Absolutely Free Multialgebras

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.

Mapping Property of Bilateration and Its Application to Human-Following Robot

Objective of this paper is twofold. The first one is to study the mapping property and unified form of the component equations of the unknown node in bilateration, and the second one is to introduce the concept model for human-following robot based on bilateration. Bilateration needs only two known nodes and two distances’ data. Because of the simple sensor arrangement in bilateration, it needs less computation and uses less number of unavoidable erroneous distances compared to the trilateration.

Localization in Algebra

This chapter provides a digression concerning the all-important technique of localization in algebra. Localization generally means formally inverting a multiplicatively closed subset in a ring. However, the chapter focuses on the particular case of inverting all nonnegative powers of a variable u in an ℝ[u]-module. Localization of an ℝ[u]-module with respect to a variable u kills the torsion elements and preserves exactness. The chapter then looks at the proposition that localization preserves the direct sum. The simplest proof for this proposition is probably one that uses the universal mapping property of the direct sum. The chapter also considers antiderivations under localization.

The macro-event property and the layered structure of the clause

We ask whether there is a “macro-event phrase,” a uniform level of syntax at which complex scenarios may be described as single events under the Macro-Event Property (MEP). The MEP is a form-meaning mapping property that constrains the compatibility of event descriptions with time-positional modifiers. An examination of English infinitival complements, Ewe serial verb constructions, and Japanese converb constructions suggests that the putative crosslinguistic “macro-event phrase” is the verbal core of the Layered Structure of the Clause theory of Role and Reference Grammar. Across languages, simple cores necessarily have the MEP, whereas complex cores have it if and only if they are integrated by ‘cosubordinate’ nexus, defined as a symmetric union of two cores that together behave like a single core. We furthermore argue that this connection between core cosubordinations and the MEP may help explain why cosubordinate cores seem to always share an argument through control.