A GENTLE INTRODUCTION TO CLONES, AND THEIR APPLICATIONS.

Ponente(s): Edith Mireya Vargas Garcia, Mike Behrisch y John K. Truss

En esta pl\'atica mencionar\'e la importancia del estudio de \emph{Clones} tanto en las matem\'aticas (\'algebra universal), como en las ciencias de la computaci\'on. Empezar\'e con una introducci\'on gentil a clones, resumiendo los resultados en la teor\'\i a de clones y concentr\'andome en los m\'as recientes. Finalmente, mencionar\'e la conexi\'on de clones con la identificaci\'on de las subclases solubles en tiempo polinomial de los \emph{Problemas de Satisfacci\'on de Restricciones}(Constraint Satisfaction Problems (CSPs)) y si el tiempo lo permite, dar\'e~ los resultados obtenidos en la reconstrucci\'on de la topolog\'\i a natural que se encuentra en los clones. {\bf{Introduction}}\par \emph{Clones} (also known as \emph{function algebras}) are sets of finitary operations on a fixed carrier set that contain all projections and are closed under composition. They play an important role in modern universal algebra due to the fact that the set of all term operations of a universal algebra $\bfa{A}$, always forms a clone. Clones can be seen as higher arity generalisations of \emph{transformation monoids}. Moreover, clones carry a natural topological structure, provided by the \emph{topology of point-wise convergence}, which is the same as the \emph{product topology} if each factor space is equiped with the \emph{discrete topology}, under this topology the corresponding clone becomes a topological clone.\par A \emph{Constraint Satisfaction Problem} ($\mathrm{CSP}$) of a finite relational structure $\mathbb{B}$, denoted by $\mathrm{CSP}\left(\mathbb{B}\right)$, can be expressed as the problem of deciding whether there exists a homomorphism $\phi$ from a finite relational structures $\mathbb{A}$ to $\mathbb{B}$. In this talk a gentle introduction to clones and their connection to $\mathrm{CSP}$s is given.