This is from Birkhoff, the "father of lattice theory" in Trends Lattice Theory. Van Nostrand 1970: "Lattices can do things for you, no matter what kind mathematician you are!". The aim this paper to show that $le$-semigroups (lattice ordered semigroups possessing a greatest element) play main role studying hypersemigroups. From many results on corresponding be obtained. converse also possible b...

Let S be an ordered semigroup. In this paper we first introduce the concepts of (∈,∈ ∨qk)-fuzzy ideals, (∈,∈ ∨qk)-fuzzy bi-ideals and (∈,∈ ∨qk)-fuzzy generalized bi-ideals of an ordered semigroup S by the ordered fuzzy points of S, and investigate their related properties. Furthermore, characterizations of regular ordered semigroups by the properties of (∈,∈ ∨qk)-fuzzy left ideals, (∈,∈ ∨qk)-fu...

Ordered semigroups are understood through their subsets. The aim of this article is to study ordered some new substructures. In regard, quasi-filters and (m,n)-quasi-filters introduced as types filters. Some properties the concepts investigated, different examples constructed, relations between quasi-ideals well (m,n)-quasi-ideals discussed.