We consider abstract interpretation, and in particular the basic operators of reduced product and complementation (see here) of abstract domains, as a tool to systematically derive denotational semantics by composition and decomposition. Reduced product allows to perform the logical conjunction of semantics, while complementation characterizes what is left from a semantics when the information concerning a given observable property is ``subtracted'' from it. We apply this idea to the case of logic programming, characterizing in a uniform algebraic setting, the interaction between a number of well known declarative semantics for logic programs.