Scott information system

In domain theory, a branch of mathematics and computer science, a Scott information system is a primitive kind of logical deductive system often used as an alternative way of presenting Scott domains.
Definition.
A Scott information system, "A", is an ordered triple formula_1
satisfying
Here formula_10 means formula_11
Examples.
Natural numbers.
The return value of a partial recursive function, which either returns a natural number or goes into an infinite recursion, can be expressed as a simple Scott information system as follows:
That is, the result can either be a natural number, represented by the singleton set formula_15, or "infinite recursion," represented by formula_16.
Of course, the same construction can be carried out with any other set instead of formula_17.
Propositional calculus.
The propositional calculus gives us a very simple Scott information system as follows:
Scott domains.
Let "D" be a Scott domain. Then we may define an information system as follows
Let formula_25 be the mapping that takes us from a Scott domain, "D", to the information system defined above.
Information systems and Scott domains.
Given an information system, formula_26, we can build a Scott domain as follows. 
Let formula_30 denote the set of points of "A" with the subset ordering. formula_30 will be a countably based Scott domain when "T" is countable. In general, for any Scott domain "D" and information system "A"
where the second congruence is given by approximable mappings.