What is $$-Q$$ - Q for a poset Q?

In the context of combinatorial reciprocity, it is a natural question to ask what “\(-Q\)” is for a poset Q. In a previous work, the definition “\(-Q:=Q\times \mathbb {R}\) with lexicographic order” was proposed based on the notion of Euler characteristic of semialgebraic sets. In fact, by using this definition, Stanley’s reciprocity for order polynomials was generalized to an equality for the Euler characteristics of certain spaces of increasing maps between posets. The purpose of this paper is to refine this result, that is, to show that these spaces are homeomorphic if the topology of Q is metrizable.