The idea of a homomorphism extends neatly to general signatures. A function between two objects with the same signature is a homomorphism if it preserves all function and relation symbols. So *φ* is a homomorphism if for each *n*-ary function symbol *f*

φ( f(x_{1}, …, x_{n}) ) = f( φ(x_{1}, …, φ(x_{n}) )

and each *n*-ary relation symbol *R*

R(ximplies_{1}, …, x_{n})R(φ(x_{1}, …, φ(x_{n}))

This coincides with the usual definition of homomorphism for groups and rings. For partially-ordered sets, homomorphisms correspond to order-preserving maps.

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