proset of integers w.r.t. divisibility

  • 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 proset, not a poset, 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).

Satisfied Properties

Properties from the database

Deduced properties

Unsatisfied Properties

Properties from the database

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

—

Special objects

  • terminal object: 00
  • initial object: 11
  • products: greatest common divisor
  • coproducts: the least common multiple, can be 00 for infinite families

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
  • regular monomorphisms: same as isomorphisms
  • regular epimorphisms: same as isomorphisms