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Substructures and quotients in the Algebra.* hierarchy #1899

@jamesmckinna

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@jamesmckinna

The recent discussion in issue #1888 concerning the correct definition of Module drew my attention to the (almost?) complete (?) lack of any treatment of algebraic substructures, and the corresponding notions of 'things-which-give-rise-to-quotients':

  • submonoid of a monoid;
  • (normal) subgroup of a group;
  • (left- and right-) ideal of a (non-commutative) ring;
  • others? (eg do we care about submagmas? subsemirings etc.)

together with the associated 'free' things, viz.

  • the submonoid generated by a subset;
  • the subgroup generated by ...;
  • the ideal generated by...;
  • the terminal ('zero'?) (sub)object as the free thing on the empty type of generators...; EDITED apologies for my ignorance of Algebra.Construct.Zero which does define these (but none of the associated homomorphisms); see also PR initial+terminal algebras #1902
  • etc.

So this is (the beginnings of) a shopping list for the above, and some proposals for how to represent them.

A left- (resp. right-) ideal of a Ring R with Carrier given by A should be given by:

  • a left- (resp. right-) R-module, with carrier type I for representing the subset in question;
  • an injective map h : I -> A which is a left- (resp. right-) R-module homomorphism

TODO:

  • work out the details! (eg: injective map or monomorphism? are they same for Setoids? etc. plus: level issues?)
  • related matters: quotients, plus short/long exact sequences to characterise things?
  • what else?

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