Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid

Abstract

If F is an ordered field and M is a finite-rank torsion-free monoid, then one can embed M into a finite-dimensional vector space over F via the inclusion M↪gp(M)↪F⊗Zgp(M), where gp(M) is the Grothendieck group of M. Let C be the class consisting of all monoids (up to isomorphism) that can be embedded into a finite-rank free commutative monoid. Here we investigate how the atomic structure and arithmetic properties of a monoid M in C are connected to the combinatorics and geometry of its conic hull cone(M)⊆F⊗Zgp(M). First, we show that the submonoids of M determined by the faces of cone(M) account for all divisor-closed submonoids of M. Then we appeal to the geometry of cone(M) to characterize whether M is a factorial, half-factorial, and other-half-factorial monoid. Finally, we investigate the cones of finitary, primary, finitely primary, and strongly primary monoids in C. Along the way, we determine the cones that can be realized by monoids in C and by finitary monoids in C.