A Category of Ordered Algebras Equivalent to the Category of Multialgebras

It is well known that there is a correspondence between sets and complete, atomic Boolean algebras ($\textit{CABA}$s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of $\textbf{Set}$ and the category of $\textit{CABA}$s. We modify this result by taking multialgebras over a signature $\Sigma$, specifically those whose non-deterministic operations cannot return the empty-set, to $\textit{CABA}$s with their zero element removed (which we call a $\textit{bottomless Boolean algebra}$) equipped with a structure of $\Sigma$-algebra compatible with its order (that we call $\textit{ord-algebras}$). Conversely, an ord-algebra over $\Sigma$ is taken to its set of atomic elements equipped with a structure of multialgebra over $\Sigma$. This leads to an equivalence between the category of $\Sigma$-multialgebras and the category of ord-algebras over $\Sigma$. The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures

Weakly 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

('jota','ômega')-Algebras and polynomial identities of Zariski-closed representable algebras

Orientador: Lucio CentroneDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Desenvolvemos neste trabalho aspectos de duas áreas da álgebra moderna e a conexão entre elas. Em álgebra abstrata, generalizamos a noçãoo de álgebra com esquemas de operadores de Higgins de "Algebras with a Scheme of Operators", que aqui se tornam $( I,\Omega)-$álgebras, usando uma linguagem própria de álgebras universais. Dados conjuntos de índices $I$ e $\Omega$ e funções $dom:\Omega\rightarrow \mathcal{P}(\bigcup_{n\in\mathbb{N}} I^{n})$ e $codom:\Omega\rightarrow\mathcal{P}( I)$, definimos uma $( I,\Omega)-$álgebra é um par $\mathcal{A}=(\{A_{ i}\}_{ i\in I}, \{f^{\omega}_{\mathcal{A}}\}_{\omega\in\Omega})$ tal que as $f^{\omega}_{\mathcal{A}}$ são funções de $\bigcup_{( i_{1}, ... , i_{n})\in dom(\omega)}A_{ i_{1}}\times\cdots\times A_{ i_{n}}$ em $\bigcup_{ i\in codom(\omega)}A_{ i}$. E na teoria de identidades polinomiais estudamos aquelas das álgebras representáveis Zariski-fechadas. Por $A$ ser representável queremos dizer que a $F-$álgebra $A$ admite um monomorfismo $\rho:A\rightarrow B$ em uma $K-$álgebra $B$ finitamente gerada, com $F$ e $K$ corpos. E por Zariski-fechada, sendo $B$ e $K^{n}$ isomorfos como espaços vetoriais, queremos dizer que o fecho topológico de $\rho(A)$ em $B$ é o próprio $\rho(A)$ quando munimos $B$ da topologia de Zariski herdada de $K^{n}$, assumindo $K$ algebricamente fechado. A conexão entre ambos, desenvolvida minunciosamente ao longo do texto, se baseia nas álgebras universais quando as definimos de maneira apropriada para nossos intentos. Para isso, divergimos da definição clássica de álgebra universal para múltiplos tipos onde, dados conjunto de índices $I$ e conjunto de símbolos funcionais $\Omega$ munidos de funções $dom:\Omega\rightarrow \bigcup_{n\in\mathbb{N}} I^{n}$ e $codom:\Omega\rightarrow I$, uma $\Omega-$álgebra universal de múltiplos tipos é um par $\mathcal{A}=(\{A_{ i}\}_{ i\in I}, \{f^{\omega}_{\mathcal{A}}\}_{\omega\in\Omega})$ tal que se $dom(\omega)=( i_{1}, ... , i_{n})$ e $codom(\omega)= i$, então $f^{\omega}_{\mathcal{A}}:A_{ i_{1}}\times\cdots\times A_{ i_{n}}\rightarrow A_{ i}$. Esperamos com esse texto formalizar os elementos de álgebras universais necessários ao estudo de identidades polinomiais em álgebras representáveis que nos permite o cálculo das codimensões no caso Zariski-fechado, além de fornecer uma prova do teorema de Birkhoff, ou seu análogo apropriado, para uma vasta classe de $( I,\Omega)-$álgebras, que chamaremos parciais, e que ainda generalizam a noção de Higgins. Para a prova do teorema de Birkhoff, desenvolvemos alguns resultados elementares de homomorfismos, termos e suas avaliações, passamos para operadores de classe e finalmente para álgebras $K-$livres. Já para o cálculo das já referidas codimensões primeiro fazemos uma breve introdução aos elementos da topologia de Zariski que nos são necessários, provando também que se $A$ é uma $F-$subálgebra de $K^{n}$ então seu fecho topológico é uma $K-$álgebra. Então trabalhamos a noção de