On ordered $\Gamma$-hypersemigroups and their relation to lattice ordered semigroups


Abstract: The concept of $\Gamma$-hypersemigroup has been introduced in Turk J Math 2020; 44 (5): 1835-1851 in which it has in which it has been shown that various results on $\Gamma$-hypersemigroups can be obtained directly as corollaries of more general results from the theory of $le$-semigroups (i.e. lattice ordered semigroups having a greatest element) or $poe$-semigroups. As a continuation of the paper mentioned above, in the present paper, the concept of ordered $\Gamma$-hypersemigroups has been introduced, and their relation to lattice ordered semigroups is given. It has been shown that although the results on ordered $\Gamma$-hypersemigroups cannot be obtained as corollaries to the corresponding results of $le$ or $poe$-semigroups, still the main idea comes from the $le$-semigroups or $poe$-semigroups, and the proofs go along the lines of the $le$ or $poe$-semigroups.

Keywords: Lattice ordered semigroup, ordered $\Gamma$-hypersemigroup, regular, intra-regular, left (right) regular

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