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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.

In this paper, concepts of generalized algebraic information system (in short, GA-inf) and weak algebraic information system (in short, wA-inf) are introduced. Relationships among several algebraicities of information systems are explored. Main results are: (1) A dcpo D is an algebraic domain iff D is represented by an algebraic information system (in short, A-inf), and iff D is represented by a GA-inf; (2) every A-inf is a GA-inf and a wA-inf; (3) every C-inf S ( D , B ) induced by a domain with a basis B is a wA-inf; (4) the category WAINF of wA-infs and approximable mappings is equivalent to the category DOM of domains and Scott continuous functions; (5) every wA-inf S satisfying Condition (MT) is a retract of w S (the information structure induced by S ) in category WAINF of wA-infs and approximable mappings. Particularly, C-inf S ( D , B ) induced by a domain D with a basis B is a retract of A-inf w S ( D , B ) in category WAINF.

The consistent Smyth powerdomain R Q C ( X ) of a topological space X means the family of all nonempty relatively compact-connected saturated subsets of X, ordered by the reverse inclusion and endowed with the upper Vietoris topology. In this paper, we study properties of consistent Smyth powerdomains of certain topological spaces (especially, quasicontinuous domains equipped with the Scott topology) from topological, order theoretical and categorical aspects. Main results are: (i) a topological space X is sober iff R Q C ( X ) is sober; (ii) if X is locally compact-connected, well-filtered and coherent, then R Q C ( X ) is coherent; (iii) every dcpo equipped with the Scott topology is locally connected, and if a dcpo L is finitely up-generated, locally compact, well-filtered and coherent, then R Q C ( L ) is a Lawson compact L-domain; (iv) if L is a quasicontinuous domain (resp., a Lawson compact quasicontinuous domain, a quasialgebraic domain), then R Q C ( L ) is a continuous dcpo- ∧ ↑ -semilattice (resp., a Lawson compact L-domain, an algebraic dcpo- ∧ ↑ -semilattice); (v) it is proved that R Q C ( L ) is a free continuous dcpo- ∧ ↑ -semilattice over a quasicontinuous domain L.

In this paper, we are mainly concerned with semicontinuity of complete lattices and their distributive reflections, introduced by Rav in 1989. We prove that for a complete lattice L, the distributive reflection Ld is isomorphic to the lattice of all radicals determined by principal ideals of L in the set-inclusion order, obtaining a method to depict the distributive reflection of a given lattice. It is also proved that if a complete lattice L is semicontinuous and every semiprime element x ∈ L is the largest in d(x), then Ld is continuous whenever the distributive reflector d is Scott continuous. We construct counterexamples to confirm a conjecture and solve two open problems posed by Zhao in 1997.

In terms of Scott-closed sets, the concept of QC-continuous posets, a generalization of C-continuous lattices is introduced. With this new concept, the following results are obtained: (1) a complete lattice is generalized completely distributive (GCD) iff it is quasicontinuous and QC-continuous; (2) a dcpo L is quasicontinuous (resp., quasialgebraic) iff the Hoare powerdomain H ( L ) is quasicontinuous (resp., quasialgebraic); (3) the Hoare powerdomain H ( L ) of a QFS-domain L is still a QFS-domain.

In 2016, Hamid et al. [1] introduced the thought of the AP-Henstock integrals of intervalvalued functions and fuzzy-number-valued functions and obtained a number of their properties. The aim of this paper is to introduce the thought of the McShane delta integrals of interval-valued functions and fuzzy-number-valued functions and discuss some of their properties.

Rough set theory is a useful tool for dealing with fuzzyness and uncertainty of knowledge. In rough set theory, knowledge reductions and generatings are important research topics and critical steps of knowledge acquisition. This paper generalize knowledge bases to abstract knowledge bases and study (abstract) knowledge bases on infinite universe by considering the problem of existence of finite reductions of infinite knowledge bases. For abstract knowledge bases, the concept of saturations and saturation reductions are introduced. Global properties of saturations and saturation reductions of abstract knowledge bases are investigated. It is proved that for a given abstract knowledge base which is closed w.r.t. arbitrary unions on a finite universe U, its saturation augmented the unverse U forms a topology, whereas a counterexample is constructed to show that this may not be true if U is infinite. Making use of the saturation of an abstract knowledge base, some sufficient and/or necessary conditions for existence of finite reductions of an infinite abstract knowledge base are given. It is proved that for an abstract knowledge base on finite universe, there is one and only one saturation reduction. Some examples are constructed to reveal various cases of existence of knowledge reductions. Simple applications of saturation reductions are also given.

In this note, the quasicontinuity of a dcpo and its principal ideals/filters are considered. It is shown that every principal ideal, as a subposet, in a quasicontinuous domain is also a quasicontinuous domain. A dcpo is quasicontinuous if every principal ideal is quasicontinuous and it has finitely many maximal elements. For a bc-domain being quasicontinuous, the requirement of directedness of the approximation family of each element can be omitted.