Weakly Free Multialgebras

Abstract

In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in exploring the foundations of multialgebras applied to the study of logic systems. 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 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 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 submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles 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.

A Category of Ordered Algebras Equivalent to the Category of Multialgebras

“…And, despite the fact we use the same construction, not only none of the aforementioned papers delve in the study of Stone-dualities or any related categorical equivalences, they use markedly different definitions of homomorphisms (except Walicki and Meldal, that use no definition of homomorphism as their work is mostly devoted to generalizing identities to the context of multialgebras): more specifically, they use what some logicians call in today's literature full homomorphisms (see [7]); that is not unexpected, since that definition is very appropriate when dealing with the theories of multigroups and hyperrings, but not when dealing with Nmatrices. [4] also use power algebras of non-partial multialgebras, as long as one considers the obvious connection between the latter and multigraphs, see [9] for a definition of this generalization of graphs (and its applications to logic), and [10] for a few constructions of multialgebras from multigraphs.…”