conjunto teste: no caso geral, um conjunto teste para uma $( I,\Omega)-$álgebra $\mathcal{A}=(\{A_{ i}\}_{ I}, \{f^{\omega}_{\mathcal{A}}\}_{\Omega})$ á uma família $S=\{S_{ i}\}_{ i\in I}$ de subconjuntos $S_{ i}\subseteq A_{ i}$ tais que $(\tau_{1}, \tau_{2})$ é uma identidade de $\mathcal{A}$ se e somente se $\tau_{1}^{\chi}=\tau_{2}^{\chi}$ para toda avaliação $\chi$ com imagem em $S$. Provamos o teorema principal do trabalho ao restringir nossas assinaturas ao caso multilinear, nos permitindo achar um limitante superior para as codimensõesAbstract: Through this work we develop aspects of two areas of modern algebra and the connection between them. In abstract algebra, we generalize algebras with a scheme of operators, as defined by Higgins in "Algebras with a Scheme of Operators", to $( I,\Omega)-$algebras, defining them in a way similar to the one used to define universal algebras. Given a set of indexes $I$ and a set of functional symbols $\Omega$ and functions $dom:\Omega\rightarrow \mathcal{P}(\bigcup_{n\in\mathbb{N}} I^{n})$ and $codom:\Omega\rightarrow\mathcal{P}( I)$, we define an $( I,\Omega)-$algebra is a pair $\mathcal{A}=(\{A_{ i}\}_{ i\in I}, \{f^{\omega}_{\mathcal{A}}\}_{\omega\in\Omega})$ such that each $f^{\omega}_{\mathcal{A}}$ is a function from $\bigcup_{( i_{1}, ... , i_{n})\in dom(\omega)}A_{ i_{1}}\times\cdots\times A_{ i_{n}}$ to $\bigcup_{ i\in codom(\omega)}A_{ i}$. Furthermore, in the theory of polynomial identities we study those of the representable Zariski-closed algebras. By $A$ being representable we mean that the $F-$algebra $A$ admits a monomorphism $\rho:A\rightarrow B$ from $A$ to a finitely generated $K-$algebra $B$, wihere $F$ and $K$ are fields. And by Zariski-closed, where $B$ and $K^{n}$ are isomorphic as vector spaces, we mean that $\rho(A)$ is closed in $B$ when we provide $B$ of the Zariski topology inherited from $K^{n}$, assuming $K$ algebraically closed. The connection between the two, developed meticulously throughout the text, is based on universal algebras when we define them accordingly to our objectives. To properly define these algebras as needed, we diverge from the classic definition of universal algebra over multiple sets, where given a set of indexes $I$ and a set of functional symbols $\Omega$ provided with functions $dom:\Omega\rightarrow \bigcup_{n\in\mathbb{N}} I^{n}$ and $codom:\Omega\rightarrow I$, an universal $\Omega-$algebra over multiple sets is a pair $\mathcal{A}=(\{A_{ i}\}_{ i\in I}, \{f^{\omega}_{\mathcal{A}}\}_{\omega\in\Omega})$ such that if $dom(\omega)=( i_{1}, ... , i_{n})$ and $codom(\omega)= i$, then $f^{\omega}_{\mathcal{A}}:A_{ i_{1}}\times\cdots\times A_{ i_{n}}\rightarrow A_{ i}$. Thus, this text aims: firstly, to formalize the elements of universal algebras useful to the study of polynomial identities in representable algebras that allows us to calculate the codimensions in the Zariski-closed case; secondly, to provide a proof of the adequate analogue of Birkhoff's theorem to a vast class of $( I,\Omega)-$algebras, the ones we shall call partial, that still generalize Higgins's definition. For the proof of Birkhoff's theorem we use classes of operators and $K-$ free algebras, and in order to better understand these concepts we first develop some simple results on homomorphisms, terms and their evaluations. To calculate the aforementioned codimensions we first make a brief introduction to the elements of Zariski's topology that are necessary to us, proving also that if $A$ is a subalgebra of $K^{n}$ then its topological closure is an $K-$algebra. We subsequently study the notion of test set: in general, a test set for a $( I,\Omega)-$algebra $\mathcal{A}=(\{A_{ i}\}_{ I}, \{f^{\omega}_{\mathcal{A}}\}_{\Omega})$ is a family $S=\{S_{ i}\}_{ i\in I}$ of subsets $S_{ i}\subseteq A_{ i}$ such that $(\tau_{1}, \tau_{2})$ is an identity of $\mathcal{A}$ if and only if $\tau_{1}^{\chi}=\tau_{2}^{\chi}$ for every evaluation $\chi$ with image in $S$. We prove the main theorem of our work by restricting our signatures to the multilinear case, allowing us to find an upper limit for codimensionsMestradoMatematicaMestre em Matemática2016/08708-0;2016/131886-9FAPESPCNP