Arithmetic of finite ordered sets: Cancellation of exponents, II

Garrett Birkhoff conjectured in 1942 that when A, B, P are finite posets satisfying A(P)congruent toB(P), then A congruent toB. 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, AxB, and C(A(B)). Every finite x-indecomposable and x-indecomposable poset A of more than one element is expressible for unique (up to isomorphism) E and P as A congruent toC(E-P) where P is connected and E is indecomposable for all three operations.