It seems like what you have in mind is components of the sentence. (In semantics, "implications" refers to a broader range of things: all the propositions that must be true if a given sentence is true.)
The phenomenon you're describing is known as "compositionality": that the syntax and the meaning of the parts combine to yield the meaning of the whole.
Since that process is called "composition", we might call the reverse process "decomposition".
Note that in modern syntax and semantics, there's no guarantee that all the elements that appear to transparently combine actually do. The underlying structures might not match the surface ones.
John and Derek love cake.
⇒ John loves cake.
⇒ Derek loves cake.
John and Derek met.
⇏ * John met.
⇏ * Derek met.