Absolutely Free Hyperalgebras

TL;DR

This paper extends the concept of absolutely free algebras to multialgebras, establishing properties of multialgebras of terms and demonstrating the non-existence of universal mapping property multialgebras and left adjoints for the forgetful functor.

Contribution

It generalizes free algebra concepts to multialgebras, introduces multialgebras of terms, and proves the non-existence of universal mapping property multialgebras.

Findings

Multialgebras satisfying the universal mapping property do not exist.

The forgetful functor from multialgebras to Set lacks a left adjoint.

Multialgebras of terms are generated by a strong basis called the ground.

Abstract

It is well known from universal algebra that, for every signature $\Sigma$, there exist algebras over $\Sigma$ which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of $\Sigma$-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of $\Sigma$-algebras to $\textbf{Set}$, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor $\mathcal{U}$, from the category of $\Sigma$-multialgebras to $\textbf{Set}$, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), it corresponds a unique homomorphism, which ressembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that $\mathcal{U}$ does not have a left adjoint.