I’ve rewritten my book in LyX
Previously I wrote my research monograph with TeXmacs word processors. TeXmacs is a very good program. However annoying bugs of TeXmacs (incorrect file “saved” status, failure to work well when multiple windows with the same document are opened, etc.) and also its slowness when working with a long (300 pages) document, forced me to switch […]
An example of a separable poset with certain property
With help of sci.math crowd it was demonstrated an example that there exist a (finite) poset, which is separable (in the sense defined in this book), but $latex \star x \subseteq \star y$ does not imply $latex x \sqsubseteq y$ (where $latex \sqsubseteq$ denotes our order) for elements $latex x$, $latex y$ of this poset. […]
Modular hyperlinked mathematics (as a replacement of books)
The following are suggestions how to replace math books with hyperlinked modules, in general, and particularly suggestions how to support it in a future version of TeXmacs software. TeXmacs is an advanced math writing software, and the easiest way to implement my ideas are to modify this existing software, rather than writing a completely new […]
About topological structures corresponding to partial order
Intuitively (not in the sense of comparing cardinalities, but in some other sense), the set of natural numbers is less than the set of whole numbers, which is less than the set of rational numbers, which is less than the set of real numbers, which is less than the set of complex numbers. First, we […]
Generalized Fréchet filters
Just a few minutes ago I conceived a definition of generalized Fréchet filters with definition for every poset on which filters are considered (however, I have not yet calculated the class of posets for which generalized Fréchet filter is defined; it should be easy but I am busy with other business). Generalized Fréchet filter on […]
Proximities are reflexive, symmetric, transitive funcoids
I’ve done a little discovery today: Proximities are the same reflexive, symmetric, transitive funcoids. For now I leave to prove this as an exercise for a reader. But later I am going to include this theorem into the book I am writing.
I withdrew my article from a journal
My article was accepted for publication in European Journal of Pure and Applied Mathematics, but it didn’t compile with their LaTeX templates. After waiting a reasonable time until they would tackle the problem, I have withdrawn my article and sent it to another journal. I would search for the bug in their LaTeX template myself, […]
Another star-category of funcoids
I’ve introduced another version of cross-composition of funcoids. This forms a category with star-morphisms. It is conjectured that this category is quasi-invertible, because I have failed to prove it. This should be included in the next version of my book.
The set of funcoids is a co-frame (without axiom of choice)
A mathematician named Todd Trimble has helped me to prove that the set of funcoids between two given sets (and more generally certain pointfree funcoids) is always a co-frame. (I knew this for funcoids but my proof required axiom of choice, while Todd’s does not require axiom of choice.) He initially published his proof here […]
Lattices ComplFCD and ComplRLD are co-brouwerian
I’ve found today earlier stated conjecture that lattices $latex \mathrm{Compl}\mathsf{FCD}(A;B)$ and $latex \mathrm{Compl}\mathsf{RLD}(A;B)$ are co-brouwerian. Exercise: Prove this fact.