(s,a_{i},1^{(n-2)})\in
I$. Thus we conclude that $I$ is an $n$-ary $S$-prime hyperideal of $R$. ∎

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

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

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

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

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

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

######  Theorem 3.14.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and let $I$ and $J$ be a hyperideals of $R$ such that $J\subseteq I$. Let
$J\cap S=\varnothing$ and $I/J\cap\bar{S}=\varnothing$ with
$\bar{S}=\\{f(s_{1}^{i-1},J,s_{i+1}^{n})\ |\ s_{1}^{i-1},s_{i+1}^{n}\in S\\}$.
If $I$ is an n-ary $S$-prime hyperideal of $R$, then $I/J$ is an n-ary
$\bar{S}$-prime hyperideal of $R/J$.

###### Proof.

Let $I$ be an n-ary $S$-prime hyperideal of $R$. Then there exists some $s\in
S$ such that if $g(x_{1}^{n})\in I$ for $x_{1}^{n}\in R$, then we get
$g(s,x_{i},1^{(n-2)})\in I$ for some $1\leq i\leq n$. Let

$g(f(x_{11}^{1(i-1)},J,x_{1(i+1)}^{1m}),...,f(x_{n1}^{n(i-1)},J,x_{n(i+1)}^{nm}))\in
I/J$
for some
$f(x_{11}^{1(i-1)},J,x_{1(i+1)}^{1m}),...,f(x_{n1}^{n(i-1)},J,x_{n(i+1)}^{nm})\in
R/J$.
This means

$f(g(x_{11}^{n1}),...,g(x_{1(i-1)}^{n(i-1)}),J,g(x_{1(i+1)}^{n(i+1)}),...,g(x_{1m}^{nm}))\in
I/J$.
Then
$f(g(x_{11}^{n1}),...,g(x_{1(i-1)}^{n(i-1)}),0,g(x_{1(i+1)}^{n(i+1)}),...,g(x_{1m}^{nm}))\subseteq
I$ which implies

$g(f(x_{11}^{1(i-1)},0,x_{1(i+1)}^{1m}),...,f(x_{n1}^{n(i-1)},0,x_{n(i+1)}^{nm}))\subseteq
I$.
Since $I$ is an n-ary $S$-prime hyperideal of $R$, then, for some $1\leq j\leq
n$, we obtain

$g(s,f(x_{j1}^{j(i-1)},0,x_{j(i+1)}^{jm}),1^{n-2})\subseteq I$
Therefore

$f(g(s,f(x_{j1}^{j(i-1)},0,x_{j(i+1)}^{jm}),1^{n-2}),J,0^{(m-2)})\in I/J$
and so

$f(g(g(s,1^{(n-2)}),f(x_{j1}^{j(i-1)},0,x_{j(i+1)}^{jm}),1^{n-2}),J,0^{(m-2)})\in
I/J$.
Thus we conclude that

$g(f(s,J,1^{(n-2)}),f(x_{j1}^{j(i-1)},J,x_{j(i+1)}^{jm}),1_{R/J}^{(n-2)})\in
I/J$
Consequently, $I/J$ is an n-ary $\bar{S}$-prime hyperideal of $R/J$.

∎

Suppose that $I$ is a normal hyperideal of Krasner $(m,n)$-hyperring
$(R,f,g)$. Then the set of all equivalence classes $[R:I^{*}]=\\{I^{*}[x]\ |\
x\in R\\}$ is a Krasner $(m,n)$-hyperring with the m-ary hyperoperation $f/I$
and the n-ary operation $g/I$, defined as follows:

$f/I(I^{*}[x_{1}],\cdots,I^{*}[x_{m}])=\\{I^{*}[z]\ |\ z\in
f(I^{*}[x_{1}],\cdots,I^{*}[x_{m}])\\},\ \ \forall x_{1}^{m}\in R$

$g/I(I^{*}[x_{1}],\cdots,I^{*}[x_{n}])=I^{*}[g(x_{1}^{n})],\ \ \forall
x_{1}^{n}\in R$
(for more details refer to [25]). Now, we establish the following result.

######  Theorem 3.15.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring
$(R,f,g)$ and let $I$ be a normal hyperideal of $R$. If $J$ is an n-ary
$S$-prime hyperideal of $R$ such that $I\subseteq J$, then $[J:I^{*}]$ is an
n-ary $[S:I^{*}]$-prime hyperideal of $[R:I^{*}]$

###### Proof.

First of all, notice that $S\cap J=\varnothing$ if and only if
$[S:I^{*}]\cap[J:I^{*}]=\varnothing$. Let
$g/I(I^{*}[x_{1}],\cdots,I^{*}[x_{n}])\in[J:I^{*}]$ for some $x_{1}^{n}\in R$.
Then $I^{*}[g(x_{1}^{n})]\in[J:I^{*}]$. This means
$I^{*}[g(x_{1}^{n})]\subseteq J$. So

$I^{*}[g(x_{1}^{n})]=f(I,g(x_{1}^{n}),0^{(m-2)})=f(I,g(x_{1}^{n}),g(0^{(n)})^{(m-2)})$

