preorder of integers w.r.t. divisiblity

  • notation: (Z,∣)(\mathbb{Z},\mid)
  • objects: integers
  • morphisms: a unique morphism (a,b):a→b(a,b) : a \to b if aa divides bb
  • Related categories: (N,≤)(\mathbb{N},\leq)

This is a preorder, not a partial order, because aa and −a-a divide each other, but are not equal for a≠0a \neq 0. Notice that this category is equivalent (but not isomorphic) to (N,∣)(\mathbb{N},\mid).

Properties

Properties from the database

Deduced properties

Non-Properties

Non-Properties from the database

Deduced Non-Properties*

*This also uses the deduced properties.

Unknown properties

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Special morphisms

  • Isomorphisms: the identities (a,a):a→a(a,a) : a \to a and the isomorphisms (a,−a):a→−a(a,-a) : a \to -a for a∈Za \in \mathbb{Z}
  • Monomorphisms: every morphism
  • Epimorphisms: every morphism