Relationship between lattice ordered semigroups and ordered hypersemigroups

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.

PDF

___

[1] Birkhoff G. Lattice Theory. Revised ed. American Mathematical Society Colloquium Publications, Vol. XXV American Mathematical Society, Providence, R.I. 1961.
[2] Birkhoff G. Lattice Theory. Corrected reprint of the 1967 third ed. American Mathematical Society Colloquium Publications, 25. American Mathematical Society, Providence, R.I. 1979.
[3] Dubreil-Jacotin ML, Lesieur L, Croisot R. Leçons sur la Théorie des Treillis des Structures Algébriques Ordonnées et des Treillis Géométriques. Gauthier-Villars, Paris 1953.
[4] Fuchs L. Partially Ordered Algebraic Systems. Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London 1963.
[5] Grätzer G. General Lattice Theory. Pure and Applied Mathematics, 75. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], NewYork-London 1978.
[6] Gu Z. On hyperideals of ordered semihypergroups. Italian Journal of Pure and Applied Mathematics. 2018; 40: 692–698.
[7] Kehayopulu N. On intra-regular ∨e-semigroups. Semigroup Forum 1980; 19 (2): 111–121. doi:10.1007/BF02572508
[8] Kehayopulu N. On prime, weakly prime ideals in ordered semigroups. Semigroup Forum 1992; 44 (3): 341–346. doi:10.1007/BF02574353
[9] Kehayopulu N. On hypersemigroups. Pure Mathematics and Applications (PU.M.A.) 2015; 25 (2): 151–156. doi:10.1515/puma-2015-0015
[10] Kehayopulu N. On le-semigroups. Turkish Journal of Mathematics 2016; 40 (2): 310–316. doi:10.3906/mat-1503-74
[11] Kehayopulu N. Hypersemigroups and fuzzy hypersemigroups. European Journal of Pure and Applied Mathematics 2017; 10 (5): 929–945.
[12] Kehayopulu N. Left regular and intra-regular ordered hypersemigroups in terms of semiprime and fuzzy semiprime subsets. Scientiae Mathematicae Japonica 2017; 80 (3): 295–305.
[13] Kehayopulu N. How we pass from semigroups to hypersemigroups. Lobachevskii Journal of Mathematics 2018; 39 (1): 121–128. doi:10.1134/S199508021801016X
[14] Kehayopulu N. On ordered hypersemigroups with idempotent ideals, prime or weakly prime ideals. European Journal of Pure and Applied Mathematics 2018; 11 (1): 10–22. doi:10.29020/nybg.ejpam.v11i1.3085
[15] Kehayopulu N. On ordered hypersemigroups given by a table of multiplication and a figure. Turkish Journal Mathematics 2018; 42 (4): 2045–2060. doi:10.3906/mat-1711-53
[16] Kehayopulu N. Lattice ordered semigroups and hypersemigroups. Turkish Journal Mathematics 2019; 43 (5): 2592– 2601. doi:10.3906/mat-1907-86
[17] Kehayopulu N. On the paper “On hyperideals of ordered semihypergroups” by Ze Gu in Ital J Pure Appl Math. European Journal of Pure and Applied Mathematics 2019; 12 (4): 1771–1778. doi:10.29020/nybg.ejpam.v12i4.3592