Algebraic Number Systems & Symmetry

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Abstractions about extended number systems and their symmetry structure, spanning hypercomplex number systems (octonion, Hurwitz quaternion, Pauli matrices), functional-algebraic structures (Banach algebra), and elementary arithmetic or group-action concepts (casting out nines, integral part, orbit in group theory).

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Banach Algebra — A Banach algebra is called unital if it has an identity element for the multiplication whose norm is 1, and commutative if its multiplication is commutative.
• Casting out nines — Casting out nines is any of three arithmetical procedures.
• Hurwitz quaternion — In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded).
• Integral part — The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations).
• Octonion — In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system.
• Orbit (group theory) — The class of 2-transitive groups (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally multiply transitive groups is well-studied in finite group theory.
• Pauli Matrices — In mathematical physics and mathematics, the Pauli matrices are a set of three 2 \times 2 complex matrices that are traceless, Hermitian, involutory and unitary.
• Unit-Quaternion Rotation Representation — A two-to-one representation of proper three-dimensional rotations by unit quaternions, where q and -q denote the same rotation and multiplication composes rotations under a declared convention.