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Based on the definition taken from Foundations of Semantic Web Technologies by Hitzler et al, a role hierarchy is called regular if there exists a strict partial order ≺ on R, such that the following hold:

a) ○ S ≺ R if and only if S− ≺ R

b) ○ every role inclusion axiom is of one of the forms:

■ R ◦ R ⊑ R, R− ⊑ R, S1 ◦ · · · ◦ Sn ⊑ R, R ◦ S1 ◦ · · · ◦ Sn ⊑ R, S1 ◦ · · · ◦ Sn ◦ R ⊑ R

such that R is a non-inverse role name, and Si ≺ R for i = 1, . . . , n.

I do not understand what strict partial order on roles exactly means and how I should interpret the notation "≺" representing it. How should I read "S ≺ R"? I'd appreciate any sort of explanation/example.

Thanks in advance!

Sara
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    Strict partial orders are asymmetric transitive relations. The main outcome of the statement is that you can't do anything as long as you want to stay decidable. Roles itself are sets of pairs that's why you can define a partial order on those sets. That's basically all. – UninformedUser Jun 05 '19 at 12:00
  • Intuitively, there should be no non-trivial cyclic relationships between roles. – UninformedUser Jun 05 '19 at 12:07
  • Thanks for the response! Yes, I understand the motivation behind it, but I don't know if I'm understanding "S ≺ R" right in the context of roles; e.g. is it right to interpret "≺" as subrole relation? i.e. can I say: _italic_S SubroleOf R iff S- SubroleOf R_italic_ – Sara Jun 05 '19 at 12:14
  • I am confused as how to interpret "≺" when it comes to roles, in a nutshell. – Sara Jun 05 '19 at 12:14
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    yes, it ok to think of subsumption resp. subrole here. I mean, in the end a role `r` does nothing more then denoting a set of pairs `(x,y)` with being in the interpretation of `r`, i.e. `x r y` – UninformedUser Jun 05 '19 at 12:35
  • Thanks a lotAKSW! – Sara Jun 05 '19 at 17:39

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