Product cardinality equality for arbitrary locally quasi-convex groups

Establish the product equality for compatible-poset cardinalities of arbitrary Hausdorff locally quasi-convex abelian groups, namely determine whether the cardinality of the compatible-poset of a product equals the maximum of the cardinalities of the compatible-posets of its factors.

Background

The paper proves the corresponding product formula for locally compact abelian groups: the cardinality of the compatible-poset of a finite product is the maximum of the cardinalities of the compatible-posets of the factors. This result relies on the discrete reduction available in the locally compact setting and on the exact cardinality formula for discrete abelian groups.

For arbitrary locally quasi-convex abelian groups, the paper’s order-embedding theorem supplies a large lower bound for each compatible-poset but does not provide the upper bound needed to compare the compatible-poset of a product with those of its factors. Consequently, the general product equality remains unresolved.