It has been shown in Turk J Math 2019; 43 (5): 2592-2601 that many results on hypersemigroups can be obtained directly as corollaries of more general results from the theory of lattice ordered semigroups, $\vee e$-semigroups or $poe$-semigroups. The present note shows that although this is not exactly the case for ordered hypersemigroups, even in this case various results may be suggested from analogous results for $le$, $\vee e$ or $poe$-semigroups and direct proofs derive along the lines of those $le$, $\vee e$ or $poe$-semigroups setting as well; the sets in the investigation provides a further indication that the results on this structure come from the lattice ordered semigroups or ordered semigroups in general. In many cases, whenever we have a look at any result on lattice ordered semigroups, we immediately know if can be transferred to ordered hypersemigroups. We never work on ordered hypersemigroups directly.
"Relationship between lattice ordered semigroups and ordered hypersemigroups,"
Turkish Journal of Mathematics: Vol. 45:
4, Article 19.
Available at: https://journals.tubitak.gov.tr/math/vol45/iss4/19