Given a type $T$, the partial function classifier $T_\bot$ is inductively generated by
and the partial order type family $\leq$ is simultaneously inductively generated by
a family of dependent terms
representing that each type $a \leq b$ is a proposition.
a family of dependent terms
representing the reflexive property of $\leq$.
a family of dependent terms
representing the transitive property of $\leq$.
a family of dependent terms
representing the anti-symmetric property of $\leq$.
a family of dependent terms
representing that $\bot$ is initial in the poset.
a family of dependent terms
a family of dependent terms
representing that denumerable/countable joins exist in the poset.
Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type (abs:1610.09254)
Martin Escardó, Cory Knapp, Partial Elements and Recursion via Dominances in Univalent Type Theory (pdf)