Poset algebras over well quasi-ordered posets

A new class of partial order-types, class G bqo is defined and investigated here. A poset P is in the class G bqo iff the poset algebra F (P ) is generated by a better quasi-order G that is included in L(P ). The free Boolean algebra F (P ) and its free distrivutive lattice L(P ) were defined in [ABKR]. The free Boolean algebra F (P ) contains the partial order P and is generated by it: F (P ) has the following universal property. If B is any Boolean algebra and f is any order-preserving map from P into a Boolean algebra B, then f can be extended to an homomorphism f̂ of F (P ) into B. We also define L(P ) as the sublattice of F (P ) generated by P . We prove that if P is any well quasi-ordering, then L(P ) is well founded, and is a countable union of well quasi-orderings. We prove that the class G bqo is contained in the class of well quasiordered sets. We prove that G bqo is preserved under homomorphic image, finite products, and lexicographic sum over better quasi-ordered index sets. We prove also that every countable well quasi-ordered set is in G bqo. We do not know, however if the class of well quasi-ordered sets is contained in G bqo. Additional results concern homomorphic images of posets algebras. This work is supported by the Center for Advanced Studies in Mathematics (Ben Gurion University). This work is supported by the Israel Science Foundation (Post-Doctoral positions at Ben Gurion University 2000–2002), the Fields Institute (Toronto 2002–2004), and by the Nato Science Fellowship (University Paris VII, CNRS-UMR 7056, 2004).