Lagrange-like interpolation in unitary rings, Boolean algebras and Boolean posets

It is known that every function with a finite support over a given field can be interpolated by means of the Lagrangian polynomial. The question is if a similar interpolation is possible if one considers a unitary ring or a Boolean algebra instead of a field. We get a positive answer to this question provided the similarity type of the algebra in question is enriched with one more unary operation, the so-called Baaz delta. We get an explicit construction of this interpolation polynomial in both the cases. When going to Boolean posets, we have a lack of operations but these can be substituted by the operators [Formula: see text] and [Formula: see text]. Hence, we generalize also the Baaz delta for posets as an operator and then we can derive an explicit interpolation term constructed by means of these operators also for Boolean posets.