Ternary Gamma Semirings: From Neural Implementation to Categorical Foundations

• URL: http://arxiv.org/abs/2603.19317v1
• Date: Sun, 15 Mar 2026 14:56:33 GMT
• Title: Ternary Gamma Semirings: From Neural Implementation to Categorical Foundations
• Abstract summary: This paper establishes a theoretical framework connecting neural network learning with abstract algebraic structures.<n>We first present a minimal counterexample demonstrating that standard neural networks completely fail on compositional generalization tasks.<n>We prove that this learned feature space constitutes a finite commutative ternary $$-semiring, whose ternary operation implements the majority vote rule.
• Score: 0.14504054468850663
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper establishes a theoretical framework connecting neural network learning with abstract algebraic structures. We first present a minimal counterexample demonstrating that standard neural networks completely fail on compositional generalization tasks (0% accuracy). By introducing a logical constraint -- the Ternary Gamma Semiring -- the same architecture learns a perfectly structured feature space, achieving 100% accuracy on novel combinations. We prove that this learned feature space constitutes a finite commutative ternary $Γ$-semiring, whose ternary operation implements the majority vote rule. Comparing with the recently established classification of Gokavarapu et al., we show that this structure corresponds precisely to the Boolean-type ternary $Γ$-semiring with $|T|=4$, $|Γ|=1$}, which is unique up to isomorphism in their enumeration. Our findings reveal three profound conclusions: (i) the success of neural networks can be understood as an approximation of mathematically ``natural'' structures; (ii) learned representations generalize because they internalize algebraic axioms (symmetry, idempotence, majority property); (iii) logical constraints guide networks to converge to these canonical forms. This work provides a rigorous mathematical framework for understanding neural network generalization and inaugurates the new interdisciplinary direction of Computational $Γ$-Algebra.

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