Relational Lattice Foundation for Algebraic Logic

TL;DR

This paper advances the relational lattice model for algebraic logic by introducing a bilattice structure, unary negation, and connecting it to database dependencies and cylindric algebras.

Contribution

It extends relational lattice theory with a bilattice framework, unary negation, and demonstrates the model's completeness and relevance to database and algebraic structures.

Findings

Uncovered complementary lattice operators forming a bilattice.

Introduced unary negation with proven laws like double negation and De Morgan.

Established the model's completeness and its application to database dependency theory.

Abstract

Relational Lattice is a succinct mathematical model for Relational Algebra. It reduces the set of six classic relational algebra operators to two: natural join and inner union. In this paper we push relational lattice theory in two directions. First, we uncover a pair of complementary lattice operators, and organize the model into a bilattice of four operations and four distinguished constants. We take a notice a peculiar way bilattice symmetry is broken. Then, we give axiomatic introduction of unary negation operation and prove several laws, including double negation and De Morgan. Next we reduce the model back to two basic binary operations and twelve axioms, and exhibit a convincing argument that the resulting system is complete in model-theoretic sense. The final parts of the paper casts relational lattice perspective onto database dependency theory and into cylindric algebras.