FOUNDATIONS OF ORDERED (SEMI)HYPERRINGS

Bijan Davvaz, S. Omidi


Abstract


In this paper, we introduce the notion of general hyperring $(R,+,\cdot)$ besides a binary relation $\le $, where $\le $ is a partial order such that satisfies the conditions: (1) If $a \le b$, then $a+c \le b+c$, meaning that for any $x \in a+c$, there exists $y \in b+c$ such that $x\le y$. The case $c+a\le c+b$ is defined similarly. (2) If $a \le b$ and $c \in R$, then $a\cdot c \le b\cdot c$, meaning that for any $x\in a\cdot c$, there exists $y\in b\cdot c$ such that $x\le y$. The case $c\cdot a \le c\cdot b$ is defined similarly. This structure is called an ordered general hyperring. Also, we present several examples of ordered general hyperrings and prove some results in this respect. By using the notion of pseudoorder on an ordered general hyperring $(R,+,\cdot,\le)$, we obtain an ordered ring. Moreover, we study some properties of pseudoorder on an ordered general hyperring.

DOI : http://dx.doi.org/10.22342/jims.22.2.233.131-150


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p-ISSN:2086-8952e-ISSN:2460-0245