A finite dynamical system is a system of multivariate functions over a finite alphabet. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables. The rank of a finite dynamical system is the cardinality of its image; the asymptotic rank is the number of its periodic points. In this paper, we determine the maximum ran...

We discuss the Feferman-Vaught Theorem in the setting of abstract model theory for finite structures. We look at sum-like and product-like binary operations on finite structures and their Hankel matrices. We show the connection between Hankel matrices and the FefermanVaught Theorem. The largest logic known to satisfy a Feferman-Vaught Theorem for product-like operations is CFOL, first order log...

An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU -rank. Every definable subgroup is commensurable with an acl(∅)definable subgroup. Every finitely based regular type in a CM-trivial ω-categorical simple theory is non-orthogonal to a type of SU -rank 1. In particular, a supersimple ω-categorical CM-trivial theory has finite SU -rank.

The algebraicity conjecture for simple groups of finite Morley rank, also known as the Cherlin-Zilber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last 15 years, the main line of attack on this problem has been Borovik’s program of transferring methods from finite group theory. Borovik’s program has led to consi...