A cartesian closed subcategory of CONT which contains all continuous domains

Exponentials in a Cartesian Closed Category Which Contains all Algebraic Domains

“…Yang [9] introduced the relation ≺ on [P → Q] which is defined by: for any f, g ∈ [P → Q], f ≺ g if and only if f (x) g(y) whenever x y. In terms of the relation ≺, Yang [10] defined the concept of a long final ideal, which is a special kind of ideal on [P → Q]. He proved that the set of all long final ideals in [P → Q] ordered by set inclusion is a continuous domain and is the exponential from P to Q (denoted by Q P ) in the category CONT .…”

“…He proved that the set of all long final ideals in [P → Q] ordered by set inclusion is a continuous domain and is the exponential from P to Q (denoted by Q P ) in the category CONT . Definition 2.1 [10]. Suppose I ∈ I D([P → Q]).…”

“…From the point of view of preserving approximation of information, it is interesting to require the mapping to preserve the way-below relations. Zhongqiang Yang [10] considered the category CONT which includes all continuous domains and all mappings preserving directed suprema and the way-below relation and proved that the category CONT is Cartesian closed. Yang [9] remarked that the category ALG which includes all algebraic domains and all mappings preserving directed suprema and the way-below relation is Cartesian closed as it is isomorphic to the category of posets.…”

“…It is natural to ask if exponentials in ALG are the same as exponentials in the bigger category CONT . So Yang [10] posed the following problem:…”

“…In Section 4, we study the relationship between the interpolation property and the relation ≺ introduced by Yang [10].…”