Successor-invariance in the finite

A first-order sentence /spl theta/ of vocabulary /spl sigma/ /spl cup/ {S} is successor-invariant in the finite if for every finite /spl sigma/-structure M and successor relations S/sub 1/ and S/sub 2/ on M, (M, S/sub 1/) /spl vDash/ /spl theta/ /spl hArr/ (M, S/sub 2/) /spl vDash/ /spl theta/. In this paper I give an example of a non-first-order definable class of finite structures, which is, however, defined by a successor-invariant first-order sentence. This strengthens a corresponding result for order-invariant in the finite, due to Y. Gurevich.