ON ALGEBRAIC ENTANGLED ALGEBRAS AND FIELDS

Abstract

Algebraic entanglement arises when replacing the usual second-order symmetry between positive and negative real numbers by a symmetry of order three related to the definition of the identity element for addition of numbers having this symmetry. With this notion, we form a field $\mathcal{T}$, where the additive associativity is assisted (or aaa), meaning that it is slightly more demanding than the current one. This aaa-field contains three distinct entangled copies of $\mathbf{R}$. The Cayley-Dickson doubling procedure applied three times successively to $\mathcal{T}$ gives three sets of numbers $\mathcal{E}, \mathcal{H}$ and $\mathcal{O}$, which are division and quadratically normed aaa-algebras of $3^2, 3^4$ and $3^8$ entangled real dimensions, respectively. The multiplication of $\mathcal{H}$ is noncommutative, and that of $\mathcal{O}$ is nonassociative. Basic mathematical analysis on $\mathcal{T}$ shows that differentiability of a function at a point does not imply its continuity at that point. Within a geometric representation of $\mathcal{E}$ in $\mathbf{R}^3$, we also find key analogies between its basic mathematical analysis and that on C , including an analog of the Cauchy integral formula.

Received: April 10, 2024;
Accepted: June 5, 2024