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July 30, 2015 / porton

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!

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