  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.
