Pointfree binary relations (a short article)
In a short (4 pages) article I define pointfree binary relations, a generalization of binary relations which does not use “points” (elements). In a certain special case (of endo-relations) pointfree binary relations are essentially the same as binary relations. It seems promising to research filters on sets of pointfree relations, generalizing the notion of reloids […]
A more abstract way to define reloids
We need a more abstract way to define reloids: For example filters on a set $latex A\times B$ are isomorphic to triples $latex (A;B;f)$ where $latex f$ is a filter on $latex A\times B$, as well as filters of boolean reloids (that is pairs $latex (\alpha;\beta)$ of functions $latex \alpha\in (\mathscr{P}B)^{\mathscr{P}A}$, $latex \beta\in (\mathscr{P}B)^{\mathscr{P}A}$ such […]
“Open maps between funcoids” rewritten
There were several errors in the section “Open maps” of my online book. I have rewritten this section and also moved the section below in the book text. However, the new proof of the theorem stating that composition of open maps between funcoids is an open map now uses a proof referring to a particular […]
Pointfree funcoids between join-semilattices conjecture
Today I’ve come up with the following easy to prove theorem (exercise!) for readers of my book: Theorem If there exists at least one pointfree funcoid from a poset $latex \mathfrak{A}$ to a poset $latex \mathfrak{B}$ then either both posets have least element or none of them. This provokes me to the following conjecture also: […]
My book is now available free of charge
Today I’ve took the bold decision to put my math research book online free (under Creative Commons license), with LaTeX source available for editing by anyone at a Git hosting. Because of conflict of licensing, it seems not that my book will be never published officially. However publishing in Git has some advantages: If I […]
Another definition of pointfree reloids
In previous post I stated that pointfree reloids can be defined as filters on pointfree funcoids. Now I suggest also an alternative definition of pointfree reloids: Pointfree reloids can be defined as filters on products $latex \mathrm{atoms}\,\mathfrak{A} \times \mathrm{atoms}\,\mathfrak{B}$ of atoms of posets $latex \mathfrak{A}$ and $latex \mathfrak{B}$. In the case if $latex \mathfrak{A}$ and […]
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 […]