Conjugation properties in incidence algebras

Incidence algebras can be regarded as a generalization of full matrix algebras. We present some conjugation properties for incidence functions. The list of results is as follows: a criterion for a convex-diagonal function $f$ to be conjugated to the diagonal function $fe$; conditions under which the conjugacy $f\sim Ce+\zeta_{\lessdot}$ holds (the function $Ce+\zeta_{\lessdot}$ may be thought of as an analog for a Jordan box from matrix theory); a proof of the conjugation of two functions $\zeta_<$ and $\zeta_{\lessdot}$ for partially ordered sets that satisfy the conditions mentioned above; an example of a partially ordered set for which the conjugacy $\zeta_<\sim \zeta_{\lessdot}$ does not hold. These results involve conjugation criteria for convex-diagonal functions of some partially ordered sets.