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Kernels and images in Algebra #1966

@jamesmckinna

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@jamesmckinna

Discussing @Taneb 's most recent comments on PR #1767 , it is clear that we are (also!) missing some fairly crucial definitions from the Algebra library/hierarchy: ie given a XHomomorphism h, define

  • kernel, ie via the property InKernel = λ x → h x ≈ 0 (suitably interpreted)
  • image, ie via the property InImage = λ y → ∃ x → h x ≈ y

together with their evident closure properties as substructures, cf. issue #1899 .

Note also that while we have definitions of structures (but not yet bundles! #1960) defining IsXMonomorphism and IsXIsomorphism based on Function.Definitions.Injective and Function.Definitions.Core2.Surjective, these might better (best?) be rephrased/proved equivalent to, properties involving the kernel and image, eg.

  • IsXMonomorphism h iff ∀ {x} → InKernel h x → x ≈ 0, etc.

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