This just a question about concept and nomenclature; and I haven't been able to find an adequate answer in the literature so far. Suppose $\Bbb{P}$ is locally finite poset with a unique minimal element $\emptyset$. Recall that a real-valued function $\varphi: \Bbb{P} \longrightarrow \Bbb{R}$ is harmonic if
\begin{equation} \varphi(x) = \sum_{x \lhd \, y} \varphi(y) \end{equation}
where $x \lhd \, y$ indicates that $x$ is covered by $y$ with respect to the partial order.
Question: Is the following the correct definition of minimal harmonic function; as used, for example, in defining the minimal part of the Martin boundary of $\Bbb{P}$?
Definition: A harmonic function $\varphi: \Bbb{P} \longrightarrow \Bbb{R}$ is said to be minimal if it is non-negative (i.e. $\varphi(x) \geq 0$ for all $x \in \Bbb{P}$), normalised (in the sense that $\varphi(\emptyset) = 1$), and has the property that whenever we have a decomposition $\varphi = s\varphi_1 + (1-s)\varphi_2$ for some pair of non-negative, normalised harmonic functions $\varphi_1$ and $\varphi_2$ with $0 < s < 1$ then either $\varphi = \varphi_1 = \varphi_2$.
thanks, Ines.
