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

Author(s)

Guilherme Vicentin de Toledo

• Document type: Master's Dissertation
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2017-10-09

Examining board members

Lucio Centrone; Plamen Emilov Kochloukov; Ivan Chestakov

• Advisor: Lucio Centrone

Abstract

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 codimensions (AU)

• FAPESP's process: 16/08708-0 - Algebras in varieties of universal algebras
• Grantee: Guilherme Vicentin de Toledo
• Support Opportunities: Scholarships in Brazil - Master