Poset algebras over well quasi-ordered posets

Abstract.

A new class of partial order-types, class ${\mathcal{G}}^{+}_{bqo}$ is defined and investigated here. A poset P is in the class ${\mathcal{G}}^{+}_{bqo}$ 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 distributive lattice L(P) have been 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 a homomorphism ${\hat f}$ 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 ${\mathcal{G}}^{+}_{bqo}$ is contained in the class of well quasi-ordered sets. We prove that ${\mathcal{G}}^{+}_{bqo}$ 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 ${\mathcal{G}}^{+}_{bqo}$. We do not know, however if the class of well quasi-ordered sets is contained in ${\mathcal{G}}^{+}_{bqo}$. Additional results concern homomorphic images of posets algebras.