Homomorphisms: how the exponential function turns addition into multiplication
Feed the sum of two numbers into the exponential function and you get the product of their separate results. That tidy trade, addition on one side and multiplication on the other, is a homomorphism: a map between two mathematical systems that carries their structure across intact, even when the operations wear different names.
The word joins Greek roots meaning same and shape, yet it seems to have entered mathematics through a slip, a German word for similar being rendered as same. It was in use by 1892, credited to the German mathematician Felix Klein.
The rule is simple to state. If a structure has an operation, applying the map after combining two elements must give the same answer as combining their images. Constants count as operations too, so when a structure demands an identity element, the map must send one identity to the other. A group homomorphism automatically does this and also sends inverses to inverses. Maps between vector spaces that respect addition and scaling are called linear maps, the backbone of linear algebra.
Examples reveal the subtleties. Sending each real number r to the two-by-two matrix with r on its diagonal respects both addition and multiplication, so it links two rings. Taking the absolute value of nonzero complex numbers respects multiplication, since the size of a product equals the product of sizes, but it fails for addition, because the size of a sum is generally not the sum of sizes. It is therefore a homomorphism of multiplicative groups and not of rings. Likewise, a map between monoids that preserves the operation but not the identity only qualifies at the weaker semigroup level.
When a homomorphism is also a one-to-one correspondence, it is an isomorphism, meaning the two structures are essentially identical. The exponential function qualifies, with the natural logarithm undoing it. Stretching the idea beyond sets with operations produced the general notion of a morphism, the starting point of category theory.
Source: Homomorphism
