The level-size characterization of monoid posets for bi-infinite weak orders

A bi-infinite weak order is a weak order whose levels are indexed in both directions by the integers. A monoid poset is a poset arising from an order-preserving action of a monoid as in the paper, and a level is one of the antichains in the weak-order decomposition.

Level-size characterization. A bi-infinite weak order is a monoid poset if and only if all its levels are of the same size.

The preceding proposition establishes necessary restrictions on the level sizes, while the converse is asserted here as a belief and is not proved in the supplied text.