Two minor traditions in mathematical logic are finite model theory and many-valued logics. Despite emerging as the straightforward restriction of model theory to finite structures, finite model theory is quite distinct from its infinite counterpart and, motivated by complexity and computational interest developed over the last century into a field unto itself with its own particular focus. Simultaneously, many-valued logics have occupied a consistent space in the space of non-classical logic research, especially with the advent of algebraic logic that utilised universal algebraic methods to give a consistent and comparable treatment to the vast array of many-valued logics that have developed. Notably, the interaction of the two is quite understudied, with only a small interest in non-classical finite model theory restricted to a specific semantics and a very recent, if burgeoning interest in the full model theory for many-valued logics.
In this talk we explore the beginning of a project to unite these two traditions. Focusing on the generalisation of the finite homomorphism preservation theorem - an meta-mathematically interesting theorem in classical finite model theory - we explore the methodology behind the generalisation from the classical to the many valued.