Applications of rough soft sets to Krasner (m,n)-hyperrings and corresponding decision making methods

Abstract

If $I$ is a normal hyperideal of a  Krasner $(m,n)$-hyperring $R$, then we define the relation $\equiv_I$ by $x\equiv_I y$ if and only if $ f(x,-y,\overset{(m-2)}{0})\cap I\ne\emptyset$, which is an equivalence relation  on $R$.  By means of this idea, we propose rough soft hyperrings (hyperideals)   with respect to a normal hyperideal in  a Krasner $(m,n)$-hyperring. Some  lower and upper rough soft hyperideals with respect to a normal hyperideal are investigated, respectively. Further, we  define the $t$-level set $U(\mu,t)=\{(x,y)\in R\times R \vert \bigwedge\limits_{z\in f(x,-y,\overset{(m-2)}{0})}\mu(z)\ge t\}$ of a Krasner $(m,n)$-hyperring $R$ and prove that it is is an equivalence relation on $R$ if $\mu$ is a fuzzy normal hyperideal of $R$. Based on this novel idea, we propose  rough soft hyperideals based on fuzzy normal hyperideals in   Krasner $(m,n)$-hyperrings.
Finally, we put forth two novel kinds  of  decision making methods to rough soft Krasner $(m,n)$-hyperrings.