The L-ordered semigroups based on L-partial orders

Fuzzy (frame-valued) ordered semigroups are investigated with the orders being L-partial orders. In this framework, L-left (resp. L-right) ideals, L-bi-ideals and L-quasi-ideals of an L-ordered semigroup are discussed, the idea of which is applied to the characterizations of regular and duo L-ordered semigroups. In addition, the relationship between an L-ordered semigroup and an L-topological semigroup is studied. As a consequence, an isomorphism between the category of L-ordered semigroups and the category of L-topological semigroups with the topologies being L-T0 and Alexandrov is obtained.