Homomorphism: Encyclopedia II - Homomorphism - Informal discussion

Homomorphism - Informal discussion

Because abstract algebra studies sets with operations that generate interesting structure or properties on the set, the most interesting functions are those which preserve the operations. These functions are known as homomorphisms.

For example, consider the natural numbers with addition as the operation. A function which preserves addition should have this property: f(a + b) = f(a) + f(b). Note that f(x) = 3x is a homomorphism, since f(a + b) = 3(a + b) = 3a + 3b = f(a) + f(b). Note that this homomorphism maps the natural numbers back onto themselves.

Homomorphisms do not have to map between sets which have the same operations. For example, operation-preserving operations exist between the set of real numbers with addition and the set of positive real numbers with multiplication. A function which preserves operation should have this property: f(a + b) = f(a) * f(b), since addition is the operation in the first set and multiplication is the operation in the second. Given the laws of exponents, f(x) = e^x satisfies this condition.

A particularly important property of homomorphisms is that the identity is always mapped to the identity. Note in the first example f(0) = 0, and 0 is the additive identity. In the second example, f(0) = 1, since 0 is the additive identity, and 1 is the multiplicative identity.

If we are considering multiple operations on a set, then all operations must be preserved for a function to be a considered a homomorphism. Even though the set may be the same, the same function might be a homomorphism, say, in group theory (sets with a single operation) but not in ring theory (sets with two related operations), because it fails to preserve the additional operation that ring theory considers.

Adapted from the Wikipedia article "Informal discussion", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki