Arithmetic of Finite Ordered Sets: Cancellation of Exponents,II

Garrett Birkhoff conjectured in 1942 that when A, B, P are finite posets satisfying A ^PB ^P, then AB. We show that this is true. Further, we introduce an operation C(A ^B), related to Garrett Birkhoff's exponentiation, and determine the structure of the algebra of isomorphism types of finite posets under the operations induced by A+B, A×B, and C(A ^B). Every finite +-indecomposable and ×-indecomposable poset A of more than one element is expressible for unique (up to isomorphism) E and P as AC(E ^P) where P is connected and E is indecomposable for all three operations.