Symmetric Dual Arithmetic (SDA): A Proposed Algebraic Framework

This paper details a proposed comprehensive axiomatization for Symmetric Dual Arithmetic (SDA).

SDA is proposed as an algebraic structure defined over a set S incorporating a unique minimal element 0 (zero) and a unique maximal element Omega ("anti-zero"), which are fundamentally linked by an order-reversing duality involution (<->).

We first introduce a generalized construction, S(L), demonstrating how SDA-like ordered structures with duality can be derived from any complete lattice L, and provide a concrete arithmetical example, S(B), based on the Boolean lattice, illustrating how arithmetic might be extended if L possesses a compatible semiring structure.

Subsequently, we establish the canonical arithmetical instance, S_Omega (based on non-negative integers N0), as a bounded commutative semiring equipped with this involution. For S_Omega, this paper outlines detailed proof strategies for consideration for its foundational algebraic properties (including associativity and distributivity), emphasizing the meticulous multi-level case analysis involved and the critical need for subsequent mechanized verification of these proofs.

The framework proposes a uniqueness theorem, phrased via a universal mapping property, which aims to characterize S_Omega as the free SDA generated by N0. The duality involution <-> is examined, detailing its proposed non-homomorphic nature with respect to the primary arithmetic operations but its precise adherence to De Morgan-like laws concerning the induced pairwise lattice operations (min/max).

The scope of currently defined operations, particularly subtraction, is clarified, with extensions identified as important avenues for future work. A conceptual categorical formalization of SDA, defining SDA objects and morphisms, is presented.

SDA is then compared with, and distinguished from, related existing algebraic structures such as bounded semirings and residuated lattices.

Furthermore, a compelling application in modeling bounded computation suggests SDA's practical utility. This work offers SDA as a potentially significant and robust framework for abstract algebra, inviting further mathematical scrutiny, formal proof mechanization, and application development.