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Cover (algebra)

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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 : XY. 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.; 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.


If F is some family of modules over some ring R, then an F-cover of a module M is a homomorphism XM with the following properties:

  • X is in the family F
  • XM 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
  • References

    Cover (algebra) Wikipedia

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