Homomorphism

Remarks

Example: for the signature $\set{+, 0}$, the map $h \colon \Z \to \Z/n\Z$ that sends an integer to its residue class is a homomorphism. Indeed, the class of $a + b$ is the sum of the classes of $a$ and $b$, and the class of $0$ is $0$.

Non-example: for the same signature, the map $x \mapsto 2x + 1$ from $\Z$ to $\Z$ fails clause 2, since $0 \mapsto 1$; it also fails clause 1, since $h(0 + 0) = 1 \ne 2 = h(0) + h(0)$.

For a signature with only function and constant symbols, clauses 1 and 2 are the whole condition; this is the notion of homomorphism of universal algebra. Whether a map is a homomorphism depends on the signature. In the signature of groups $\set{\cdot, {}^{-1}, e}$ the inverse is preserved by definition, while for $\set{\cdot}$ alone it follows from the group axioms.