$\hskip
42.67912pt=g(f(I,x_{1},0^{(m-2)}),\cdots,f(I,x_{n},0^{(m-2)}))\subseteq J$
Since $J$ is an n-ary $S$-prime hyperideal of $R$, then there exists $s\in S$
such that $g(s,f(I,x_{i},0^{(m-2)}),1^{(n-2)})\subseteq J$ for some $1\leq
i\leq n$ which implies

$f(I,g(s,x_{i},1^{(n-2)}),0^{(m-2)})\subseteq J$.
Hence $I^{*}[g(s,x_{i},1^{(n-2)})]\in[J:I^{*}]$ which means
$g/I(I^{*}[s],I^{*}[x_{i}],I^{*}[1]^{(n-2)})\in[J:I^{*}]$. Thus $[J:I^{*}]$ is
an n-ary $[S:I^{*}]$-prime hyperideal of $[R:I^{*}]$. ∎

Let $(R_{1},f_{1},g_{1})$ and $(R_{2},f_{2},g_{2})$ be two Krasner
$(m,n)$-hyperrings such that $1_{R_{1}}$ and $1_{R_{2}}$ be scalar identitis
of $R_{1}$ and $R_{2}$, respectively. Then the $(m,n)$-hyperring $(R_{1}\times
R_{2},f_{1}\times f_{2},g_{1}\times g_{2})$ is defined by m-ary hyperoperation
$f=f_{1}\times f_{2}$ and n-ary operation $g=g_{1}\times g_{2}$, as follows:

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

$g_{1}\times
g_{2}((x_{1},y_{1}),\cdots,(x_{n},y_{n}))=(g_{1}(x_{1}^{n}),g_{2}(y_{1}^{n}))$,
for all $a_{1}^{m},x_{1}^{n}\in R_{1}$ and $b_{1}^{m},y_{1}^{n}\in R_{2}$ [2].
Suppose that $S=S_{1}\times S_{2}$ such that $S_{1}$ and $S_{2}$ are n-ary
multiplicative subsets of $R_{1}$ and $R_{2}$, respectively. Assume that
$I_{1}$ is an n-ary $S_{1}$-prime hyperideal of $R_{1}$. It is easy to verify
that $(I_{1}\times R_{2})\cap S=\varnothing\Longleftrightarrow I_{1}\times
S_{1}=\varnothing$. In the next theorem, we characterize n-ary $S$-prime
hyperideals of $R_{1}\times R_{2}$.

######  Theorem 3.16.

Let $(R_{1},f_{1},g_{1})$ and $(R_{2},f_{2},g_{2})$ be two Krasner
$(m,n)$-hyperrings such that $1_{R_{1}}$ and $1_{R_{2}}$ be scalar identitis
of $R_{1}$ and $R_{2}$, respectively. Suppose that $S=S_{1}\times S_{2}$ such
that $S_{1}$ and $S_{2}$ are n-ary multiplicative subsets of $R_{1}$ and
$R_{2}$, respectively. Then $I_{1}$ is an n-ary $S_{1}$-prime hyperideal of
$R_{1}$ if and only if $I_{1}\times R_{2}$ is an n-ary $S$-prime hyperideal of
$R_{1}\times R_{2}$.

###### Proof.

$\Longrightarrow$ Assume that $I_{1}$ is an n-ary $S_{1}$-prime hyperideal of
$R_{1}$. Let $g_{1}\times g_{2}((x_{1},y_{1}),\cdots,(x_{n},y_{n}))\in
I_{1}\times R_{2}$ for some $x_{1}^{n}\in R_{1}$ and $y_{1}^{n}\in R_{2}$.
Then we get $g_{1}(x_{1}^{n})\in I_{1}$. By the hypothesis, there exists
$s_{1}\in S_{1}$ such that $g_{1}(s_{1},x_{i},1^{(n-2)})\in I_{1}$ for some
$1\leq i\leq n$. Then we have $g_{1}\times
g_{2}((s_{1},1_{R_{2}}),(x_{i},y_{i}),(1_{R_{1}},1_{R_{2}})^{(n-2)})\in
I_{1}\times R_{2}$. Thus we conclude that $I_{1}\times R_{2}$ is an n-ary
$S$-prime hyperideal of $R_{1}\times R_{2}$.
$\Longleftrightarrow$ Let $I_{1}\times R_{2}$ be an n-ary $S$-prime hyperideal
of $R_{1}\times R_{2}$. Assume that $g_{1}(x_{1}^{n})\in I_{1}$ for some
$x_{1}^{n}\in R_{1}$. Then $g_{1}\times
g_{2}((x_{1},1_{R_{2}}),\cdots(x_{n},1_{R_{2}}))\in I_{1}\times R_{2}$. Since
$I_{1}\times R_{2}$ is an n-ary $S$-prime hyperideal of $R_{1}\times R_{2}$,
then there exists an element $(s_{1},s_{2})$ in $S$ such that $g_{1}\times
g_{2}((s_{1},s_{2}),(x_{i},1_{R_{2}}),(1_{R_{1}},1_{R_{2}})^{(n-2)})\in
I_{1}\times R_{2}$ for some $1\leq i\leq n$. This means
$g_{1}(s_{1},x_{i},1_{R_{1}}^{(n-2)})\in I_{1}$. Consequently, $I_{1}$ is an
n-ary $S_{1}$-prime hyperideal of $R_{1}$. ∎

We give the following results obtained by the previous theorem.

