If $M$ is a monoid, then an $M$-graded algebra over $k$ is the same thing as a $k[M]$ comodule algebra. To see this, if $\delta$ is a coaction of $k[M]$ on an algebra $A$, for each $m \in M$ define
$$A_m = \{ a \in A : \delta(a) = m \otimes a \}.$$
One can then check that if $\delta$ is compatible with the multiplication in $A$, this defines an $M$-grading on $A$. Conversely, if $A$ is an $M$-graded algebra, then each $a \in A$ can be written uniquely in the form $a = \sum_{m \in M} a_m$, with $a_m \in A_m$. So we can define the coaction
$$\delta(a) = \sum_{m \in M} m \otimes a_m,$$
and one can check that this makes $A$ into a $k[M]$ comodule algebra. The details are for example in Hopf Algebras and Their Actions on Rings chapter 4.
I'm interested in whether there is a similar definition of an $M$-filtered algebra in terms of the coalgebraic structure of $k[M]$. Of course the first issue with this is that an $M$-filtration doesn't make sense for an arbitrary monoid $M$, so we should probably require that $M$ has some sort of compatible ordering. (This may also mean that we need to modify our definition of $k[M]$ to take this ordering into account.) The main property I'd hope for from such a definition is a generalization of the usual notion of associated graded algebra of an $\mathbb{N}$-filtered algebra. I'd also be interested in any non-trivial special cases where this can be done. So overall the questions are:
Is there a bialgebra associated to a (partially / linearly / lattice / well) ordered monoid $M$? Or perhaps an additional structure on the usual $k[M]$ that keeps track of the ordering?
Is there a way to define $M$-filtered algebras in terms of this bialgebra?
Does this yield a nice generalization of associated graded algebras?
P.S. For an ordered monoid $M$, by an $M$-filtered algebra I mean an algebra $A$ with subspaces $A_m$ for each $m \in M$ such that
- $\cup_{m \in M}A_m = A$
- $n < m \implies A_n \subseteq A_m$
- $A_n A_m \subseteq A_{nm}$
In particular, its not obvious what one would mean by associated graded algebra when $M \neq \mathbb{N}$.
