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).

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:

1. If I want to add to the books something new, it is easy; no need for official edition-2.
2. It is easy to correct errors as soon as they are found and reported.

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 $\mathrm{atoms}\,\mathfrak{A} \times \mathrm{atoms}\,\mathfrak{B}$ of atoms of posets $\mathfrak{A}$ and $\mathfrak{B}$.

In the case if $\mathfrak{A}$ and $\mathfrak{B}$ are powerset lattices, this definition coincides with the definition of reloids (and with the definition of pointfree reloids given in the previous post).

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 $A$ and $B$ denoted $\mathsf{FCD}(A;B)$ are essentially the same as pointfree funcoids $\mathsf{pFCD}(\mathfrak{F}(A);\mathfrak{F}(B))$ (where $\mathfrak{F}(A)$ denotes filters on a set $A$).

Reloids between sets $A$ and $B$ denoted $\mathsf{RLD}(A;B)$ are essentially the same as filters $\mathfrak{F}(\mathbf{Rel}(A;B))$ (where $\mathbf{Rel}$ is the category of binary relations between sets.)

But, as I’ve recently discovered (see my book), $\mathbf{Rel}(A;B)$ is essentially the same as $\mathsf{pFCD}(\mathscr{P}A;\mathscr{P}B)$. So $\mathsf{RLD}(A;B) = \mathfrak{F}(\mathbf{pFCD}(\mathscr{P}A;\mathscr{P}B))$.

This way (for every posets $\mathfrak{A}$, $\mathfrak{A}$) $\mathfrak{F}\mathbf{pFCD}(\mathscr{A};\mathscr{B})$ corresponds to $\mathbf{pFCD}(\mathfrak{F}(\mathscr{A});\mathfrak{F}(\mathscr{B}))$ in the same way as $\mathsf{RLD}(A;B)$ corresponds to $\mathsf{FCD}(A;B)$. In other words, $\mathfrak{F}\mathbf{pFCD}(\mathscr{A};\mathscr{B})$ are the pointfree reloids corresponding to pointfree funcoids $\mathbf{pFCD}(\mathfrak{F}(\mathscr{A});\mathfrak{F}(\mathscr{B}))$.

Yeah!

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 $\mathfrak{A}$ to $\mathfrak{B}$ for every poset of pointfree funcoids between filters on $\mathfrak{A}$ and filters on $\mathfrak{B}$, by analogy with the filtrator of funcoids whose core is a set of binary relations (the same as a pointfree funcoids, by the above bijective correspondence). This way the theory of filtrators of funcoids generalizes for pointfree funcoids.

By the way, this bijective correspondence is a functor.