Poset algebras over well quasi-ordered posets

A new class of partial order-types, class $\gbqo^+$ is defined and investigated here. A poset $P$ is in the class $W^+$ iff the free poset algebra $F(P)$ is generated by a better quasi-order $G$ that is included in the free lattice $L(P)$. 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 $W^+$ is contained in the class of well quasi-ordered sets. We prove that $W^+$ 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 $W^+$. We do not know, however if the class of well quasi-ordered sets is contained in $W^+$. Additional results concern homomorphic images of posets algebras.