A Maximal Monoidal Closed Category of Distributive Algebraic Domains

Abstract 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 objects, we realize a categorical semantics of linear logic in SUP. The multiplicatives are fully distributive w.r.t. the additives. Given PRIME, the full subcategory of BC with prime-algebraic dcpos as objects, we introduce a prime-algebraic quotient ΠA which preserves all the logical operations in SUP up to isomorphism. Therefore, if C is a categorical semantics in BC, then ΠC is a categorical semantics in PRIME. In particular, ΠSUP, the full subcategory of BC with all prime-algebraic lattices as objects, is such a categorical semantics. PRIME is symmetric monoidal closed and maximal with respect to being closed under [formula] and ⊥, if we demand that all objects are algebraic and distributive. Thus, ΠSUP is a maximal categorical semantics with respect to these conditions. We discuss the modalities !(—) and ?(—) in ΠSUP.