[In a comment to the OP, I asked whether a statement along the lines of Corollary 1 below would count as an "interesting structure theorem", and this post expands on John Baez's yes to that question.]
Let $H$ be a multiplicatively written monoid; unless stated otherwise, $H$ need not be commutative, cancellative, or whatever. An atom of $H$ is a non-unit $a \in H$ such that $a \ne xy$ for all non-units $x, y \in H$. We write $H^\times$ and $\mathscr A(H)$, resp., for the group of units and the set of atoms of $H$.
Denote by $\mathscr F(X)$ the free monoid on a basis $X$; we'll refer to the elements of $\mathscr F(X)$ as $X$-words. Given a non-unit $x \in H$, we define $\mathcal Z_H(x)$ as the set of all $\mathscr A(H)$-words $\mathfrak a$ such that $x = \pi_H(\mathfrak a)$, where $\pi_H$ is the canonical (monoid) epimorphism $\mathscr F(H) \to H$ (see the section "Pills of factorization theory" in this answer if something in the terminology or in the notation doesn't look familiar). The $\mathscr A(H)$-words in $\mathcal Z_H(x)$ are the atomic factorizations of $x$ (relative to $H$).
We write $\preceq_H$ for the "shuffling preorder" induced on $\mathscr F(\mathscr A(H))$ by the divisibility preorder${}^{(1)}$ $\mid_H$ on $H$; roughly speaking, this means that an $\mathscr A(H)$-word $\mathfrak a$ is $\preceq_H$-smaller than an $\mathscr A(H)$-word $\mathfrak b$ if and only if $\mathfrak a$ is a subword of a permutation of $\mathfrak b$ modulo $\mid_H$-equivalence${}^{(2)}$ on the letters.
We call $H$ atomic (resp., an FF-monoid) if $\mathcal Z_H(x)$ is non-empty (resp., finite and non-empty modulo $\preceq_H$-equivalence) for every non-unit $x \in H$; and an FmF-monoid if, for each non-unit $x \in H$, the set of $\preceq_H$-minimal $\mathscr A(H)$-words in $\mathcal Z_H(x)$ is finite and non-empty modulo $\preceq_H$-equivalence.
Examples. (i) If $H$ is the multiplicative monoid of the ring of integers, then $\mathscr A(H)$ is the set of (positive or negative) primes, and the $H$-words $(-2) \ast 3 \ast (-5)$ and $(-3) \ast 5 \ast 2$ are $\preceq_H$-equivalent atomic factorizations of $-30$.
(ii) If $p \in \mathbf N^+$ is a prime, $n$ is an integer $\ge 2$, and $H$ is the multiplicative monoid of the ring of integers modulo $p^n$, then the $H$-word $\bar{p}^{\ast (n+1)}$ is an atomic factorization of $\bar{0}$ but is not $\preceq_H$-minimal (here, $\bar{x}$ is the residue class mod $p^n$ of an integer $x$).
We say that $H$ is finitely generated modulo units (or shortly, an f.g.u. monoid) if there is a finite subset $A \subseteq H$ such that $H \setminus H^\times$ is contained in the subsemigroup generated by the set $H^\times A H^\times$; and is locally finitely generated modulo units (or shortly, an l.f.g.u. monoid) if, for each $a \in H$, the smallest divisor-closed submonoid${}^{(3)}$ of $H$ containing $a$ is an f.g.u. monoid.
Examples. (iii) The monoid $H$ considered in Example (i) is not atomic: $0$ is a non-unit of $H$ with no atomic factorizations. Note, though, that $0$ is an irreducible of $H$ (see the definition before Theorem 2 below), and $H$ has the weaker property that every non-unit is a product of irreducibles.
(iv) The monoid $H$ considered in Example (ii) is FmF, but not FF; in particular, if $k$ an integer $\ge n$ and $u_1, \ldots, u_k \in H$ are units, then the $H$-word $\bar{p}u_1 \ast \cdots \bar{p}u_k$ is an atomic factorization of $\bar{0}$. (Note that two $\mathscr A(H)$-words are $\preceq_H$-equivalent only if they have the same number of letters.)
With these premises in place, we have:
Theorem 1. Every acyclic${}^{(4)}$, l.f.g.u. monoid is FmF.
Most notably, this implies the following:
Corollary 1. If $H$ is a unit-cancellative${}^{(5)}$, commutative monoid and the quotient monoid $H/H^\times$ is finitely generated, then $H$ is FF.
