# 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 $A$ 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!