### 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…

### 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…

### “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…

### Offtopic: John 3:16 – Christ having an universal property

Bible, John 3:16: (CJB) “For God so loved the world that he gave his only and unique Son, so that everyone who trusts in him may have eternal life, instead of being utterly destroyed.” (ISV) “For this is how God loved the…

### The set of ideals on an infinite join-semilattice is a boolean algebra

The below is wrong, because pointfree funcoids between boolean algebras are not the same as 2-staroids between boolean algebras. It was an error. I have just discovered that the set of ideals on an infinite join-semilattice is a boolean algebra (moreover it…

### 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…

### Regular funcoids, a generalization of regular topospaces (rewritten because of an error)

Both my definition and description of properties of regular funcoids were erroneous. (The definition was not compatible with the customary definition of regular topological spaces due an error in the definition, and its properties included mathematical errors.) I have rewritten the erroneous…

### 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…

### 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…

### 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…