Hyperalgebras of terms

Absolutely Free Hyperalgebras

We generalize, in a natural way, algebras of terms to hyperalgebras of terms, which display many properties of freely generated algebras : they extend uniquely to homomorphisms, not functions, but pairs of functions and collections of choices, which select how an homomorphism approaches indeterminacies ; and they are generated by a set that fits a strong definition of basis, which we call the ground of the hyperalgebra.With these definitions at hand, we offer simplified proofs that freely generated hyperalgebras do not exist and that does not have a left adjoint.