A more abstract way to define reloids

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).

2 comments

  1. We can identify one-element relations as atoms of our poset. It remains to prove that our structure determines binary relations up to re-order (bijections) of variables x and y.

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