2 research outputs found
Free dcpo-algebras via directed spaces
Directed spaces are natural topological extensions of dcpos in domain theory
and form a cartesian closed category. We will show that the D-completion of
free algebras over a Scott space , on the context of directed spaces,
are exactly the free dcpo-algebras over dcpo , which reveals the close
connection between directed powerspaces and powerdomains. By this result, we
provide a topological representation of upper, lower and convex powerdomains of
dcpos uniformly.Comment: 18 page
One-step Closure, Ideal Convergence and Monotone Determined Space
Monotone determined spaces are natural topological extensions of dcpo. Its
main purpose is to build an extended framework for domain theory. In this
paper, we study the one-step closure and ideal convergence on monotone
determined space. Then we also introduce the equivalent characterizations of
c-spaces and locally hypercompact space. The main results are:1.Every c-space
has one-step closure and every locally hypercompact space has weak one-step
closure;2.A monotone determined space has one-step closure if and only if it is
d-meet continuous and has weak one-step closure. 3.IS-convergence(resp.
IGS-convergence) is topological iff X is a c-space (resp. locally hypercompact
space); 4.If X is a d-meet continuous space, then the following three
conditions are equivalent to each other: (i) X is c-space; (ii) The net (xj )
ISL-converges to x iff (xj ) I-converges to x with respect to Lawson topology;
(iii) The net (xj ) IGSL-converges to x iff (xj ) I-converges to x with respect
to Lawson topology.Comment: 14page