######  Corollary 3.17.

Let $(R_{i},f_{i},g_{i})$ be a Krasner $(m,n)$-hyperring for each $1\leq i\leq
t$ such that $1_{R_{i}}$ is scalar identity of $R_{i}$. Assume that
$S=S_{1}\times\cdots\times S_{t}$ such that $S_{i}$ is an n-ary multiplicative
subset of $R_{i}$ for each $1\leq i\leq t$. If $I_{i}$ is an n-ary
$S_{i}$-prime hyperideal of $R_{i}$ for some $1\leq i\leq t$, then
$R_{1}\times\cdots\times R_{i-1}\times I_{i}\times R_{i+1}\times\cdots\times
R_{t}$ is an n-ary $S$-prime hyperideal of $R_{1}\times\cdots\times R_{t}$.

##  4\. $n$-ary S-primary hyperideals

The aim of this section is to define the notion of n-ary $S$-primary
hyperideals in a Krasner $(m,n)$-hyperring. The overall framework of the
structure is then explained.

######  Definition 4.1.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I$ be a hyperideal of $R$ with $I\cap S=\varnothing$. $I$ refers to an
n-ary $S$-primary hyperideal if there exists an $s\in S$ such that for all
$x_{1}^{n}\in R$ with $g(x_{1}^{n})\in I$, we have $g(s,x_{i},1^{(n-2)})\in I$
or $g(x_{1}^{i-1},s,x_{i+1}^{n})\in\sqrt{I}^{(m,n)}$.

######  Example 4.2.

In Example 3.2, the hyperideal $P=\\{0\\}$ is a 3-ary $S$-primary hyperideal
of $R$.

The following is a direct consequence and can be proved easily and so the
proof is omited.

######  Theorem 4.3.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
such that $S\subseteq U(R)$ and $I$ be a hyperideal of $R$. Then $I$ is an
n-ary $S$-primary hyperideal of $R$ if and only if $I$ is an n-ary primary
hyperideal of $R$.

######  Theorem 4.4.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I$ be a hyperideal of $R$ with $I\cap S=\varnothing$. Then $I$ is an
n-ary $S$-primary hyperideal of $R$ if and only if $(I:s)=\\{x\in R\ |\
g(x,s,1^{(n-2)})\in I\\}$ is an n-ary primary hyperideal of $R$ for some $s\in
S$.

###### Proof.

$\Longrightarrow$ Let $I$ be an n-ary $S$-primary hyperideal of $R$. Then
there exists $s\in S$ such that if $g(a_{1}^{n})\in I$, then
$g(s,a_{i},1^{(n-2)})\in I$ or
$g(a_{1}^{i-1},s,a_{i+1}^{n})\in\sqrt{I}^{(m,n)}$. Now we show $(I:s)=\\{x\in
R\ |\ g(s,x,1^{(n-2)})\in I\\}$ is an n-ary primary hyperideal of $R$. Suppose
that $g(x_{1}^{n})\in(I:s)$ for some $x_{1}^{n}\in R$. This means
$g(g(x_{1}^{n}),s,1^{(n-2)})=g(x_{1}^{i-1},g(s,x_{i},1^{(n-2)}),x_{i+1}^{n})\in
I$. Then we have
$g(s,g(s,x_{i},1^{(n-2)}),1^{(n-2)})=g(s^{(2)},x_{i},1^{(n-3)})\in I$ or
$g(x_{1}^{i-1},s,x_{i+1}^{n})\in\sqrt{I}^{(m,n)}$. In the first case, we have
$g(s,x_{i},1^{(n-2)})\in I$ or $g(s^{(3)},1^{(n-3)})\in\sqrt{I}^{(m,n)}$. From
$g(s^{(3)},1^{(n-3)})\in\sqrt{I}^{(m,n)}$, it follows that there exists
$t\in\mathbb{N}$ such that $g(g(s^{(3)},1^{(n-3)})^{(t)},1^{(n-t)})\in I$ for
$t\leq n$ or $g_{(l)}(g(s^{(3)},1^{(n-3)})^{(t)})\in I$ for $t=l(n-1)+1$. In
both possibilities, we conclude that $I\cap S\neq\varnothing$ which is a
contradiction. Hence get $g(s,x_{i},1^{(n-2)})\in I$ which implies
$x_{i}\in(I:s)$. In the second case, there exists $t\in\mathbb{N}$ such that
$g(g(x_{1}^{i-1},s,x_{i+1}^{n})^{(t)},1^{(n-t)})\in I$ for $t\leq n$ or
$g_{(l)}(g(x_{1}^{i-1},s,x_{i+1}^{n})^{(t)})\in I$ for $t=l(n-1)+1$. If $t\leq
n$, then we obtain
$g(s,g(s^{(t)},1^{(n-t)}),1^{(n-2)})=g(s^{(t+1)},1^{(n-t-1)})\in\sqrt{I}^{(m,n)}$
or $g(s,g(x_{1}^{i-1},1,x_{i+1}^{n})^{(t)},1^{(n-t-1)})\in I$. From
$g(s^{(t+1)},1^{(n-t-1)})\in\sqrt{I}^{(m,n)}$, it follows that $I\cap
S\neq\varnothing$. Thus we conclude that
$g(s,g(x_{1}^{i-1},1,x_{i+1}^{n})^{(t)},1^{(n-t-1)})\in I$ which means
$g(g(x_{1}^{i-1},1,x_{i+1}^{n})^{(t)},1^{(n-t)})\in(I:s)$. Therefore
$g(x_{1}^{i-1},1,x_{i+1}^{n})\in\sqrt{(I:s)}$. Similar for other case.
Consequently, $(I:s)$ is an n-ary primary hyperideal of $R$.

