# Algebra and Logic (AL)

- Thursday from 10:15 to 12:00

### Description:

This seminar will focus on both the algebraic property of amalgamation and the logical property of interpolation. Amalgamation for a class of algebras means that whenever two algebras in the class share a common subalgebra, they embed into an "amalgamating" member of the class in such a way that the common subalgebra is preserved. Interpolation for a logic means that whenever some formula A is a consequence of another formula B, there exists an "interpolating'' formula C whose variables occur in both A and B, that is a consequence of A and has B as a consequence. Surprisingly, these two seemingly very different properties are closely related, forming a bridge between the realms of algebra and logic. We will study these properties first in the context of intuitionistic logic and Heyting algebras, considering also Pitt's uniform interpolation proof for the logic and its algebraic analogue, before turning our attention to the more general setting.

### Requirements:

The seminar is particularly suitable for students with some experience of either universal algebra or proof theory, but will be accessible to all students familiar with basic notions from algebra and logic.

### References:

N. Bezhanishvili and D. de Jongh. Intuitionistic logic, ESSLLI Course 2005.

G. Metcalfe. Amalgamation and interpolation, Course notes, 2017.

A. Pitts. On an interpretation of second-order quantification in first-order intuitionistic propositional logic, Journal of Symbolic Logic 57 (1992), 33-52.