Incidence algebras and dot products

Question

Central question: since an arbitrary poset (or lattice) is not necessarily (comprised within) a vector space, how does one think about the structural similarity of convolutions on incidence algebras to dot products (and, generalizing, inner products)?

Incidence algebra: IIRC, on a poset $X$, we can define an algebra by taking a function $f(x,y)$ for $x < y$ and defining the convolution of $f(x,y)$ and $g(x,y)$ to be

$(f {\cdot} g)(x,y) = {\sum_{x \le z \le y}f(x,z)g(z,y)}$

This looks structurally similar to the old dot product,

${\mathbf f {\cdot} \mathbf g} = {\sum_{i=1}^nf_ig_i}$

with the exception that $z$ ranges over the subset between $x$ and $y$ and the elements $x, y, z \in X$ do not have to be in $\mathbb R$, $\mathbb C$, etc. In applied settings my instinct is to use the linearity of convolution liberally in the same way I would with dot products (or inner products, more carefully), but is that safe? Advisable? Interesting?

And is there an abstract generalization of either dot products or inner products that includes incidence algebras?

[Corrected based on first comment.]

Answer 1

In applied settings my instinct is to use the linearity of convolution liberally in the same way I would with dot products (or inner products, more carefully), but is that safe? Advisable? Interesting?

I think by "linearity" you mean:

1. $c (f*g) = (c f)*g$
2. $(c + f)*g = c*g + f*g$

We immediately run into some questions about what exactly multiplication and addition are meant to represent here.

One thing you could be thinking is that $c$ is a function $X\to Y$ just like $f, g$, and multiplication is just the convolution. If that's the case, then these "linearity" properties just basic properties of a ring, and therefore hold if the algebra is a ring.

Alternatively, you might be thinking that $c$ is a scalar, in which case $c*f$ doesn't really make sense. The first linearity property might make sense though, depending on how exactly you define multiplication: $$\begin {align} c (f*g) & = c{\sum_{x \le z \le y}f(x,z)g(z,y)} \\ & = {\sum_{x \le z \le y} (c f(x,z)) g(z,y)}\\ & = (c f)*g \end {align} $$

Lastly, you could instead mean that the convolution is "linear" in the sense that $c (f*g)(x,y) = (f*g)(c x,y)$. Again, we have to wonder what multiplication is as the multiplication in $c (f*g)$ is (potentially) different from the multiplication in $c x$. It seems to me in general that it would be challenging to define multiplication in a way that this identity holds, but perhaps it is true for some algebras. I don't know if these algebras have a name though.