$\Longleftarrow$ Let $(I:s)=\\{x\in R\ |\ g(x,s,1^{(n-2)})\in I\\}$ be an
n-ary primary hyperideal of $R$ for some $s\in S$. Suppose that
$g(x_{1}^{n})\in I$ for some $x_{1}^{n}\in R$. Then we get
$g(x_{1}^{n})\in(I:s)$. Since $(I:s)$ is an n-ary primary hyperideals of $R$,
then $x_{i}\in E_{S}$ which implies $g(s,x_{i},1^{(n-2)})\in I$ or
$g(x_{1}^{i-1},1,x_{i+1}^{n})\in\sqrt{(I:s)}$ which means there exists
$t\in\mathbb{N}$ such that
$g(s,g(x_{1}^{i-1},1,x_{i+1}^{n})^{(t)},1^{(n-t-1)})\in I$ for $t\leq n$ or
$g(s,g_{(l)}(x_{1}^{i-1},1,x_{i+1}^{n})^{(t)}),1^{(n-2)})\in I$ for
$t=l(n-1)+1$. If $t\leq n$, then we have
$g(g(x_{1}^{i+1},s,x_{i+1}^{n})^{(t)},1^{(n-t)})\in I$ which implies
$g(x_{1}^{i-1},s,x_{i+1}^{n})\in\sqrt{I}$. Similar for other case. This means
$I$ is an n-ary $S$-primary hyperideal of $R$. ∎

In the following, we consider the relationship between an n-ary $S$-primary
hyperideal and its radical.

######  Theorem 4.5.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I$ be an n-ary $S$-primary hyperideal of $R$. Then $\sqrt{I}^{(m,n)}$ is
an n-ary $S$-prime hyperideal of $R$.

###### Proof.

Since $I\cap S=\varnothing$ then we conclude that $\sqrt{I}^{(m,n)}\cap
S=\varnothing$. Now, let $g(x_{1}^{n})\in\sqrt{I}$ and for all
$j\in\\{1,...,i-1,i+1,...,n\\}$, $g(s,x_{j},1^{(n-2)})\notin\sqrt{I}^{(m,n)}$
for each $s\in S$. Since $g(x_{1}^{n})\in\sqrt{I}^{(m,n)}$, there exists
$t\in\mathbb{N}$ such that $g(g(x_{1}^{n})^{(t)},1^{(n-t)})\in I$ for $t\leq
n$ or $g_{(l)}(g(x_{1}^{n})^{(t)})\in I$ for $t=l(n-1)+1$. If
$g(g(x_{1}^{n})^{(t)},1^{(n-t)})\in I$, then there exists $s\in S$ such that
$g(s,x_{i}^{(t)},1^{(n-t-1)})\in I$ or
$g(s,g(x_{1}^{i-1},1,x_{i+1}^{n})^{(t)},1^{(n-t-1)})$ as $I$ is a n-ary
$S$-primary hyperideal of $R$. Hence we get
$g(g(s,x_{i},1^{(n-2)})^{(t)},1^{(n-t)})\in I$ which implies
$g(s,x_{i},1^{(n-2)})\in\sqrt{I}^{(m,n)}$ or
$g(g(x_{1}^{i-1},s,x_{i+1}^{n})^{(t)},1^{(n-t)})\in I$. In the second case, we
get

$\hskip
19.91684ptg(g(x_{1}^{(t)},g(x_{2}^{i-1},s,1,x_{i+1}^{n})^{(t)},1^{(n-2t)})\in
I$

$\Longrightarrow
g(g(x_{1}^{(t)},1^{(n-t)}),g(x_{2}^{i-1},s,1,x_{i+1}^{n})^{(t)},1^{(n-t-1)})\in
I$

$\Longrightarrow g(s,g(x_{1}^{(t)},1^{(n-t)}),1^{(n-2)})\in I\ \text{or}\
g(s,g(x_{2}^{i-1},s,1,x_{i+1}^{n})^{(t)},1^{(n-t-1)})\in\sqrt{I}^{(m,n)}$

$\Longrightarrow g(g(s,x_{1},1^{(n-2)})^{(t)},1^{(n-t)})\in I\ \text{or}\
g(g(x_{2}^{i-1},s^{(2)},x_{i+1}^{n})^{(t)},1^{(n-t)})\in\sqrt{I}^{(m,n)}$

