If a lattice L is distributive, then L is also modular . By assuming x y, we have x y = y . Hence, from the distributive property we get: x (y z) ( x y) (x z) y ( x z) 14 Definition A.7. Congruence Lattice Reflexivity: For every element a in A, a is related to itself, denoted as a a; Symmetry: For any elements a and b in A, if a b, then b a; Transitivity: For any elements a, b, and c in A, if a b and b c, then a c . An algebra with no other congruences is called simple . A type F is defined as a set of operation symbols along with their respective arities.

The rise of Artificial Intelligence (AI) recently empowered researchers to investigate hard mathematical problems which eluded traditional approaches for decades. Y et, the use of AI in Universal Algebra (UA)--one of the fields laying the foundations of modern mathematics--is still completely unexplored.

If a lattice L is distributive, then L is also modular . By assuming x y, we have x y = y . Hence, from the distributive property we get: x (y z) ( x y) (x z) y ( x z) 14 Definition A.7. Congruence Lattice Reflexivity: For every element a in A, a is related to itself, denoted as a a; Symmetry: For any elements a and b in A, if a b, then b a; Transitivity: For any elements a, b, and c in A, if a b and b c, then a c . An algebra with no other congruences is called simple . A type F is defined as a set of operation symbols along with their respective arities.

Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative justification-based notion of similarity based on the observation that sets of generalizations encode important properties of elements. We show that similarity defined in this way has appealing mathematical properties. As we construct our notion of similarity from first principles using only elementary concepts of universal algebra, to convince the reader of its plausibility, we show that it can be naturally embedded into first-order logic via model-theoretic types.

The rise of Artificial Intelligence (AI) recently empowered researchers to investigate hard mathematical problems which eluded traditional approaches for decades. Yet, the use of AI in Universal Algebra (UA) -- one of the fields laying the foundations of modern mathematics -- is still completely unexplored. This work proposes the first use of AI to investigate UA's conjectures with an equivalent equational and topological characterization. While topological representations would enable the analysis of such properties using graph neural networks, the limited transparency and brittle explainability of these models hinder their straightforward use to empirically validate existing conjectures or to formulate new ones. To bridge these gaps, we propose a general algorithm generating AI-ready datasets based on UA's conjectures, and introduce a novel neural layer to build fully interpretable graph networks. The results of our experiments demonstrate that interpretable graph networks: (i) enhance interpretability without sacrificing task accuracy, (ii) strongly generalize when predicting universal algebra's properties, (iii) generate simple explanations that empirically validate existing conjectures, and (iv) identify subgraphs suggesting the formulation of novel conjectures.