### A more abstract way to define reloids

We need a more abstract way to define reloids:

For example filters on a set $latex A\times B$ are isomorphic to triples $latex (A;B;f)$ where $latex f$ is a filter on $latex A\times B$, as well as filters of boolean reloids (that is pairs $latex (\alpha;\beta)$ of functions $latex \alpha\in (\mathscr{P}B)^{\mathscr{P}A}$, $latex \beta\in (\mathscr{P}B)^{\mathscr{P}A}$ such that $latex y\sqcap \alpha x\neq\bot \Leftrightarrow x\sqcap \beta y\neq\bot$ (for all $latex x\in\mathscr{P}A$ and $latex y\in\mathscr{P}B$).

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 thoughts on “A more abstract way to define reloids”

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 $latex x$ and $latex y$.

2. Correction: It is not a semigroup, it is a precategory.