Absolutely Free Hyperalgebras

Marcelo E. Coniglio, Guilherme V. Toledo

Published 2021-01-11Version 1

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.