Homomorphisms and Horn Clauses

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(x1, …, xn) ) = f( φ(x1, …, φ(xn) )

and each n-ary relation symbol R

R(x1, …, xn) implies R(φ(x1, …, φ(xn))

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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