Groups and monoids of Pythagorean triples connected to conics

TL;DR

This paper introduces new algebraic structures on Pythagorean triples using matrix-based operations, characterizes the resulting monoids, and uncovers novel connections with groups over conics.

Contribution

It defines and characterizes monoids of Pythagorean triples via matrix operations and explores their relationship with groups over conics, providing new algebraic insights.

Findings

Characterization of injections leading to commutative monoids

Identification of Pythagorean triple preserving matrices

Establishment of groups over conics related to Pythagorean triples

Abstract

We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and $3 \times 3$ matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean triples with the studied operations.