Every torsion-free atomic monoid M can be embedded into a real vector space via the inclusion M → gp(M ) → R⊗ Z gp(M ), where gp(M ) is the Grothendieck group of M . Let C be the class consisting of all submonoids (up to isomorphism) that can be embedded in a finite-rank free commutative monoid. Here we investigate how the atomic structure and factorization properties of members of C reflect in the combinatorics and geometry of their conic hulls cone(M ) ⊆ R ⊗ Z gp(M ). First, we establish geometric characterizations in terms of cone(M ) for a monoid M in C to be factorial, half-factorial, and other-half-factorial. Then we show that the submonoids of M determined by the faces of cone(M ) amount for all divisor-closed submonoids of M . Finally, we investigate the cones of finitary, primary, finitely primary, and strongly primary monoids in C (monoids in these classes have been relevant in the development of factorization theory). Along the way, we study the cones that can be realized by monoids in C and by finitary monoids in C.An atomic monoid M is half-factorial (or an HFM) provided that for all noninvertible x ∈ M , any two factorizations of x have the same number of irreducibles (counting repetitions). In addition, an integral domain is half-factorial (or an HFD) if its multiplicative monoid is an HFM. The concept of half-factoriality was first investigated by L. Carlitz in the context of algebraic number fields; he proved that an algebraic number field is an HFD if and only if its class group has size at most two [6]. However, the term "half-factorial domain" is due to A. Zaks [45]. In [46], Zaks studied Krull domains that are HFDs in terms of their divisor class groups. Parallel to this, L. Skula [42] and J.Śliwa [43], motivated by some questions of W. Narkiewicz on algebraic number theory [39, Chapter 9], carried out systematic studies of HFDs. Since then HFMs and HFDs have been actively studied (see [7] and references therein).Other-half-factoriality, on the other hand, is a dual version of half-factoriality, and it was introduced by J. Coykendall and W. Smith [12]. An atomic monoid M is called other-half-factorial (or an OHFM) provided that for all non-invertible x ∈ M no two distinct factorizations of x in M contain the same number of irreducibles (counting repetitions). Although an integral domain is a UFD if and only if its multiplicative monoid is an OHFM [12, Corollary 2.11], OHFMs are not always factorial or halffactorial, even in the class C. In the second part of this paper, we offer geometric and combinatorial characterizations for the HFMs in C and for the OHFMs in C.The study of primary monoids was initiated by T. Tamura [44] and M. Petrich [40] in the 1970s and has received a great deal of attention since then [34,35,20]. Primary monoids naturally appear in commutative algebra: an integral domain is 1-dimensional and local if and only if its multiplicative monoid is primary. One of the most useful subclasses of primary monoids in factorization theory is that one consisting of finitely primary ...
