The purpose of the course is to introduce listeners to universal algebra and its applications in the study of families of logical systems. The course material can be divided into three parts:

I) elements of universal algebra;
II) Boolean algebras and their representations;
III) algebraic semantics for non-classical logics.

I. By an (abstract) algebra we mean an arbitrary structure in a signature whose only predicate symbol is equality. Classes of algebras that are axiomatizable by identities, i.e. by equalities between terms, are called varieties. For example, one can talk about the variety of all groups or rings. At first glance universal algebra is a study of varieties. We shall become familiar with the basic notions and methods of universal algebra used in logic. In particular, using the so-called free algebras (which are interesting in their own right) we shall prove the famous Birkhoff theorem: a class of algebras is a variety if and only if it is closed under homomorphic images, subalgebras and direct products.

II. By a lattice is meant a partially ordered set such that every non-empty finite subset of it has a supremum and an infimum. In effect, lattices can be viewed as algebra. Moreover, they occupy a central place in universal algebra. Boolean algebras are a special class of lattices, which play an important role in mathematics. We shall prove the main results related to Boolean algebras. In particular, among these results will be different representation theorems for Boolean algebras, one of which establishes an intimate connection between Boolean algebras and topological spaces of a special sort. This connection is called Stone duality.

III. It is known that relational semantics, also known as possible world semantics or Kripke semantics, often fails to fully characterize logical systems: for such systems no completeness theorems with respect to relational semantics can be obtained. On the other hand, practically all reasonable systems are strongly complete with respect to suitable algebraic semantics. Despite the fact that this kind of semantics is significantly more abstract, it turns out to be very convenient from a mathematical point of view, since it opens the way for wide applications of methods of universal algebra. We shall discuss algebraic semantics for (propositional) modal logic $\mathrm{K}$ and intuitionistic logic $\mathrm{Int}$. These semantics induce close connections between logical and algebraic properties. For instance, it turns out that an extension of $\mathrm{Int}$ has Craig’s interpolation property if and only if the corresponding variety of algebras is amalgamable. Such connections allow us to obtain numerous striking results.

Preliminary programme

1. A brief digression into classical first-order logic.
2. Homomorphisms, subalgebras and congruences. Homomorphism theorem.
3. Direct products. Direct and subdirect decompositions.
4. Free algebras. Birkhoff's theorem on varieties.
5. Lattices. The Knaster–Tarski theorem.
6. Boolean algebras and Boolean rings.
7. Representation of Boolean algebras. Filters.
8. Ultrafilters and Stone duality.
9. A brief digression into non-classical logics.
10. Modal algebras and algebraic semantics for propositional modal logic $\mathrm{K}$.
11. Heyting (pseudo-Boolean) algebras and algebraic semantics for propositional intuitionistic logic $\mathrm{Int}$.
12. Application of universal algebra in the study of the lattices of extensions of $\mathrm{K}$ and $\mathrm{Int}$.

Main references
S. Burris, H.P. Sankappanavar. A Course in Universal Algebra. Springer, 1981.
S. Givant, P. Halmos. Introduction to Boolean Algebras. Springer, 2009.
A. Chagrov, M. Zakharyaschev. Modal Logic. Oxford University Press, 1997.
D.M. Gabbay, L. Maksimova. Interpolation and Definability: Modal and Intuitionistic Logics. Oxford University Press, 2005.

Lecturer
Speranski Stanislav Olegovich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no.  075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center