Absolutely Free Hyperalgebras

Marcelo E. Coniglio, Guilherme V. Toledo

It is well know from universal algebra that, for every signature $\Sigma$,
there exist algebras over $\Sigma$ which are freely generated. Furthermore,
they are, up to isomorphisms, unique, and equal to algebras of terms.
Equivalently, the forgetful functor, from the category of $\Sigma$-algebras to
$\textbf{Set}$, has a left adjoint.
This result does not extend to hyperalgebras, which generalize algebras by
allowing the result of an operation to assume a non-empty set of values. Not
only freely generated hyperalgebras do not exist, but the forgetful functor
$\mathcal{U}$, from the category of $\Sigma$-hyperalgebras to $\textbf{Set}$,
does not have a left adjoint.
In this paper 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 $\mathcal{U}$ does not have a left adjoint.