homomorphism between algebraic systems
Let (A,O),(B,O) be two algebraic systems with operator set O. Given operators ωA on A and ωB on B, with ω∈O and n= arity of ω, a function f:A→B is said to be compatible with ω if
f(ωA(a1,…,an))=ωB(f(a1),…,f(an)). |
Dropping the subscript, we now simply identify ω∈O as an operator for both algebras A and B. If a function f:A→B is compatible with every operator ω∈O, then we say that f is a homomorphism
from A to B. If O contains a constant operator ω such that a∈A and b∈B are two constants assigned by ω, then any homomorphism f from A to B maps a to b.
Examples.
-
1.
When O is the empty set
, any function from A to B is a homomorphism.
-
2.
When O is a singleton consisting of a constant operator, a homomorphism is then a function f from one pointed set (A,p) to another (B,q), such that f(p)=q.
-
3.
A homomorphism defined in any one of the well known algebraic systems, such as groups, modules, rings, and lattices (http://planetmath.org/Lattice
) is consistent with the more general definition given here. The essential thing to remember is that a homomorphism preserves constants, so that between two rings with 1, both the additive identity 0 and the multiplicative identity
1 are preserved by this homomorphism. Similarly, a homomorphism between two bounded lattices (http://planetmath.org/BoundedLattice) is called a {0,1}-lattice homomorphism (http://planetmath.org/LatticeHomomorphism) because it preserves both 0 and 1, the bottom and top elements of the lattices.
Remarks.
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•
Like the familiar algebras, once a homomorphism is defined, special types of homomorphisms can now be named:
-
–
a homomorphism that is one-to-one is a monomorphism
;
-
–
an onto homomorphism is an epimorphism
;
-
–
an isomorphism
is both a monomorphism and an epimorphism;
-
–
a homomorphism such that its codomain is its domain is called an endomorphism;
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–
finally, an automorphism is an endomorphism that is also an isomorphism.
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–
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•
All trivial algebraic systems (of the same type) are isomorphic.
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•
If f:A→B is a homomorphism, then the image f(A) is a subalgebra
of B. If ωB is an n-ary operator on B, and c1,…,cn∈f(A), then ωB(c1,…,cn)=ωB(f(a1),…,f(an))=f(ωA(a1,…,an))∈f(A). f(A) is sometimes called the homomorphic image of f in B to emphasize the fact that f is a homomorphism.
Title | homomorphism between algebraic systems |
Canonical name | HomomorphismBetweenAlgebraicSystems |
Date of creation | 2013-03-22 15:55:36 |
Last modified on | 2013-03-22 15:55:36 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 8 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 08A05 |
Defines | compatible function |
Defines | homomorphism |
Defines | monomorphism |
Defines | epimorphism |
Defines | endomorphism |
Defines | isomorphism |
Defines | automorphism |
Defines | homomorphic image |