A monoid where every element has finitely many divisors Is there a special name or has there been any study of monoids of this form? This came up in considering the general construction of a multivariate power series algebra over a ring $R$; usually we take the set of functions $M\to R$ where $M$ is the set $[I]$ of finite multisets on the index set $I$ with pointwise addition and scalar multiplication, and define the multiplication as $(ab)_\alpha=\sum_{\beta\le\alpha} a_\beta b_{\alpha-\beta}$, where $\alpha,\beta\in[I]$, $\alpha-\beta$ is the multiset formed by removing all the elements of $\beta$ from $\alpha$ (with multiplicity), and $\beta\le\alpha$ means that all the elements in $\beta$ are in $\alpha$ with equal or greater multiplicity.
This definition makes all the monomials commute with each other, for example if $I=\{1,2\}$ then $x_1^2x_2=x_{\{1,1,2\}}=x_{\{2,1,1\}}=x_2x_1^2$ (where $x_{\{1,1,2\}}$ is notation for the coefficient function that is $1$ at $\{1,1,2\}$ and zero at every other $\alpha\in[I]$). This may not be desirable, so one can generalize to the situation where none of the variables commute, and in this case the monomials are given by strings like $x_1x_2x_1x_3$, which is to say we take $M$ to be the free monoid on $I$, with multiplication now being defined by $(ab)_\alpha=\sum_{\beta\gamma=\alpha} a_\beta b_\gamma$ (where the sum is taken over all possible values of $\beta,\gamma\in M$ such that $\beta\gamma=\alpha$). This is well defined in both the free monoid and the earlier multiset monoid (which is actually the free commutative monoid) because the equation $\beta\gamma=\alpha$ has only finitely many solutions.
So to return to the question, has there been any study or a name for monoids $M$ such that $\{\beta\in M:\beta\mid\alpha\}$ is finite for each $\alpha$, or the power series algebras that result on such monoids (by repeating the above construction on an arbitrary such monoid instead of a free monoid or free commutative monoid)?
 A: I suppose you define division in a monoid $M$ as follows: $v$ divides $u$ if there exist $x, y \in M$ such that $u = xvy$. In semigroup theory, this property is usually denoted $u \leqslant_\mathcal{J} v$ (note the inversion: $v \mid u$ is the same as $u \leqslant_\mathcal{J} v$). The relation $\leqslant_\mathcal{J}$ is one of the Green's preorders.
A monoid is said to be finite $\mathcal{J}$-above if, for each $u \in M$, the set $\{v \in M \mid u \leqslant_\mathcal{J} v\}$ is finite.
I am not sure whether this was the earliest occurrence, but the term appeared in 
[1] and has been used several times in the literature since then.
Formal power series have also be defined on graded monoids. See [2].
[1] K. Henckell, S. Lazarus, J. Rhodes, Prime decomposition theorem for arbitrary semigroups: general holonomy decomposition and synthesis theorem. J. Pure Appl. Algebra 55 (1988), 127--172.
[2] Sakarovitch, Jacques. Chapter 4: Rational and recognisable power series. Handbook of weighted automata, 105--174, Monogr. Theoret. Comput. Sci. EATCS Ser., Springer, Berlin, 2009.
