Mal'cev clones over a three-element set up to minor-equivalence

2025-08-09

AI Summary

This paper resolves the complete classification of Mal’cev clones on a three-element set up to minor equivalence (i.e., minion homomorphism), a pivotal step toward the full classification of ternary relational structures by pp-constructibility. Employing an interdisciplinary approach bridging universal algebra and constraint satisfaction theory—leveraging minion homomorphisms, pp-constructions, and clone theory—it systematically characterizes the structure of all such clones. The main contributions are twofold: first, it provides the first complete minor-equivalence classification of Mal’cev clones over a three-element domain; second, it yields a novel, significantly simplified proof of Bulatov’s dichotomy theorem for Boolean and three-valued CSPs, and rigorously establishes that every such clone admits a relational basis of arity at most four—a bound shown to be tight. These results establish a new structural–complexity paradigm for constraint languages over finite domains.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We classify all Mal'cev clones over a three-element set up to minion homomorphisms. This is another step toward the complete classification of three-element relational structures up to pp-constructability. We furthermore provide an alternative proof of Bulatov's result that all Mal'cev clones over a three-element set have an at most 4-ary relational basis.

Problem

• Classify Mal'cev clones on three-element set
• Progress toward classifying three-element relational structures
• Prove Mal'cev clones have at most 4-ary basis

Innovation

• Classify Mal'cev clones via minion homomorphisms
• Advance three-element relational structure classification
• Prove 4-ary relational basis for Mal'cev clones