In abstract algebra, a cover is one instance of some mathematical structure mapping onto another instance, such as a group (trivially) covering a subgroup.
The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances.
In order to be interesting, the cover is usually endowed with additional properties, which are highly dependent on the context.
A classic result in semigroup theory due to D. B. McAlister states that every inverse semigroup has an E-unitary cover; besides being surjective, the homomorphism in this case is also idempotent separating, meaning that in its kernel an idempotent and non-idempotent never belong to the same equivalence class.
If F is some family of modules over some ring R, then an F-cover of a module M is a homomorphism X→M with the following properties: In general an F-cover of M need not exist, but if it does exist then it is unique up to (non-unique) isomorphism.