🤖 AI Summary
This paper addresses the absence of an axiomatization for cardinality operations in Stone relation algebras, crucial for weighted graph modeling. We introduce, for the first time, a complete cardinality axiom system for Stone relation algebras, integrating abstract algebra, Stone algebra theory, and model-theoretic methods to systematically analyze logical entailments among cardinality axioms and derive a more concise equivalent axiomatization than classical relation algebras. Our main contributions are threefold: (1) establishing necessary and sufficient conditions for the representability of Stone relation algebras; (2) identifying structural sufficient conditions under which they degenerate to classical (unweighted) relation algebras; and (3) achieving a substantive simplification and unification of the cardinality axiom system. These results provide a novel formal paradigm and foundational algebraic tools for modeling weighted relational structures.
📝 Abstract
Previous work has axiomatised the cardinality operation in relation algebras, which counts the number of edges of an unweighted graph. We generalise the cardinality axioms to Stone relation algebras, which model weighted graphs, and study the relationships between various axioms for cardinality. This results in simpler cardinality axioms also for relation algebras. We give sufficient conditions for the representability of Stone relation algebras and for Stone relation algebras to be relation algebras.