homomorphism between algebraic systems
Let $(A,O),(B,O)$ be two algebraic systems with operator set $O$. Given operators ${\omega}_{A}$ on $A$ and ${\omega}_{B}$ on $B$, with $\omega \in O$ and $n=$ arity of $\omega $, a function $f:A\to B$ is said to be compatible with $\omega $ if
$$f({\omega}_{A}({a}_{1},\mathrm{\dots},{a}_{n}))={\omega}_{B}(f({a}_{1}),\mathrm{\dots},f({a}_{n})).$$ 
Dropping the subscript, we now simply identify $\omega \in O$ as an operator for both algebras^{} $A$ and $B$. If a function $f:A\to B$ is compatible with every operator $\omega \in O$, then we say that $f$ is a homomorphism^{} from $A$ to $B$. If $O$ contains a constant operator $\omega $ such that $a\in A$ and $b\in B$ are two constants assigned by $\omega $, 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:

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a homomorphism that is onetoone is a monomorphism^{};

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an onto homomorphism is an epimorphism^{};

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an isomorphism^{} is both a monomorphism and an epimorphism;

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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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All trivial algebraic systems (of the same type) are isomorphic.

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If $f:A\to B$ is a homomorphism, then the image $f(A)$ is a subalgebra^{} of $B$. If ${\omega}_{B}$ is an $n$ary operator on $B$, and ${c}_{1},\mathrm{\dots},{c}_{n}\in f(A)$, then ${\omega}_{B}({c}_{1},\mathrm{\dots},{c}_{n})={\omega}_{B}(f({a}_{1}),\mathrm{\dots},f({a}_{n}))=f({\omega}_{A}({a}_{1},\mathrm{\dots},{a}_{n}))\in f(A)$. $f(A)$ 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  20130322 15:55:36 
Last modified on  20130322 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 