  In this paper we intend to study implications in their most general form,
generalizing different classes of implications including the Heyting
implication, sub-structural implications and weak strict implications.
Following the topological interpretation of the intuitionistic logic, we will
introduce non-commutative spacetimes to provide a more dynamic and subjective
interpretation of an intuitionistic proposition. These combinations of space
and time are natural sources for well-behaved implications and we will show
that their spatio-temporal implications represent any other reasonable abstract
implication. Then to provide a faithful well-behaved syntax for abstract
implications, we will develop a logical system for the non-commutative
spacetimes for which we will present both topological and Kripke semantics.
These logics unify sub-structural and sub-intuionistic logics by embracing them
as their special fragments.
