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.

## Contents

* 1 Examples
* 2 Modules
* 3 See also
* 4 Notes
* 5 References

## 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

* Embedding

## Notes

1. ↑ Lawson p. 230
2. ↑ Grilett p. 360
3. ↑ 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.

## References

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

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