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

TL;DR

This paper extends Lagrange interpolation to unitary rings, Boolean algebras, and Boolean posets by introducing additional operators like Baaz delta, providing explicit construction methods for these generalized interpolations.

Contribution

It introduces a generalized Lagrange interpolation method for algebraic structures beyond fields, using the Baaz delta and Min U/Max L operators for posets.

Findings

Explicit interpolation polynomials constructed for rings and Boolean algebras.

Generalization of Baaz delta operator to Boolean posets.

Interpolation applicable in structures lacking traditional polynomial operations.

Abstract

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 Min U and Max L. 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.