Bridging Equational Properties and Patterns on Graphs: an AI-Based Approach

AI-assisted solutions have recently proven successful when applied to Mathematics and have opened new possibilities for exploring unsolved problems that have eluded traditional approaches for years or even centuries. Following this direction, this paper presents an innovative approach aiming at establishing correlations between equational properties of algebraic structures that can be represented through graphs and specific subportions of their topological representation. The methodology incorporates the utilization of graph neural architectures to validate theorems or conjectures, complemented by Explainability (XAI) metrics that lend support to these statements. In particular, we examine the distributive and modular properties of algebraic lattices, whose characterization is well-known in universal algebra, hence using these properties as an experimental test bench. The findings of this study demonstrate the effectiveness of the proposed approach in identifying and retrieving established subpatterns that characterize the equational properties under investigation. Moreover, the approach exhibits the capability to generate novel and noteworthy candidates as theorem suggesters, thereby offering valuable prospects for further exploration by mathematicians.