A $C_0$-semigroup is a strongly continuous family $\{T(t)\}$ of bounded linear operators on a normed space $X$, indexed in $\mathbb R_+$ and with two additional properties that make it look like an operator-valued version of the exponential function: $T(t)T(s)=T(t+s)$ for all $t,s\in \mathbb R_+$ and $T(0)=\hbox{Id}_X$. On the basis of two concrete examples I was wondering whether there is room to generalise this notions to operator families indexed on *general directed sets*. For this purpose, defining a sum-like operation on a directed set seems to be unavoidable and I was wondering whether this has been done in the past.

(I admit that the only two examples I can think of are rather trivial: $\mathbb R$ viewed as a collection of two halflines directed towards 0 (which returns the definition of $C_0$-group), as well as the positive hyperoctant in $\mathbb R^n$ (which for $n=2$ reminds of how analytic $C_0$-semigroups are defined). But what I am striving for is the definition of $C_0$-semigroups indexed on some kind of tree-like sets.)