Minimizing the fundamental set of indecomposables and generating relations

Determine whether and how the fundamental set of indecomposables used to generate relations up to semigroup automorphisms can be reduced, possibly by replacing the relations associated with individual indecomposables by different relations.

Background

The paper constructs a CT-presentation of a semilattice using one relation for every indecomposable element and then shows that, in the totally real number-field setting, these relations can be organized into finitely many orbits under multiplication by totally positive units. The chosen fundamental set I_0 may contain redundant representatives, such as unit translates of basis indecomposables. The authors explicitly leave unresolved whether this set can be made smaller while still producing a generating family of relations, including the possibility of using relations different from the canonical relations r_α.