# On Hoehnke Ideal in Ordered Semigroups

## Authors

• Niovi Kehayopulu Professor Docent Dr. University of Athens

## Keywords:

ordered semigroup, semiprime subset (right ideal), completely semiprime ideal, prime subset (right ideal), completely prime ideal, Hoehnke ideal

## Abstract

For a proper subset $A$ of an ordered semigroup $S$, we denote by $H_A(S)$ the subset of $S$ defined by $H_A(S):=\{h\in S \mbox { such that if } s\in S\backslash A, \mbox { then } s\notin (shS]\}$. We prove, among others, that if $A$ is a right ideal of $S$ and the set $H_A(S)$ is nonempty, then $H_A(S)$ is an ideal of $S$; in particular it is a semiprime ideal of $S$. Moreover, if $A$ is an ideal of $S$, then $A\subseteq H_A(S)$. Finally, we prove that if $A$ and $I$ are right ideals of $S$, then $I\subseteq H_A(S)$ if and only if $s\notin (sI]$ for every $s\in S\backslash A$. We give some examples that illustrate our results. Our results generalize the Theorem 2.4 in Semigroup Forum 96 (2018), 523--535.

2018-10-24

## How to Cite

Kehayopulu, N. (2018). On Hoehnke Ideal in Ordered Semigroups. European Journal of Pure and Applied Mathematics, 11(4), 911–921. https://doi.org/10.29020/nybg.ejpam.v11i4.3341

Computer Science

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