The corollary is essentially due to A. Geroldinger and F. Halter-Koch, cf. Proposition 2.7.8.4 in their monograph:
- Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, Boca Raton (FL), 2006.
These results are basically a generalization of the existence part of the fundamental theorem of arithmetic and, at least to some extent, bring about the same kind of "structural content" (though atomic factorizations are, in general, far from being unique in any sensible way). In fact, they are part of a bigger picture; and though the OP asked only about finitely generated (commutative) monoids, it is perhaps worth seeing what can be done in a greater generality.
In particular, taking an irreducible of $H$ to be a non-unit $a \in H$ such that $a \ne xy$ for all non-units $x, y \in H$ with $HxH \ne HaH \ne HyH$ leads to:
Theorem 2. If $H$ is a Dedekind-finite${}^{(6)}$ monoid satisfying the ACCP${}^{(7)}$, then every non-unit of $H$ factors as a product of irreducibles.
To my knowledge, a proof of Theorem 2 for commutative rings was first published by D.D. Anderson and S. Valdes-Leon in
- Factorization in Commutative Rings with Zero Divisors, Rocky Mountain J. Math. 26 (1996), No. 2, 439-480;
their proof (see loc. cit., Theorem 3.2) carries over verbatim to commutative monoids (and is, in some sense, straightforward from the definitions). In a different direction, we have:
Theorem 3. The conditions below are equivalent:
- $H$ is acyclic and satisfies the ACCP.
- $H$ is unit-cancellative and satisfies the ACCPR and the ACCPL${}^{(8)}$.
Moreover, each of these conditions implies that $H$ is atomic.
Note that every atom is irreducible, and the converse is true if $H$ is acyclic. Insofar as I'm aware, the terms "atom" and "atomic" were first coined by P.M. Cohn, whom I believe deserves credit also for the statement and the proof of the following (see Question 395980 for discussion on this "controversial point"):
Corollary 2. If $H$ is cancellative and satisfies the ACCPR and the ACCPL, then it is atomic.
There is more to the story. For further details and generalizations, I can only address the interested reader to an article of mine on monoids and preorders (here) and to a couple of related papers with Austin A. Antoniou (here) and Laura Cossu (here).
Notes.
(1) $a \mid_H b$, for some $a, b \in H$, if and only if $b \in HaH$.
(2) Given a preorder $\preceq$ on a set $X$, we say that two elements $x, y \in H$ are $\preceq$-equivalent if $x \preceq y \preceq x$.
(3) A submonoid $K$ of $H$ is divisor-closed if $x \mid_H y$ for some $x \in H$ and $y \in K$ implies that $x \in K$ (among other things, this guarantees that $H$ and $K$ have the same units).
(4) $H$ is acyclic if $xyz \ne y$ for all $x, y, z \in H$ such that either $x$ or $z$ is not a unit.
(5) $H$ is unit-cancellative if $xy \ne x \ne yx$ for all $x, y \in H$ such that $y$ is a non-unit. Obviously, every cancellative monoid is unit-cancellative; moreover, every acyclic monoid is unit-cancellative and the two notions coincide on the level of commutative monoids. On the other hand, Adian's embedding theorem (that is, the special case of Guba's embedding theorem where the left and right graphs of a finite semigroup presentation are cycle-free) implies at once the existence of a finitely generated, cancellative monoid with trivial group of units that is not acyclic.
(6) $H$ is Dedekind-finite if every left- or right-invertible element is a unit (or equivalently, if $xy = 1_H$ for some $x, y \in H$ implies $yx = 1_H$). It is fairly clear that every unit-cancellative or commutative monoid is Dedekind-finite.
(7) $H$ satisfies the ACCP if there is no (infinite) sequence $a_1, a_2, \ldots$ of elements of $H$ such that $Ha_nH \subsetneq Ha_{n+1}H$ for each $n \in \mathbf N^+$ (a set of the form $HaH$ with $a \in H$ is called a principal ideal of $H$).
(8) $H$ satisfies the ACCPR if there is no (infinite) sequence $a_1, a_2, \ldots$ of elements of $H$ such that $a_nH \subsetneq a_{n+1}H$ for each $n \in \mathbf N^+$ (a set of the form $aH$ with $a \in H$ is called a principal right ideal of $H$); and it satisfies the ACCPL if the opposite monoid of $H$ satisfies the ACCPR.
