We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them, and introduce a new characteristic of relations, ternarity, that measures their `complexity of relating' and allows to refine reduction results. In particular, we refine Peirce's controversial reduction thesis, and show that reducibility behavior is dramatically different on finite and infinite domains.

翻译：本文研究通过布尔积或相对积（即对谓词应用合取与存在量词所得的原始正公式）将关系逻辑约简（分解）为低元数关系的方法。我们的代数框架统一了数据库理论中的自然连接与数据依赖、克隆理论中的关系代数，以及C.S.皮尔士的联结代数。我们还提出了新的约简构造方法，系统研究了不可约关系及其约简过程，并引入关系的新特征量——三元性，用以度量关系的“关联复杂度”并优化约简结果。特别地，我们优化了皮尔士存在争议的约简论题，并证明在有限域与无限域上关系的可约简性行为存在显著差异。