homomorphism between algebraic systems
Dropping the subscript, we now simply identify as an operator for both algebras and . If a function is compatible with every operator , then we say that is a homomorphism from to . If contains a constant operator such that and are two constants assigned by , then any homomorphism from to maps to .
When is the empty set, any function from to is a homomorphism.
When is a singleton consisting of a constant operator, a homomorphism is then a function from one pointed set to another , such that .
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 -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:
All trivial algebraic systems (of the same type) are isomorphic.
If is a homomorphism, then the image is a subalgebra of . If is an -ary operator on , and , then . is sometimes called the homomorphic image of in to emphasize the fact that is a homomorphism.
|Title||homomorphism between algebraic systems|
|Date of creation||2013-03-22 15:55:36|
|Last modified on||2013-03-22 15:55:36|
|Last modified by||CWoo (3771)|