Identities with involutions on incidence algebras

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 lambda 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. Let (lambda) over cap and sigma(lambda) denote, respectively, the lambda-orthogonal and the lambda-symplectic involutions of I(P, F). For the case that P has length at most 2 and vertical bar P vertical bar >= 4, we show that the (lambda) over cap -identities and the sigma(lambda)-identities of I(P, F) follow fromthe ordinary identity [x(1), x(2)][x(3), x(4)]. In that context, passing to the particular case I(C-2n, F), where C-2n is a poset called crown with 2n elements, and using the classification of the involutions on I(C-2n, F), we show that, for all involutions rho on I(C-2n, F), every rho-identity also follows from the ordinary identity [x(1), x(2)][x(3), x(4)]. For the case that P has length at most 3 and vertical bar P vertical bar >= 4, we determine the generators of the T((lambda) over cap)-ideal Id((lambda) over cap) (I(P, F)) when every element of P that is neither minimal nor maximal is fixed by lambda and, for such an element, there exists a unique minimal element of P that is comparable with it.