  Let $X$ be a finite connected poset, $K$ a field of characteristic zero and
$I(X,K)$ the incidence algebra of $X$ over $K$ seen as a Lie algebra under the
commutator product. In the first part of the paper we show that any
$\frac{1}{2}$-derivation of $I(X,K)$ decomposes into the sum of a
central-valued $\frac 12$-derivation, an inner $\frac{1}{2}$-derivation and a
$\frac{1}{2}$-derivation associated with a map $\sigma:X^2_<\to K$ that is
constant on chains and cycles in $X$. In the second part of the paper we use
this result to prove that any transposed Poisson structure on $I(X,K)$ is the
sum of a structure of Poisson type, a mutational structure and a structure
determined by $\lambda:X^2_e\to K$, where $X^2_e$ is the set of $(x,y)\in X^2$
such that $x<y$ is a maximal chain not contained in a cycle.
