This conjecture appeared to be false.

Now I propose an alternative conjecture:

Let $A$, $B$ be sets.

Conjecture Funcoids $f$ from $A$ to $B$ bijectively corresponds to the sets $R$ of pairs
$(\mathcal{X}; \mathcal{Y})$ of filters (on $A$ and $B$ correspondingly) that

1. $R$ is nonempty.
2. $R$ is a lower set.
3. every $\left\{ \mathcal{X} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}$ is a dcpo for every $\mathcal{Y} \in \mathfrak{F}B$ and every $\left\{ \mathcal{Y} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}$ is a dcpo for every $\mathcal{X} \in \mathfrak{F}A$

by the mutually inverse formulas:
$(\mathcal{X} ; \mathcal{Y}) \in R \Leftrightarrow \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \sqsubseteq f \quad \mathrm{and} \quad f = \bigsqcup^{\mathsf{FCD}} \left\{ \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}$.

Just a few minutes ago I’ve formulated a new important conjecture about funcoids:

Let $A$, $B$ be sets.

Conjecture Funcoids $f$ from $A$ to $B$ bijectively corresponds to the sets $R$ of pairs
$(\mathcal{X}; \mathcal{Y})$ of filters (on $A$ and $B$ correspondingly) that

1. $R$ is nonempty.
2. $R$ is a lower set.
3. $R$ (ordered pointwise) is a dcpo

by the mutually inverse formulas:
$(\mathcal{X} ; \mathcal{Y}) \in R \Leftrightarrow \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \sqsubseteq f \quad \mathrm{and} \quad f = \bigsqcup^{\mathsf{FCD}} \left\{ \mathcal{X} \times^{\mathsf{FCD}} \mathcal{Y} \mid (\mathcal{X} ; \mathcal{Y}) \in R \right\}$.

Conjecture Join of a set $S$ on the lattice of transitive reloids is the join (on the lattice of reloids) of all compositions of finite sequences of elements of $S$.

It was expired by theorem 2.2 in “Hans Weber. On lattices of uniformities”.

There is a similar conjecture for funcoids (instead of reloids).

The terminology used is from my free ebook.

I have added a new section “Properties preserved by relationships” to my math research book.

This section considers (in the form of theorems and conjectures) whether properties (reflexivity, symmetry, transitivity) of funcoids and reloids are preserved an reflected by their relationships (functions $(\mathsf{FCD})$, $(\mathsf{RLD})_{\mathrm{in}}$, $(\mathsf{RLD})_{\mathrm{out}}$ which map between funcoids and reloids).

I’ve released a new version of my free math ebook.

The main feature of this new release is chapter “Alternative representations of binary relations” where I essentially claim that the following are the same:

• binary relations
• pointfree funcoids between powersets
• Galois connections between powersets
• antitone Galois connections between powersets

This theorem is presented with a commutative diagram.

I have added also some other minor theorems.

I introduce a new math abstraction, categories of sides, in order to generalize two theorems into one.

Category of sides $\Upsilon$ is an ordered category whose objects are (small) bounded lattices and whose morphisms are maps between lattices such that every Hom-set is a bounded lattice and (for all relevant variables):

1. $(a \sqcup b) X = a X \sqcup b X$
2. $(a \sqcap b) X \sqsubseteq a X \sqcap b X$
3. $(\lambda x \in \mathfrak{A}: x \sqcap c) \in \Upsilon (\mathfrak{A}; \mathfrak{A})$ for every $c \in \mathfrak{A}$
4. $a \bot = \bot$
5. $\top X = \top$ unless $X = \bot$

I call morphisms of such categories sides.

The category of pointfree funcoids between boolean lattices is a category of sides. Also it seems (not checked yet) that the category of Galois connections between boolean lattices is a category of sides.

This way, it seems that I’ve found a common generalization of two theorems:

Theorem For category of pointfree funcoids, the set of morphisms between a non-atomic boolean lattice and itself is not a boolean lattice.

Theorem For category of Galois connections, the set of morphisms between a non-atomic boolean lattice and itself is not a boolean lattice.

The last theorem is a slight reformulation of theorem 3.8 in “Zahava Shmuely. The tensor product of distributive lattices. algebra universalis, 9(1):281–296.” (I borrowed the proof idea from that Zahava’s article.)

Common generalization:

Theorem For every category of sides, the set of morphisms between a non-atomic boolean lattice and itself is not a boolean lattice.

It is also conceivable to define pointfree reloids as filers on a (fixed) category of sides.

Note that the definition of “categories of sides” is preliminary, I may probably add more axioms in the future, if found convenient.

I have proved the following negative result:

Theorem $\mathsf{pFCD} (\mathfrak{A};\mathfrak{A})$ is not boolean if $\mathfrak{A}$ is a non-atomic boolean lattice.

The theorem is presented in this file.

$\mathsf{pFCD}(\mathfrak{A};\mathfrak{B})$ denotes the set of pointfree funcoids from a poset $\mathfrak{A}$ to a poset $\mathfrak{B}$ (see my free ebook).

The theorem and its proof were modeled after theorem 3.8 in this article (December 1979) by Zahava Shmuely.

It would be probably interesting to a find a common generalization of my theorem and the original Zahava Shmuely’s one.