On the irreducible Non First normal form Relations

The algebraic properties of Non First normal form Relations (NFR) are discussed. To introduce Canonical forms and Irreducible forms, two operations, Nest and Composition, are separately used. We investigate the condition under which both classes are matched. Regularity condition is proposed in which there exists only one irreducible form. Testing regularity is also discussed, and efficient procedures are presented. As the application, the fixed attributes on NFR, which extend key concept, are characterized.