Thursday 21 September 2023, 14:00, room 1Z31: Applications of Homological Algebra to Algebraic Theories, Mirai Ikebuchi (Kyoto university).
It is well-known that some algebraic theories such as groups or Boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. In fact, one can define abelian groups, called the homology groups, associated with the given algebraic theory, and then a lower bound on the number of axioms is obtained from the information of those groups. The same methodology applies to equational unification, the problem of solving an equation modulo equational axioms. I provide a relationship between equational unification and homological algebra for algebraic theories: Equational unifiability implies the surjectivity of a homomorphism of abelian groups associated with the algebraic theory and the equation to solve. From this, one can show the non-unifiability by checking that the associated homomorphism is not surjective. In the talk, we will see concrete examples and surjectivity can be checked through simple matrix operations.