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:

gollark: How's that?
gollark: Hold on.
gollark: How about:Create a new section "Bees" %bees.Create a rule "Bee utilization part 1" (%bees-1) in %bees:> If bees are deployed, they may be used against any player, if a Bee Poll indicating this target player is passed. The deployment status of bees is to be considered part of the Game State. If bees are used on a player they lose 1 point. Bees are not considered a resource and if they are deployed an unlimited amount of bee-related actions may be taken.Create a rule "Bee Poll" (%bee-poll) in %polls:> A Bee Poll is required to authorize bees to perform actions, as described in %bees. The default allowed reactions for a Bee Poll are 👍 (representing a vote for) and 👎 (representing a vote against). Bee Polls may be ended if they have existed for 12 hours, rather than the usual 24. When a Bee Poll ends, if there are more votes for the Bee Poll than against it, the Bee Poll passes. Players are permitted to use multiple reactions on a Bee Poll.
gollark: What? I'm going to just cancel the existing proposal and make one creating the bee section and bee rules section 1.
gollark: Wait, maybe it should create a bee *section* too.

See also

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. Zbl 1145.12001.

References
  • Howie, John M. (1995). Fundamentals of Semigroup Theory. Clarendon Press. ISBN 0-19-851194-9.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.