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 ( T , C o n , ⊢ ) {\displaystyle (T,Con,\vdash )} T is a set of tokens (the basic units of information) {\displaystyle T{\mbox{ is a set of tokens (the basic units of information)}}} C o n ⊆ P f ( T ) the finite subsets of T {\displaystyle Con\subseteq {\mathcal {P}}_{f}(T){\mbox{ the finite subsets of }}T} ⊢ ⊆ ( C o n ∖ { ∅ } ) × T {\displaystyle {\vdash }\subseteq (Con\setminus \lbrace \emptyset \rbrace )\times T} satisfying If a ∈ X ∈ C o n then X ⊢ a {\displaystyle {\mbox{If }}a\in X\in Con{\mbox{ then }}X\vdash a} If X ⊢ Y and Y ⊢ a , then X ⊢ a {\displaystyle {\mbox{If }}X\vdash Y{\mbox{ and }}Y\vdash a{\mbox{, then }}X\vdash a} If X ⊢ a then X ∪ { a } ∈ C o n {\displaystyle {\mbox{If }}X\vdash a{\mbox{ then }}X\cup \{a\}\in Con} ∀ a ∈ T : { a } ∈ C o n {\displaystyle \forall a\in T:\{a\}\in Con} If X ∈ C o n and X ′ ⊆ X then X ′ ∈ C o n .