ALGEBRAIC DOMAINS OF NATURAL TRANSFORMATIONS

Motivated by the semantics of polymorphic programming languages and ty ped lambda-calculi, by formal methods in functor category semantics, a nd by well-known categorical and domain-theoretical constructs, we stu dy domains of natural transformations F --> G of functors F, G:OMEGA - -> C with a small category OMEGA as source and a cartesian closed cate gory of Scott-domains C as target. We put constraints on the image arr ows of the functors to obtain that F --> G is an object in C. Inf-fait hful domains F --> G allow that infima in F --> G can be computed in e ach component [FA --> GA] separately. If F, G:OMEGA --> SCOTT are two functors such that for all f in mor(OMEGA) the maps F (f) preserve fin ite elements and G(f) preserve all nonempty infima, then F --> G is in f-faithful, and all inf-faithful domains are Scott-domains. Familiar n otions like ''inverse limits'', ''small products'', and ''strict funct ion spaces'' are special instances of functors that meet the condition s above. We extend these results to retracts of Scott-domains.