# 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

 $f(\omega_{A}(a_{1},\ldots,a_{n}))=\omega_{B}(f(a_{1}),\ldots,f(a_{n})).$

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. 1.

When $O$ is the empty set, any function from $A$ to $B$ is a homomorphism.

2. 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. 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.

• 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;

• finally, an automorphism is an endomorphism that is also an isomorphism.

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

• 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 $c_{1},\ldots,c_{n}\in f(A)$, then $\omega_{B}(c_{1},\ldots,c_{n})=\omega_{B}(f(a_{1}),\ldots,f(a_{n}))=f(\omega_{% A}(a_{1},\ldots,a_{n}))\in f(A)$. $f(A)$ 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 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