Pointfree reloids discovered
After I defined pointfree funcoids which generalize funcoids (see my draft book) I sought for pointfree reloids (a suitable generalization of reloids, see my book) long time. Today I have finally discovered pointfree reloids. The idea is as follows: Funcoids between sets $latex A$ and $latex B$ denoted $latex \mathsf{FCD}(A;B)$ are essentially the same as […]
Binary relations are essentially the same as pointfree funcoids between powersets
After this Math.StackExchange question I have proved that binary relations are essentially the same as pointfree funcoids between powersets. Full proof is available in my draft book. The most interesting aspect of this is that is that we can construct filtrator with core being pointfree funcoids from $latex \mathfrak{A}$ to $latex \mathfrak{B}$ for every poset […]
Ideals, free stars, and mixers in my book
I wrote a section on ideals, free stars, and mixers in my book. Now free stars (among with ideals and mixers) are studied as first class objects, being shown isomorphic to filters on posets. In (not so far) future it should allow to extend our research of pointfree funcoids and staroids/multifuncoids, using now comprehensive theory […]
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 […]
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 […]
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.