# 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 *f* : *X* → *Y*. 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.^{[1]} McAlister's covering theorem generalizes to orthodox semigroups: every orthodox semigroup has a unitary cover.^{[2]}

Examples from other areas of algebra include the Frattini cover of a profinite group^{[3]} 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:

*X*is in the family*F**X*→*M*is surjective- Any surjective map from a module in the family
*F*to*M*factors through*X* - Any endomorphism of
*X*commuting with the map to*M*is an automorphism.

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:

- Projective covers (always exist over perfect rings)
- flat covers (always exist)
- torsion-free covers (always exist over integral domains)
- injective covers

## See also

## Notes

- ↑ Lawson p. 230
- ↑ Grilett p. 360
- ↑ Fried, Michael D.; Jarden, Moshe (2008).
*Field arithmetic*. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.**11**(3rd revised ed.). Springer-Verlag. p. 508. ISBN 978-3-540-77269-9. Zbl 1145.12001.

## References

- Howie, John M. (1995).
*Fundamentals of Semigroup Theory*. Clarendon Press. ISBN 0-19-851194-9.