$\Longrightarrow g(s,x_{1},1^{(n-2)})\in\sqrt{I}^{(m,n)}\ \text{or}\
g(g(x_{2}^{i-1},s^{(2)},x_{i+1}^{n})^{(t)},1^{(n-t)})\in\sqrt{I}^{(m,n)}$
Since $g(s,x_{1},1^{(n-2)})\in\sqrt{I}^{(m,n)}$ is a contradiction, then

$\hskip
19.91684ptg(g(x_{2}^{i-1},s^{(2)},x_{i+1}^{n})^{(t)},1^{(n-t)})\in\sqrt{I}^{(m,n)}$

$\Longrightarrow\exists
w\in\mathbb{N};g(g(g(x_{2}^{i-1},s^{(2)},x_{i+1}^{n}),1^{(n-t)})^{(w)},1^{(n-w)})\in
I$

$\Longrightarrow
g(g(x_{2}^{(t+w)},1^{(n-t-w)},g(g(x_{3}^{i-1},s^{(2)},1,x_{i+1}^{n})^{(w)},1^{(n-w-1)})\in
I$

$\Longrightarrow g(s,x_{2},1^{(n-2)})\in\sqrt{I}^{(m,n)}\ \text{or}\
g(g(g(x_{3}^{i-1},s^{(3)},x_{i+1}^{n})^{(t)},1^{(n-t)})^{(w)},1^{(n-w)})\in$

$\hskip 19.91684pt\sqrt{I}^{(m,n)}$

$\vdots$

$\Longrightarrow\cdots\ \text{or}\ g(s,x_{n},1^{(n-2)})\in\sqrt{I}^{(m,n)}$
which is contradiction with $g(s,x_{j},1^{(n-2)})\notin\sqrt{I}^{(m,n)}$ for
all $j\in\\{1,...,i-1,i+1,...,n\\}$. Thus we have
$g(s,x_{i},1^{(n-2)})\in\sqrt{I}^{(m,n)}$. Consequently, $\sqrt{I}^{(m,n)}$ is
an n-ary $S$-prime hyperideal of $R$. If $t=l(n-1)+1$, then by using a similar
argument, one can easily complete the proof. ∎

For a hyperideal $I$ of a Krasner $(m,n)$-hyperring $R$, we refer to the n-ary
$S$-prime hyperideal $P=\sqrt{I}^{(m,n)}$ as the associated $S$-prime
hyperideal of $I$ and on the other hand $I$ is referred to as an n-ary
$P$-$S$-primary hyperideal of $R$.

######  Theorem 4.6.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I_{1}^{n}$ be n-ary $P$-$S$-primary hyperideals of $R$ for some n-ary
$S$-prime hyperideal $P$ of $R$. Then, $I=\cap_{i=1}^{n}I_{i}$ is an n-ary
$P$-$S$-primary hyperideal of $R$.

###### Proof.

Let $I_{j}$ be an n-ary $S$-primary hyperideal of $R$ for all $1\leq j\leq n$.
Then there exists $s_{j}\in S$ such that if $g(x_{1}^{n})\in I_{j}$ for
$x_{1}^{n}\in R$, then $g(s_{j},x_{i},1^{(n-2)})\in I_{j}$ or
$g(x_{1}^{i-1},s_{j},x_{i+1}^{n})\in\sqrt{I_{j}}^{(m,n)}$ for some $1\leq
i\leq n$. We put $s=g(s_{1}^{n})$. Let $g(a_{1}^{n})\in I$ for some
$a_{1}^{n}\in R$ and let us assume that $g(s,a_{i},1^{(n-2)})\notin I$ for all
$1\leq i\leq n$. This means $g(s_{j},a_{i},1^{(n-2)})\notin I_{j}$ for some
$1\leq j\leq n$. Since $I_{j}$ is an $S$-primary hyperideal of $R$, we get
$g(a_{1}^{i-1},s_{j},a_{i+1}^{n})\in\sqrt{I_{j}}^{(m,n)}$. Since $I_{1}^{n}$
are n-ary $P$-$S$-primary hyperideals of $R$ for some n-ary $S$-prime
hyperideal $P$ of $R$, then $\sqrt{I_{j}}^{(m,n)}=\sqrt{I}$. Then we obtain
$g(a_{1}^{i-1},s_{j},a_{i+1}^{n})\in\sqrt{I}^{(m,n)}$ which implies
$g(a_{1}^{i-1},s,a_{i+1}^{n})\in\sqrt{I}^{(m,n)}$ and the proof is over. ∎

######  Theorem 4.7.

Let $S$ be an n-ary multiplicative subset of a Krasner $(m,n)$-hyperring $R$
and $I_{1}^{n-1}$ be hyperideals of $R$ such that for each $1\leq i\leq n-1$,
$I_{i}\cap S\neq\varnothing$. If $I$ is an n-ary $S$-primary hyperideal of
$R$, then $g(I_{1}^{n-1},I)$ is an n-ary $S$-primary hyperideal of $R$.

