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.

• 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 $c_1,\mathrm{\dots },c_n\in f(A)$, 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	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