Narrow Results

In this paper, we deal with $(m,n)$-hyperideals of ordered semihyperrings. First, we provide some properties about these $(m,n)$-hyperideals. Next, we provide nontrivial examples of $(m,n)$-hyperideals that are not left/right hyperideals. Finally, we apply our results to ordered semirings and provide nontrivial examples on $(m,n)$-ideals that are not left/right ideals.

An ordered semiring $(S,+,\cdot ,\le )$ is called ordered $k$-regular if for every element $a$ of $S$ there exist $x,y,s,t\in S$ with $x\le asa,y\le ata$ such that $a+x\le y$. An ordered ideal $A$ of $S$ is called an ordered $k$-ideal, if $x\in S$ and $x+a=b$ for some $a,b\in A$ then $x\in A$. In this work, we characterize ordered $k$-regular semirings using their ordered $k$-ideals. Moreover, characterizations of left(right) ordered $k$-regular semirings and left(right) ordered $k$-weakly regular semirings are investigated.

In this paper, we consider the set of all generalized inverses of an $m\times n$ matrix $A$, denoted by $A\{1\}$ and define two binary operations, ⊕ and ⋆ with some properties which offer the structure of a semi ring to $A\{1\}$. The compatibility of some partial orders like Sussman’s order, Conrad’s order, etc. is also checked which make it an ordered matrix semiring.