###### Proof.

Since $g(I_{1}^{n-1},I)\subseteq I$ and $I\cap S=\varnothing$, we get
$g(I_{1}^{n-1},I)\cap S=\varnothing$. Let $I$ be an n-ary $S$-primary
hyperideal of $R$. Then there exists $s\in S$ such that if $g(a_{1}^{n})\in I$
for $a_{1}^{n}\in R$, then $g(s,a_{i},1^{(n-2)})\in I$ or
$g(a_{1}^{i-1},s,a_{i+1}^{n})\in\sqrt{I}^{(m,n)}$. Suppose that
$g(x_{1}^{n})\in g(I_{1}^{n-1},I)$ for some $x_{1}^{n}\in R$. As
$g(I_{1}^{n-1},I)\subseteq I$, then $g(x_{1}^{n})\in I$. Since for each $1\leq
i\leq n-1$, $I_{i}\cap S\neq\varnothing$, then we have $u_{i}\in I_{i}\cap S$.
Put $u=g(u_{1}^{n-1},1)$. If $g(s,x_{i},1^{(n-2)})\in I$, then
$g(g(u,s,1^{(n-2)}),x_{i},1^{(n-2)})=g(u,g(s,x_{i},1^{(n-2)}),1^{(n-2)})\in
g(I_{1}^{(n-1)},I)$. Now, let
$g(x_{1}^{i-1},s,x_{i+1}^{n})\in\sqrt{I}^{(m,n)}$. Then there exists
$t\in\mathbb{N}$ such that $g(g(x_{1}^{i-1},s,x_{i+1}^{n})^{(t)},1^{(n-t)})\in
I$ for $t\leq n$ or $g_{(l)}(g(x_{1}^{i-1},s,x_{i+1}^{n})^{(t)})\in I$ for
$t=l(n-1)+1$. In the former case, we have

$\hskip
19.91684ptg(u^{(t)},g(g(x_{1}^{i-1},s,x_{i+1}^{n})^{(t)},1^{(n-t)})),1^{(n-t-1)})$

$\hskip
11.38092pt=g(g(x_{1}^{i-1},g(u,s,1^{(n-2)}),x_{i+1}^{n})^{(t)},1^{(n-t)})$

$\hskip 11.38092pt\in g(I_{1}^{n-1},I)$
which implies
$g(x_{1}^{i-1},g(u,s,1^{(n-2)}),x_{i+1}^{n})\in\sqrt{g(I_{1}^{n-1},I)}^{(m,n)}$.
Thus, $g(I_{1}^{n-1},I)$ is an n-ary $S$-primary hyperideal of $R$. In the
second case, a similar argument completes the proof. ∎

######  Theorem 4.8.

Let $R$ be a Krasner $(m,n)$-hyperring and $S$ be an $n$-ary multiplicative
subset of $R$ with $1\in S$. If $I$ is an $n$-ary $S$-primary hyperideal of
$R$ with $I\cap S=\varnothing$, then $S^{-1}I$ is an $n$-ary primary
hyperideal of $S^{-1}R$.

###### Proof.

Let $\frac{a_{1}}{s_{1}},...,\frac{a_{n}}{s_{n}}\in S^{-1}R$ such that
$G(\frac{a_{1}}{s_{1}},...,\frac{a_{n}}{s_{n}})\in S^{-1}I$ . Then we have
$\frac{g(a_{1}^{n})}{g(s_{1}^{n})}\in S^{-1}I$. It implies that there exists
$t\in S$ such that $g(t,g(a_{1}^{n}),1^{(n-2)})\in I$ and so
$g(a_{1}^{i-1},g(t,a_{i},1^{(n-2)}),a_{i+1}^{n})\in I$. Without destroying the
generality, we may assume that $g(a_{1}^{n-1},g(t,a_{n},1^{(n-2)}))\in I$.
Since $I$ is an $n$-ary $S$-primary hyperideal of $R$, then there exist $s\in
S$ such that at least one of the cases hold: $g(s,a_{i},1^{(n-2)})\in I$ for
some $1\leq i\leq n-1$, $g(s,g(t,a_{n},1^{(n-2)}),1^{(n-2)})\in I$,
$g(a_{1}^{i-1},s,a_{i+1}^{n-1},g(t,a_{n},1^{(n-2)}))\in\sqrt{I}^{(m,n)}$ for
some $1\leq i\leq n-1$ or $g(a_{1}^{n-1},s)\in\sqrt{I}^{(m,n)}$.
If $g(s,a_{i},1^{(n-2)})\in I$ for some $1\leq i\leq n-1$, then
$G(\frac{a_{i}}{s_{i}},\frac{1}{1}^{(n-1)})=\frac{g(a_{i},1^{(n-1)})}{g(s_{i},1^{(n-1)})}=\frac{g(s,a_{i},1^{(n-2)})}{g(s,s_{i},1^{(n-2)})}\in
S^{-1}I$. If $g(s,g(t,a_{n},1^{(n-2)}),1^{(n-2)})\in I$ then
$G(\frac{a_{n}}{s_{n}},\frac{1}{1}^{(n-1)})=\frac{g(a_{n},1^{(n-1)})}{g(s_{n},1^{(n-1)})}=\frac{g(s,t,a_{n},1^{(n-3)})}{g(s,t,s_{n},1^{(n-3)})}\in
S^{-1}I$. If
$g(a_{1}^{i-1},s,a_{i+1}^{n-1},g(t,a_{n},1^{(n-2)})\in\sqrt{I}^{(m,n)}$ for
some $1\leq i\leq n-1$, then we get
$G(\frac{a_{1}}{s_{1}},...,\frac{a_{i-1}}{s_{i-1}},\frac{1}{1},\frac{a_{i+1}}{s_{i+1}},...,\frac{a_{n-1}}{s_{n-1}},\frac{a_{n}}{s_{n}})=\break\frac{g(a_{1}^{i-1},g(s,t,1^{(n-2)}),a_{i+1}^{n-1},a_{n})}{g(s_{1}^{i-1},g(s,t,1^{(n-2)}),s_{i+1}^{n-1},a_{n})}\in
S^{-1}\sqrt{I}^{(m,n)}=\sqrt{S^{-1}I}^{(m,n)}$, by Lemma 4.7 in [5]. If
$g(a_{1}^{n-1},s)\in\sqrt{I}^{(m,n)}$, then
$G(\frac{a_{1}}{s_{1}},...,\frac{a_{n-1}}{s_{n-1}},\frac{1}{1})=\frac{g(a_{1}^{n-1},s)}{g(s_{1}^{n-1},s)}\in
S^{-1}\sqrt{I}^{(m,n)}=\sqrt{S^{-1}I}^{(m,n)}$. Thus $S^{-1}I$ is an $n$-ary
primary hyperideal of $S^{-1}R$. ∎

