  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.
