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 (that is filters on cartesian products) from my my research book

We need a more abstract way to define reloids:

For example filters on a set $A\times B$ are isomorphic to triples $(A;B;f)$ where $f$ is a filter on $A\times B$, as well as filters of boolean reloids (that is pairs $(\alpha;\beta)$ of functions $\alpha\in (\mathscr{P}B)^{\mathscr{P}A}$, $\beta\in (\mathscr{P}B)^{\mathscr{P}A}$ such that $y\sqcap \alpha x\neq\bot \Leftrightarrow x\sqcap \beta y\neq\bot$ (for all $x\in\mathscr{P}A$ and $y\in\mathscr{P}B$).

I propose a way to encompass all ways to describe reloids as follows:

Let call a filtrator of pointfree reloid a pair of a filtrator and an associative operation on its core. Then call abstract reloids pointfree reloids isomorphic (both a filtrators and as semigroups) to reloids.

I am yet unsure that this structure encompasses all essential properties of reloids (just like as primary filtrators encompass all properties of filters on posets).

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 point $x$.

It is currently an open problem to rewrite this proof in pointfree style.

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 world: He gave his uniquely existing Son so that everyone who believes in him would not be lost but have eternal life.”

I think the word “unique” here should read “having universal property” (as universal properties defined in mathematics, specifically category theory).

There are many children of God, but they are children of God only through “universal” son of God that is Christ.

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 is a complete atomistic boolean algebra).

For me, it is a very counter-intuitive theorem (after all the set of ideals “should” not be boolean algebra, except of the finite case). Is it worth to call it a paradox?

http://math.stackexchange.com/questions/1475328/when-ideals-are-a-boolean-algebra

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 $\mathfrak{A}$ to a poset $\mathfrak{B}$ then either both posets have least element or none of them.

This provokes me to the following conjecture also:

Conjecture If there exists at least one pointfree funcoid from a poset $\mathfrak{A}$ to a poset $\mathfrak{B}$, then either both or none of these two posets are join-semilattices.

If the conjecture comes up true, it would allow some simplification of some theorem conditions in my book, as there would no more necessity to claim that both source and destination are join-semilattices as I use in some of my theorems.

I do not expect that this conjecture will be particularly difficult, I have not yet invested my time into solving it.

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 section of my book.

Now it is shown that being regular for a funcoid $f$ is equivalent to each of the following formulas:

1. $\mathrm{Compl}\,(f \circ f^{-1} \circ f) \sqsubseteq \mathrm{Compl}\,f$.
2. $\mathrm{Compl}\,(f \circ f^{-1} \circ f) \sqsubseteq f$.

These formulas seem not being an example of math beautify. So I suspect that the traditional definition of regular topospaces should be amended (or rather not to produce a terminology conflict, replaced with an other algebraically more elegant concept).