######  Theorem 4.9.

Let $R$ be a Krasner $(m,n)$-hyperring, $S$ be an $n$-ary multiplicative
subset of $R$ such that $1\in S$ and $I$ be a hyperideal of $R$ with $I\cap
S=\varnothing$ . If $S^{-1}I$ is an $n$-ary primary hyperideal of $S^{-1}R$
and $S^{-1}I\cap R=(I:s)$ for some $s\in S$, then $I$ is an $n$-ary
$S$-primary hyperideal of $R$.

###### Proof.

Suppose that $S^{-1}I$ is an $n$-ary primary hyperideal of $S^{-1}R$ and
$S^{-1}I\cap R=(I:s)$ for some $s\in S$. Assume that $g(a_{1}^{n})\in I$ for
some $a_{1}^{n}\in R$. Then we obtain
$G(\frac{a_{1}}{1},\cdots,\frac{a_{n}}{1})\in S^{-1}I$. Since $S^{-1}I$ is an
$n$-ary primary hyperideal of $S^{-1}R$, we get $\frac{a_{i}}{1}\in S^{-1}I$
or
$G(\frac{a_{1}}{1},\cdots,\frac{a_{i-1}}{1},\frac{1}{1},\frac{a_{i+1}}{1},\cdots,\frac{a_{n}}{1})\in\sqrt{S^{-1}I}^{(m,n)}$
for some $1\leq i\leq n$. In the former case, we have
$a_{i}=\frac{a_{i}}{1}\in S^{-1}I\cap R$ which implies $a_{i}\in(I:s)$ by the
hypothesis. This means $g(s,a_{i},1^{(n-2)})\in I$ and we are done. In the
second case, we get
$G(G(\frac{a_{1}}{1},\cdots,\frac{a_{i-1}}{1},\frac{1}{1},\frac{a_{i+1}}{1},\cdots,\frac{a_{n}}{1})^{(t)},\frac{1}{1}^{(n-t)})\in
S^{-1}I$ for $t\leq n$ or
$G_{(l)}(G(\frac{a_{1}}{1},\cdots,\frac{a_{i-1}}{1},\frac{1}{1},\frac{a_{i+1}}{1},\cdots,\frac{a_{n}}{1})^{(t)})\in
S^{-1}I$ for $t=l(n-1)+1$. The first possibility follows that
$g(g(a_{1},\cdots,a_{i-1},1,a_{i+1},\cdots,a_{n})^{(t)},1^{(n-t)})=\frac{g(g(a_{1},\cdots,a_{i-1},1,a_{i+1},\cdots,a_{n})^{(t)},1^{(n-t)})}{g(g(1^{(n)})^{(t)},1^{(n-t)})}\in
S^{-1}I\cap R$. Hence we conclude that
$g(s,g(a_{1},\cdots,a_{i-1},1,a_{i+1},\cdots,a_{n})^{(t)},1^{(n-t-1)})\in I$,
By the hypothesis. This means
$g(g(a_{1},\cdots,a_{i-1},s,a_{i+1},\cdots,a_{n})^{(t)},1^{(n-t)})\in I$ which
implies $g(a_{1},\cdots,a_{i-1},s,a_{i+1},\cdots,a_{n})\in\sqrt{I}^{(m,n)}$,
as needed. In the second possibility, one can easily complete the proof by
using an argument similar. ∎

