Lattices of lower finite breadth.
Lattices of Scott-closed sets
A dcpo is continuous if and only if the lattice of all Scott-closed subsets of is completely distributive. However, in the case where is a non-continuous dcpo, little is known about the order structure of . In this paper, we study the order-theoretic properties of for general dcpo’s . The main results are: (i) every is C-continuous; (ii) a complete lattice is isomorphic to for a complete semilattice if and only if is weak-stably C-algebraic; (iii) for any two complete semilattices...
Locally solid topological lattice-ordered groups
Locally solid Riesz spaces have been widely investigated in the past several decades; but locally solid topological lattice-ordered groups seem to be largely unexplored. The paper is an attempt to initiate a relatively systematic study of locally solid topological lattice-ordered groups. We give both Roberts-Namioka-type characterization and Fremlin-type characterization of locally solid topological lattice-ordered groups. In particular, we show that a group topology on a lattice-ordered group is...
Lower semicontinuous functions with values in a continuous lattice
It is proved that for every continuous lattice there is a unique semiuniform structure generating both the order and the Lawson topology. The way below relation can be characterized with this uniform structure. These results are used to extend many of the analytical properties of real-valued l.s.cḟunctions to l.s.cḟunctions with values in a continuous lattice. The results of this paper have some applications in potential theory.
Łukasiewicz tribes are absolutely sequentially closed bold algebras
We show that each sequentially continuous (with respect to the pointwise convergence) normed measure on a bold algebra of fuzzy sets (Archimedean -algebra) can be uniquely extended to a sequentially continuous measure on the generated Łukasiewicz tribe and, in a natural way, the extension is maximal. We prove that for normed measures on Łukasiewicz tribes monotone (sequential) continuity implies sequential continuity, hence the assumption of sequential continuity is not restrictive. This yields...
Metrizable completely distributive lattices
The purpose of this paper is to study the topological properties of the interval topology on a completely distributive lattice. The main result is that a metrizable completely distributive lattice is an ANR if and only if it contains at most finite completely compact elements.
Natural continuity space structures on dual Heyting algebras
On atomistic lattices.
On quasi-uniform space valued semi-continuous functions
F. van Gool [Comment. Math. Univ. Carolin. 33 (1992), 505–523] has introduced the concept of lower semicontinuity for functions with values in a quasi-uniform space . This note provides a purely topological view at the basic ideas of van Gool. The lower semicontinuity of van Gool appears to be just the continuity with respect to the topology generated by the quasi-uniformity , so that many of his preparatory results become consequences of standard topological facts. In particular, when the order...
On the Scott topology on the set of continuous maps
On -continuous tolerances of -distributive lattices
Partial dcpo’s and some applications
We introduce partial dcpo’s and show their some applications. A partial dcpo is a poset associated with a designated collection of directed subsets. We prove that (i) the dcpo-completion of every partial dcpo exists; (ii) for certain spaces , the corresponding partial dcpo’s of continuous real valued functions on are continuous partial dcpos; (iii) if a space is Hausdorff compact, the lattice of all S-lower semicontinuous functions on is the dcpo-completion of that of continuous real valued...
Partially ordered sets with projections and their topology [Book]
Partitions and partially ordered sets
Perfect compactifications of frames
Perfect compactifications of frames are introduced. It is shown that the Stone-Čech compactification is an example of such a compactification. We also introduce rim-compact frames and for such frames we define its Freudenthal compactification, another example of a perfect compactification. The remainder of a rim-compact frame in its Freudenthal compactification is shown to be zero-dimensional. It is shown that with the assumption of the Boolean Ultrafilter Theorem the Freudenthal compactification...
Quasicontinuous posets.
Quasicontinuous spaces
We lift the notion of quasicontinuous posets to the topology context, called quasicontinuous spaces, and further study such spaces. The main results are: (1) A space is a quasicontinuous space if and only if is locally hypercompact if and only if is a hypercontinuous lattice; (2) a space is an -continuous space if and only if is a meet continuous and quasicontinuous space; (3) if a -space is a well-filtered poset under its specialization order, then is a quasicontinuous space...
Remarks on the sobriety of Scott topology and weak topology on posets
We give some necessary and sufficient conditions for the Scott topology on a complete lattice to be sober, and a sufficient condition for the weak topology on a poset to be sober. These generalize the corresponding results in [1], [2] and [4].
Semimodularity in lower continuous strongly dually atomic lattices
For lattices of finite length there are many characterizations of semimodularity (see, for instance, Grätzer [3] and Stern [6]–[8]). The present paper deals with some conditions characterizing semimodularity in lower continuous strongly dually atomic lattices. We give here a generalization of results of paper [7].
Sublattices of Euclidean spaces.