Cover (algebra)

In abstract algebra, a cover is one instance of some mathematical structure mapping onto another instance, such as a group (trivially) covering a subgroup. This should not be confused with the concept of a cover in topology.
When some object "X" is said to cover another object "Y", the cover is given by some surjective and structure-preserving map . 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.
Examples.
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.; something slightly stronger has actually be shown for inverse semigroups: every inverse semigroup admits an F-inverse cover. McAlister's covering theorem generalizes to orthodox semigroups: every orthodox semigroup has a unitary cover.
Examples from other areas of algebra include the Frattini cover of a profinite group and the universal cover of a Lie group.
Modules.
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. 
Examples include: