On ordered $\Gamma$-hypersemigroups, minimal bi-ideals, and minimal left ideals


Abstract: The definition of ordered $\Gamma$-hypersemigroups and the definitions of regular and intra-regular ordered $\Gamma$-hypersemigroups in the existing bibliography should be corrected. Care should be given to the definitions of bi-$\Gamma$-hyperideals and quasi-$\Gamma$-hyperideals as well. The main results are a characterization of minimal bi-ideals of an ordered $\Gamma$-hypersemigroup $S$ in terms of $\cal B$-simple bi-ideals of $S$ and a characterization of minimal left (resp. right) ideals of an ordered $\Gamma$-hypersemigroup $S$ in terms of left (resp. right) simple subsemigroups of $S$.

Keywords: Ordered $\Gamma$-hypersemigroup, regular, intra-regular, bi-$\Gamma$-hyperideal, quasi-$\Gamma$-hyperideal, ${\cal B}$-simple bi-idea

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