# 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 () 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.

 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