Admissible relation

In programming language semantics, a relation between two models of a programming language is admissible when it is well-behaved with respect to recursion - in a domain-theoretic sense.

Suppose we have partial orders (D, \sqsubseteq_D) and (E, \sqsubseteq_E) which are complete. (This could be either chain-complete, or directed-complete, but we will choose chain-completeness here for simplicity.) Moreover, suppose they have least elements \bot_D and \bot_E respectively.

A relation R \subseteq D \times E is admissible when

• it is strict, i.e. \bot_D \mathbin{R} \bot_E, and also
• it is chain-closed, i.e. if we have chains d_0 \sqsubseteq_D d_1 \sqsubseteq \dots and e_0 \sqsubseteq_E e_1 \sqsubseteq \dots with d_i \mathbin{R} e_i for all i, then (\bigsqcup_i d_i) \mathbin{R} (\bigsqcup_i e_i).