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In this paper, we study polynomial identities with involution of an incidence algebra  I ( P ,  F ) where  P  is a connected finite poset with an involution  λ  and  F  is a field of characteristic zero. At first, we also consider  P  of length at most 2 and then of length at most 3. Letanddenote, respectively, the  λ -orthogonal and the  λ -symplectic involutions of  I ( P ,  F ). For the case that  P  has length at most 2 and, we show that the-identities and the-identities of  I ( P ,  F ) follow from the ordinary identity. In that context, passing to the particular case, whereis a poset called crown with 2 n  elements, and using the classification of the involutions on, we show that, for all involutions  ρ  on, every  ρ -identity also follows from the ordinary identity. For the case that  P  has length at most 3 and, we determine the generators of the-idealwhen every element of  P  that is neither minimal nor maximal is fixed by  λ  and, for such an element, there exists a unique minimal element of  P  that is comparable with it. Communicated by Igor Klep

Identities with involutions on incidence algebras