The first requirement of a generalized role hierarchy to be regular is as follows:
$S \prec R$ if and only if $S^{-} \prec R$.
The above is used in the context of $\mathcal{SROIQ}$ RBoxes.
(For reference, please see page 168 of this book or page 83 of this book.)
Here, a role (or a relation) can be explained with an example.
The expression
hasAffiliation(rudiStuder, aifb)
describes that rudiStuder is affiliated with aifb. Here, hasAffiliation
is a role. A role is sometimes referred to as a relation or property in some literature.
If we name the inverse of role hasAffiliation as employs, we can express the same fact using the inverse role as,
employs(aifb, rudiStuder)
I was wondering whether somebody can put forth a real-life example where both a role (relation) $S$ and its inverse $S^{-}$ maintain strict partial order with another role (relation) $R$.