structure homomorphism (original) (raw)

Let Σ be a fixed signature, and 𝔄 and 𝔅 be two structures for Σ. The interesting functions from 𝔄 to 𝔅 are the ones that preserve the structure.

A function f:𝔄→𝔅 is said to be a homomorphism (or simply morphism) if and only if:

1. 1. For every constant symbol c of Σ, f⁢(c𝔄)=c𝔅.
2. 1. For every natural number n and every n-ary function symbol F ofΣ,

      f⁢(F𝔄⁢(a1,…,an))=F𝔅⁢(f⁢(a1),…,f⁢(an)).

3. 1. For every natural number n and every n-ary relation symbol Rof Σ,

      R𝔄⁢(a1,…,an)⇒R𝔅⁢(f⁢(a1),…,f⁢(an)).

Homomorphisms with various additional properties have special names: