The identities of additive binary arithmetics

TL;DR

This paper explores the algebraic identities linking two types of binary addition operations, showing they have a finite basis and relate to nilpotent rings, contributing to universal algebra theory.

Contribution

It provides a simple description of identities connecting addition modulo 2^n and bitwise addition, establishing their finite basis and algebraic equivalence to nilpotent rings.

Findings

Identities connecting the two additions have a finite basis.

The algebra with these operations is rationally equivalent to a nilpotent ring.

The algebra generates a Specht variety.

Abstract

Operations of arbitrary arity expressible via addition modulo 2^n and bitwise addition modulo 2 admit a simple description. The identities connecting these two additions have finite basis. Moreover, the universal algebra with these two operations is rationally equivalent to a nilpotent ring and, therefore, generates a Specht variety.