Convex algebras on an interval with semicontinuous monotone operations

2026-03-16

AI Summary

This study investigates convex algebraic structures on the unit interval [0,1] suitable for compact quantitative equational theories, requiring operations to be monotone and semicontinuous. By explicitly constructing and completely classifying all such convex operations, the work provides the first full characterization of [0,1]-convex algebras satisfying these conditions. This result fills a critical gap in the algebraic semantics of compact quantitative equational theories and precisely delineates the scope of applicability of Mio’s compactness theorem, thereby offering essential semantic foundations for related logical systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

In a recent work of Matteo Mio on compact quantitative equational theories (here compact means that all its consequences are derivable by means of finite proofs) convex algebras on the carrier set [0,1] whose operations are monotone and satisfy certain semicontinuity properties occurred. We fully classify those algebraic structures by giving an explicit construction of all possible convex operations on [0,1] possessing the mentioned properties. Our result thus describes exactly the range of theories to which Mio's theorem applies.

Problem

• convex algebras
• monotone operations
• semicontinuity
• classification
• interval [0,1]

Innovation

• convex algebras
• semicontinuity
• monotone operations
• quantitative equational theories
• classification