Scott domain

In denotational semantics that use domain theory, a Scott domain can mean whatever we want. This is often guided by the properties of the orders used for our particular purpose.

• For most purposes we can define a Scott domain to be a

  1. countably algebraic,
  2. consistently complete cpo.

  This is equivalent to having a cpo that is

  1. bifinite/SFP, and
  2. having continuous binary infima.

  This class of Scott domains is closed under most type constructors, except the Plotkin powerdomain.

• If we are studying nondeterminism powerdomains, then we can define a Scott domain to be a bifinite domain (aka an SFP domain). This is because SFP domains are closed under all nice constructors, including all three powerdomain constructors.

• If we are studying PCF, then we can define a Scott domain to be a coherently complete cpo.