MAXIMAL MONOIDAL CLOSED CATEGORY OF DISTRIBUTIVE ALGEBRAIC DOMAINS

We study the category BC of bounded complete dcpos and maps preserving all suprema (linear maps). BC is a symmetric monoidal closed category . If SUP denotes the full subcategory of BC with dcpos with one as obj ects, we realize a categorical semantics of linear logic in SUP. The m ultiplicatives are fully distributive w.r.t. the additives. Given PRIM E, the full subcategory of BC with prime-algebraic dcpos as objects, w e introduce a prime-algebraic quotient product A which preserves all t he logical operations in SUP up to isomorphism. Therefore. if C is a c ategorical semantics in BC, then product C is a categorical semantics in PRIME. In particular, product SUP, the full subcategory of BC with all prime-algebraic lattices as objects, is such a categorical semanti cs. PRIME is symmetric monoidal closed and maximal with respect to bei ng closed under i and perpendicular to, if we demand that all objects are algebraic and distributive. Thus, product SUP is a maximal categor ical semantics with respect to these conditions. We discuss the modali ties !(_) and ?(_) in product SUP. (C) 1995 Academic Press, Inc.