Relationship between lattice ordered semigroups and ordered hypersemigroups

Abstract

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, ∨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, ∨e or poesemigroups and direct proofs derive along the lines of those le, ∨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.