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

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 circle times(Z) gp(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 Min Care connected to the combinatorics and geometry of its conic hull cone(M) subset of F circle times(Z) gp(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 Mis 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 inCand by finitary monoids in C. (C) 2020 Elsevier Inc. All rights reserved.