Abstract
In this paper, the concepts of Scott S-sets, weakly local S-compactness and S-bases of posets are introduced. With these new concepts, some characterizations of supercontinuous (resp., superalgebraic) posets are given. In order to provide a topological interpretation of S-bases, the new concept of the S-essential topology on posets is also introduced. Properties and characterizations of S-bases via the S-essential topology are obtained. Main results are: (1) A poset L is supercontinuous iff every two different points can be separated by a principal filter and the complement of a Scott S-set; (2) A poset L is supercontinuous iff it is weakly local S-compact and for every x∈U, where U is a Scott S-set, there is a Scott S-set filter V such that x∈V⊆U; (3) A poset L is supercontinuous iff it has an S-basis; (4) In a supercontinuous poset, a subset B is an S-basis iff for every Scott S-set U and S-e-open set G, U∩G∩B≠∅ whenever U∩G≠∅. Counterexamples are constructed to show that supercontinuity is not hereditary to principal ideals, nor to Scott S-sets.
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