🤖 AI Summary
To address semantic modeling and automatic conversion of prefixed units, this paper introduces the first unified formal semantic model grounded in category and group theory: it jointly models unit prefixes, base units, and conversion rules as a ternary relation endowed with both group and category structures, and defines natural and accidental semantic operations; based on this, it establishes a hierarchy of subclasses with progressively strengthened algebraic properties. Key contributions include: (1) the first unit-semantic framework integrating category theory and group theory, enabling unified characterization of prefixed units and conversion rules; (2) a classification system for conversion relations with well-defined algebraic properties; and (3) a symbolic, deterministic, polynomial-time rewriting-based conversion algorithm that eliminates reliance on lookup tables and supports rigorously verifiable unit reasoning.
📝 Abstract
Units of measure with prefixes and conversion rules are given a formal semantic model in terms of categorial group theory. Basic structures and both natural and contingent semantic operations are defined. Conversion rules are represented as a class of ternary relations with both group-like and category-like properties. A hierarchy of subclasses is explored, each satisfying stronger useful algebraic properties than the preceding, culminating in a direct efficient conversion-by-rewriting algorithm.