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 ωO and n= arity of ω, a function f:AB is said to be compatible with ω if


Dropping the subscript, we now simply identify ωO as an operator for both algebrasMathworldPlanetmathPlanetmath A and B. If a function f:AB is compatible with every operator ωO, then we say that f is a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from A to B. If O contains a constant operator ω such that aA and bB are two constants assigned by ω, then any homomorphism f from A to B maps a to b.


  1. 1.

    When O is the empty setMathworldPlanetmath, 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/LatticeMathworldPlanetmath) 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 identityPlanetmathPlanetmath 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.


  • 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 monomorphismMathworldPlanetmathPlanetmath;

    • an onto homomorphism is an epimorphismMathworldPlanetmath;

    • an isomorphismMathworldPlanetmathPlanetmath 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:AB is a homomorphism, then the image f(A) is a subalgebraMathworldPlanetmathPlanetmath of B. If ωB is an n-ary operator on B, and c1,,cnf(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