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1.

Luoshan Xu 《Topology and its Applications》2006,153(11):1886-1894

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

- (1)
- A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;
- (2)
- A poset is continuous iff its Scott topology is completely distributive;
- (3)
- A topological
*T*_{0}space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology; - (4)
- A topological
*T*_{1}space is a discrete space iff its topology is completely distributive.

2.

In this paper the new concept of B-posets is introduced. Some properties
of B-posets and FS-posets are examined. Main results are: (1) Posets obtained
from B-posets (FS-posets) by eliminating a proper upper subset, adding two or
more finitely many incomparable maximal elements, taking vertical sums w.r.t.
a maximal element are also B-posets (FS-posets); (2) A poset is a(n) B-domain
(FS-domain) iff it is a Lawson compact B-poset (FS-poset); (3) The directed
completions of B-posets (FS-posets) are B-domains (FS-domains); (4) The category
B-POS (FS-POS) of B-posets (FS-posets) and Scott continuous maps
is cartesian closed and has the category B-DOM (FS-DOM) of B-domains
(FS-domains) and Scott continuous maps as a full reflective subcategory. 相似文献

3.

In this paper, posets which may not be dcpos are considered. In terms of the Scott topology on posets, the new concept of
quasicontinuous posets is introduced. Some properties and characterizations of quasicontinuous posets are examined. The main
results are: (1) a poset is quasicontinuous iff the lattice of all Scott open sets is a hypercontinuous lattice; (2) the directed
completions of quasicontinuous posets are quasicontinuous domains; (3) A poset is continuous iff it is quasicontinuous and
meet continuous, generalizing the relevant result for dcpos.
Supported by the NSF of China (10371106, 10410638) and by the Fund (S0667-082) from Nanjing University of Aeronautics and
Astronautics. 相似文献

4.

In this paper, consistent algebraic L-domains are considered. One algebraic and two topological characterization theorems
for their directed completions are given. It is proved that eliminating a set of maximal elements with empty interior from
an algebraic L-domain results a consistent algebraic L-domain whose directed completion is just the given algebraic L-domain
up to isomorphism. It is also proved that the category

**CALDOM**of consistent algebraic L-domains and Scott continuous maps is Cartesian closed and has the category**ALDOM**of algebraic L-domains and Scott continuous maps as a full reflective subcategory. Received January 8, 2005; accepted in final form June 15, 2005. 相似文献5.

In this paper the question of what classes

*A*of*T*_{0}-spaces should be paired with classes of domains in order that all function spaces [*A**B*] for*A**A*and*B*are -compact domains is considered. It is shown that core compact spaces are paired with bounded complete domains and a class of topological spaces called*RW*-spaces (with finitely many components) is paired with the class of -compact pointed*L*-domains (*L*-domains). 相似文献6.

In this paper, concepts of quasi-finitely separating maps and quasi-approximate identities are introduced. Based on these concepts, QFS-domains and quasicontinuous maps are defined. Properties and characterizations of QFS-domains are explored. Main results are: (1) finite products, nonempty Scott closed subsets and quasicontinuous projection images of QFS-domains, as well as FS-domains, are all QFS-domains; (2) QFS-domains are compact in the Lawson topology; (3) An L-domain is a QFS-domain iff it is an FS-domain, iff it is compact in the Lawson topology; (4) Bounded complete quasicontinuous domains, in particular quasicontinuous lattices, are all QFS-domains. 相似文献

7.

In this paper, the concept of dual normal relations on sets is introduced and generalized. Intrinsic characterizations of them are obtained. 相似文献

8.

In this paper, the concepts of conjugative relations and dual conjugative relations are introduced and generalized. Intrinsic
characterizations of them are obtained. In addition, we give algebraic and topological characterizations of posets for which
the relation

*≰*is conjugative/dual conjugative. 相似文献9.

The concept of approximation spaces is a key notion of rough set theory, which is an important tool for approximate reasoning about data. This paper concerns algebraic aspects of generalized approximation spaces. Concepts of

*R*-open sets,*R*-closed sets and regular sets of a generalized approximation space (*U*,*R*) are introduced. Algebraic structures of various families of subsets of (*U*,*R*) under the set-inclusion order are investigated. Main results are: (1) The family of all*R*-open sets (respectively,*R*-closed sets,*R*-clopen sets) is both a completely distributive lattice and an algebraic lattice, and in addition a complete Boolean algebra if relation*R*is symmetric. (2) The family of definable sets is both an algebraic completely distributive lattice and a complete Boolean algebra if relation*R*is serial. (3) The collection of upper (respectively, lower) approximation sets is a completely distributive lattice if and only if the involved relation is regular. (4) The family of regular sets is a complete Boolean algebra if the involved relation is serial and transitive. 相似文献10.

In this note, the new concepts of C-bases (resp., BC-bases, L-bases) which are special kinds of abstract bases are introduced.
It is proved that the round ideal completion of a C-basis (resp., BC-basis, L-basis) is a continuous lattice (resp., bc-domain,
L-domain). Furthermore, representation theorems of continuous lattices (resp., bc-domains, L-domains) by means of the round
ideal completions of C-bases (resp., BC-bases, L-bases) are obtained.
Supported by the NSF of China (10371106, 60774073) and by the Fund (S0667-082) from
Nanjing University of Aeronautics and Astronautics. 相似文献