The predicate completion of a partial information system

Abstract

Abstract Originally, partial information systems were introduced as a means of providing a representation of the Smyth powerdomain in terms of order convex substructures of an information-based structure. For every partial information system 𝕊, there is a new partial information system that is natrually induced by the principal lowersets of the consistency predicate for 𝕊. In this paper, we show that this new system serves as a completion of the parent system 𝕊 in two ways. First, we demonstrate that the induced system relates to the parent system 𝕊 in much the same way as the ideal completion of the consistency predicate for 𝕊 relates to the consistency predicate itself. Second, we explore the relationship between this induced system and the notion of D-completions for posets. In particular, we show that this induced system has a “semi-universal” property in the category of partial information systems coupled with the preorder analog of Scott-continuous maps that is induced by the universal property of the D-completion of the principal lowersets of the consistency predicate for the parent system 𝕊.