Timeline for Was lattice theory central to mid-20th century mathematics?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 8, 2014 at 6:30 | comment | added | user46855 | Updated the version on my site with minor corrections, more side notes, and more references | |
| Feb 15, 2014 at 14:50 | comment | added | user46855 | a frame has a canonically unique T$_0$ realization iff every meet-irreducible is completely meet-irreducible and these meet-generate the lattice iff it has a sober T$_D$ realization. (b) what happens to the main selling point, the choiceless compactness of infinite products of compact spaces? one must lose either the correct inclusion of topological spaces (inventing a infinite product of complete atomic boolen algebras which is no more so) or lose the selling point. So a distributive only study of such pointless spaces seems improbable. | |
| Feb 15, 2014 at 14:46 | comment | added | user46855 | In the distributive case, this might be less useful since (a) the main point of frames is not to be pointless, but to be point-independent. For the classical algebraic geometry of the polynomial ring $K[x,y,\dots,z]$ ($K$ algebraically closed field) this means to study common properties of the minimal topological realization (with only the classical points:the maximal spectrum) and the maximal realization (its sobrification, i.e. all irreducible subvarieties are "points", i.e. prime spectrum). On the contrary, for T$_2$ spaces there is no ambiguity: [continues] | |
| Feb 15, 2014 at 14:43 | comment | added | user46855 | the "pointful" case of orthogonality between projective points (vectorial lines) in a Hilbert space is generalized (to cover all cases of operator algebras with many projections) by meet-continuos geometries with a suitable orthogonality (Redei's pubbications of von Neumann's letters show that, to save modularity, he briefly considered the possibility of a intuitionistic-like negation in a infinite dimensional projective geometry. See also the "projectales" introduced in a more categorical language by G. Khatcherian sciencedirect.com/science/article/pii/002240499190082D [Continues] | |
| Feb 15, 2014 at 14:41 | comment | added | user46855 | Thank you for the interest. My remark about complete boolean algebras with a subframe had two objectives: (1) to introduce a case where there is no categorically adjoint contsruction, but many inequivalent "pseudoinverses" usefully exists; (2) to prepare for the nondistributive case, where it is more useful. See mathoverflow.net/questions/7250/… and mathoverflow.net/questions/152249/… [continue ...] | |
| Feb 12, 2014 at 16:48 | comment | added | Manny Reyes | Thank you for putting so much time into an extremely interesting and detailed essay! One point that I find especially intriguing is the idea of "pointless topological spaces" as complete Boolean algebras endowed with a subframe. Does this lead to a noticeably different theory than that of frames/locales? Do you know of any literature exploring this idea? | |
| Feb 12, 2014 at 15:25 | comment | added | Ryan Reich | +1 for finding a connection between "lattices" and "lattices", not to mention all your other comments. | |
| Feb 12, 2014 at 13:43 | history | answered | user46855 | CC BY-SA 3.0 |