Relationship between Dumont–Thomas complement and two’s-complement arithmetic

Determine which properties are shared by Dumont–Thomas complement numeration systems and the two’s-complement numeration system, and determine whether integer addition in Dumont–Thomas complement numeration systems can be performed analogously to addition of natural numbers.

Background

The paper introduces Dumont–Thomas complement numeration systems for all integers by analogy with the binary two’s-complement system. It establishes positionality criteria but does not analyze in detail whether arithmetic operations, especially addition, inherit comparable behavior. The authors explicitly ask both for a comparison of structural properties and for an addition procedure analogous to the natural-number setting.

References

What properties are shared between our systems and two's complement? In particular, can the addition of integers be performed similarly to the addition of natural numbers?

— Positionality of Dumont--Thomas numeration systems for integers  (2503.04487 - Kreczman et al., 6 Mar 2025) in Section 7, “Open problems”