Universal Algebra for Logics

These notes form Lecture Notes of a short course which I will give at 1st School on Universal Logic in Montreux. They cannot be recommended for self studies because, although all definitions and main ideas are included, there are no proofs and examples. I’m going to provide some of them during my lectures, leaving easy ones as exercises. In the first part we discuss some of the most important notions of universal algebra. Then we concentrate on free algebras and varieties. Our main goal is to prove Birkhoff’s Theorem, which says that a class of similar algebras is a variety iff it is definable by a set of equations. In the last part we say more about lattices and boolean algebras, as these algebraic structures which are especially important for logic. Universal algebra is sometimes seen as a special branch of model theory, in which we are dealing with structures having operations only. However, it is only one of aspects of universal algebra, which appears to be a powerful tool in many areas. Universal algebra borrows techniques and ideas from logic, lattice theory and category theory. The connections between lattice theory and the general theory of algebras are particularly strong. We assume that the reader has the basic knowledge of mathematics.