## 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, $n$-ary 2-absorbing and 2-absorbing primary hyperideals in Krasner $(m,n)$-hyperrings, Matematicki Vesnik, 71 (3) (2019) 250-262.
  * [3] M. Anbarloei, Unifing the prime and primary hyperideals under one frame in a Krasner $(m,n)$-hyperring, Comm. Algebra, 49 (2021) 3432-3446.
  * [4] M. Anbarloei, A study on a generalization of the n-ary prime hyperideals in Krasner $(m,n)$-hyperrings, Afrika Matematika, 33 (2021) 1021-1032.
  * [5] M. Anbarloei, Krasner $(m,n)$-hyperring of fractions, submitted.
  * [6] A. Asadi, R. Ameri, Direct limit of Krasner (m,n)-hyperrings,Journal of Sciences, 31 (1) (2020) 75-83.
  * [7] G. Crombez, On $(m,n)$\- rings, Abh. Math. Semin. Univ., Hamburg, 37 (1972) 180-199.
  * [8] G. Crombez, J. Timm, On $(m,n)$-quotient rings, Abh. Math. Semin. Univ., Hamburg, 37 (1972) 200-203.
  * [9] S. Corsini, Prolegomena of hypergroup theory, Second edition, Aviani editor, Italy,  (1993).
  * [10] S. Corsini, V. Leoreanu, Applications of hyperstructure theory, Advances in Mathematics , vol. 5, Kluwer Academic Publishers, (2003).
  * [11] B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, Palm Harbor, USA, (2007).
  * [12] B. Davvaz, T. Vougiouklis, n-ary hypergroups, Iran. J. Sci. Technol., 30 (A2) (2006) 165-174.
  * [13] Z. Dongsheng, $\delta$-primary ideals of commutative rings. Kyungpook Mathematical Journal, 41 (2001) 17-22.
  * [14] W. Dorente, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Math. Z., 29 (1928) 1-19.
  * [15] B. Fahid, Z. Dongsheng, 2-Absorbing $\delta$-primary ideals of commutative rings. Kyungpook Mathematical Journal, 57 (2017) 193-198.
  * [16] A. Hamed, A. Malek, $S$-prime ideals of a commutative ring, Beitr Algebra Geom,  (2019) DOI: 10.1007/s13366-019-00476-5.
  * [17] K. Hila, K. Naka, B. Davvaz, On $(k,n)$-absorbing hyperideals in Krasner $(m,n)$-hyperrings, Quarterly Journal of Mathematics, 69 (2018) 1035-1046.
  * [18] E. Kasner, An extension of the group concept (reported by L.G. Weld), Bull. Amer. Math. Soc., 10 (1904) 290-291.
  * [19] V. Leoreanu, Canonical n-ary hypergroups, Ital. J. Pure Appl. Math., 24(2008).
  * [20] V. Leoreanu-Fotea, B. Davvaz, n-hypergroups and binary relations, European J. Combin., 29 (2008) 1027-1218.
  * [21] V. Leoreanu-Fotea, B. Davvaz, Roughness in n-ary hypergroups, Inform. Sci., 178 (2008) 4114-4124.
  * [22] 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.
  * [23] F. Marty, Sur une generalization de la notion de groupe, $8^{th}$ Congress Math. Scandenaves, Stockholm, (1934) 45-49.
  * [24] E. Massaoud, $S$-primary ideals of a commutative ring, Communications in Algebra (2021) DOI: 10.1080/00927872.2021.1977939.
  * [25] S. Mirvakili, B. Davvaz, Relations on Krasner $(m,n)$-hyperrings, European J. Combin., 31(2010) 790-802.
  * [26] S. Mirvakili, B. Davvaz, Constructions of $(m,n)$-hyperrings, Matematicki Vesnik, 67 (1) (2015) 1-16.
  * [27] M. Norouzi, R.Ameri, V. Leoreanu-Fotea, Normal hyperideals in Krasner $(m,n)$-hyperrings, An. St. Univ. Ovidius Constanta 26 (3) (2018) 197-211.
  * [28] S. Omidi, B. Davvaz, Contribution to study special kinds of hyperideals in ordered semihyperrings, J. Taibah Univ. Sci., 11 (2017) 1083-1094.
  * [29] S. Ostadhadi-Dehkordi, B. Davvaz, A Note on Isomorphism Theorems of Krasner $(m,n)$\- hyperrings, Arabian Journal of Mathematics, 5 (2016) 103-115.
  * [30] E. Ozel Ay, G. Yesilot, D. Sonmez, $\delta$-Primary Hyperideals on Commutative Hyperrings, Int. J. Math. and Mathematical Sciences, (2017) Article ID 5428160, 4 pages.
  * [31] S. Visweswaran, Some results on $S$-primary ideals of a commutative ring, Beitr Algebra Geom (2021) DOI: 10.1007/s13366-021-00580-5.
  * [32] T. Vougiouklis, Hyperstructures and their representations, Hadronic Press Inc., Florida, (1994).
  * [33] M.M. Zahedi, R. Ameri, On the prime, primary and maximal subhypermodules, Ital. J. Pure Appl. Math., 5 (1999) 61-80.
  * [34] J. Zhan, B. Davvaz, K.P. Shum, Generalized fuzzy hyperideals of hyperrings, Computers and Mathematics with Applications, 56 (2008) 1732-1740.