structure homomorphism
Let be a fixed signature, and and be two structures
for . The interesting functions from to are the ones that preserve the structure.
A function is said to be a homomorphism (or simply morphism
) if and only if:
- 1.
For every constant symbol of , .
- 2.
For every natural number
and every -ary function symbol of,
- 3.
For every natural number and every -ary relation symbol of ,
Homomorphisms with various additional properties have special names:
- •
An injective
(http://planetmath.org/Injective) homomorphism is called a monomorphism
.
- •
A surjective
homomorphism is called an epimorphism
.
- •
A bijective
homomorphism is called a bimorphism
.
- •
An injective homomorphism is called an embedding if, for every natural number and every -ary relation symbol of ,
the converse
of condition 3 above, holds.
- •
A surjective embedding is called an isomorphism
.
- •
A homomorphism from a structure to itself (e.g. (http://planetmath.org/Eg), ) is called an .
- •
An isomorphism from a structure to itself is called an automorphism.
Title | structure homomorphism |
Canonical name | StructureHomomorphism |
Date of creation | 2013-03-22 12:43:22 |
Last modified on | 2013-03-22 12:43:22 |
Owner | almann (2526) |
Last modified by | almann (2526) |
Numerical id | 14 |
Author | almann (2526) |
Entry type | Definition |
Classification | msc 03C07 |
Synonym | homomorphism |
Synonym | morphism |
Synonym | monomorphism |
Synonym | epimorphism |
Synonym | bimorphism |
Synonym | embedding |
Synonym | isomorphism |
Synonym | endomorphism |
Synonym | automorphism |
Related topic | AxiomaticTheoryOfSupercategories |
Defines | structure morphism |
Defines | structure monomorphism |
Defines | structure epimorphism |
Defines | structure bimorphism |
Defines | structure embedding |
Defines | structure isomorphism |
Defines | structure endomorphism |
Defines | structure automorphism |