Incidence algebra

Remarks

The product is associative, and the function $\delta$ with $\delta(x, x) = 1$ and $\delta(x, y) = 0$ for $x < y$ is the identity. The zeta function is $\zeta(x, y) = 1$ for all $x \le y$. An element $f$ is invertible if and only if $f(x, x)$ is a unit of $K$ for every $x$. The inverse of $\zeta$ is the Möbius function $\mu$ of $P$, defined by $\mu(x, x) = 1$ and $\mu(x, y) = -\sum_{x \le z < y} \mu(x, z)$ for $x < y$.

Example: let $P = \set{1, \dots, n}$ with the usual order. Every interval is finite, and $f \mapsto (f(i, j))_{i \le j}$ identifies $I(P, K)$ with the algebra of upper triangular $n \times n$ matrices over $K$; the product $*$ becomes matrix multiplication.

Non-example: $\Q$ with its usual order is not locally finite, since $[0, 1]$ is infinite. The sum defining $(\zeta * \zeta)(0, 1)$ would have infinitely many terms equal to $1$.