A topological structure on certain initial algebras

There is a well-known natural topology on the set of compatible total orders on a group, and recent results of Clay [2], Dabkowska et al. [4], and Sikora [12] have characterized this topology for certain groups. We consider a similar topology on the set of distinct monomial algebras in polynomial and Laurent polynomial rings. We study the latter topological structure for monomial algebras that come from rings of multiplicative invariants and show that they are either finite discrete spaces or homeomorphic to the Cantor set.