A Noncommutative Version of the Natural Numbers
Tyler Foster
TL;DR
This paper introduces a novel algebraic system that generalizes natural numbers with a noncommutative addition, maintaining associative, commutative, and distributive multiplication, revealing new nontrivial arithmetic properties.
Contribution
It constructs and analyzes a noncommutative analog of natural numbers with unique algebraic properties and consistent arithmetic operations.
Findings
Addition is noncommutative but commutes with itself.
Multiplication remains associative, commutative, and distributive.
The system exhibits interesting nontrivial arithmetic properties.
Abstract
In this note, we construct and study an algebraic system similar to the natural numbers, but with noncommutative addition. The addition we introduce is a binary operation that commutes with itself in the sense of N. Durov. Neverheless, the multiplication in this system (defined by iterating the noncommutative addition) turns out to be associative, commutative, and distributive over addition, and the resulting system has interesting and nontrivial arithmetic.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Advanced Algebra and Logic