Title:

Pseudofinite rings and their generalizations

We study a class of structures admitting functions which assign to definable sets values in a linearly ordered commutative domain and satisfy some natural axioms. Such functions are called count functions. A general version of count functions was introduced by Schanuel [50]. A developed version of this concept has been proposed in a more recent work by Krajcek and Scanlon [33]. These authors further introduced closely related notions of weak and strong Euler characteristics of definable sets [33]; my definition of a count function follows their approach. Moreover, a count function is a special case of what Krajcek and Scanlon called a nontrivial strong ordered Euler characteristic. We define a tallied structure M as an algebraic structure that admits a count function. At the intuitive level, count functions measure the sizes of definable sets. Unlike finite groups, the size of a tallied group is not the cardinality of the underlying set, but belongs to a ring that is much larger than the ring of integers, such as a ring of supernatural integers. We focus on the study of algebraic properties of tallied structures. The key result is the first step towards proving a tallied version of the celebrated theorem of Wedderburn about commutativity of finite division rings, namely, the proof of a tallied version of the no less celebrated theorem by Frobenius on the structure of sharply transitive finite permutation groups. In the process, we proved necessary facts about tallied groups in general, for example, that the class of tallied structures is closed under taking ultraproducts. Further, this thesis explores how counting arguments might work when the concept of a tallied structure undergoes some natural modification.
