Activities This Week

I plan to discuss the construction, examples and some applications the Galois-type correspondence between subsemigroups of the endomorphism semigroup End(A) of an algebra A and sets of formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of End(A) on definable sets over A. Between possible applications of this Galois correspondence is a uniform approach to geometries defined by various fragments of the initial language.

The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way one can get an effective syntactical expression by semantic tools.

Yet another advantage is a common approach to generalizations of the main model theoretic concepts to the sublanguages of the first order language. It also reveals new connections between well-known concepts. One more application concerns the generalization of the unification theory or more generally Term Rewriting Theory to the logic unification theory.

Many interesting problems in additive combinatorics have a translation to geometric questions. A classical example to this is when Elekes used point-line incidence bounds on the sum-product problem of Erdos and Szemeredi. In this talk we will see more examples and will list several open problems in additive combinatorics.