| Title:
|
$\mathcal Z$-distributive function lattices (English) |
| Author:
|
Erné, Marcel |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
138 |
| Issue:
|
3 |
| Year:
|
2013 |
| Pages:
|
259-287 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is known that for a nonempty topological space $X$ and a nonsingleton complete lattice $Y$ endowed with the Scott topology, the partially ordered set $[X,Y]$ of all continuous functions from $X$ into $Y$ is a continuous lattice if and only if both $Y$ and the open set lattice $\mathcal O X$ are continuous lattices. This result extends to certain classes of $\mathcal Z$-distributive lattices, where $\mathcal Z$ is a subset system replacing the system $\mathcal D$ of all directed subsets (for which the $\mathcal D$-distributive complete lattices are just the continuous ones). In particular, it is shown that if $[X,Y]$ is a complete lattice then it is supercontinuous (i.e.\^^Mcompletely distributive) iff both $Y$ and $\mathcal O X$ are supercontinuous. Moreover, the Scott topology on $Y$ is the only one making that equivalence true for all spaces $X$ with completely distributive topology. On the way to these results, we find necessary and sufficient conditions for $[X,Y]$ to be complete, and some new, purely topological characterizations of continuous lattices by continuity conditions on their (infinitary) lattice operations. (English) |
| Keyword:
|
completely distributive lattice |
| Keyword:
|
continuous function |
| Keyword:
|
continuous lattice |
| Keyword:
|
Scott topology |
| Keyword:
|
subset system |
| Keyword:
|
$\mathcal Z$-continuous |
| Keyword:
|
$\mathcal Z$-distributive |
| MSC:
|
06B35 |
| MSC:
|
06D10 |
| MSC:
|
06F30 |
| MSC:
|
54F05 |
| MSC:
|
54H10 |
| idZBL:
|
Zbl 06260033 |
| idMR:
|
MR3136497 |
| DOI:
|
10.21136/MB.2013.143437 |
| . |
| Date available:
|
2013-09-14T11:47:17Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143437 |
| . |
| Reference:
|
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