The first appearance of separoid that I could find is from this paper from 2001. It defines the separoid, in Definition 1.1., as follows:
Given a ternary relation $\cdot\perp\kern-5pt\perp\cdot\vert\cdot$ on $\mathcal{S}$, we call $\perp\kern-5pt\perp$ a separoid (on ($\mathcal{S}$, $\leqslant$)), or the triple ($\mathcal{S}$, $\leqslant$,$\perp\kern-5pt\perp$) a separoid, if:
- S1: ($\mathcal{S}$, $\leqslant$) is a join semilattice,
and
- P1: $x\perp\kern-5pt\perp y\vert x$,
- P2: $x\perp\kern-5pt\perp y\vert z\Rightarrow y\perp\kern-5pt\perp x\vert z$,
- P3: $x\perp\kern-5pt\perp y\vert z\ \text{and}\ w\leqslant y\Rightarrow x\perp\kern-5pt\perp w\vert z$,
- P4: $x\perp\kern-5pt\perp y\vert z\ \text{and}\ w\leqslant y\Rightarrow x\perp\kern-5pt\perp y\vert(z\lor w)$,
- P5: $x\perp\kern-5pt\perp y\vert z\ \text{and}\ x\perp\kern-5pt\perp w\vert(y\lor z)\Rightarrow x\perp\kern-5pt\perp (y\lor w)\vert z$.
This definition can also be seen in the author's later papers, such as this.
However, in other materials such as wekipedia and other papers (1, 2, ...), the definition is different from the above. The following is from wikipedia:
A separoid is a set $S$ endowed with a binary relation $\vert\subseteq 2^S\times 2^S$ on its power set, which satisfies the following simple properties for $A,B\subseteq S$:
$A\vert B\iff B\vert A$
$A\vert B\Rightarrow A\cap B=\emptyset$
$A\vert B\ \text{and}\ A'\subset A \Rightarrow A'\vert B$
I could see that two authors are claiming different definitions for separoid. Still, since they have the same name, I think there should be some kind of relation between those two. Maybe one definition covers the other, but I'm really not sure.
So my question is, what is the relation between those two definitions, and which of the two is the definition I should adopt?