[Paper Review] Translatable quadratical quasigroups

R. A. R. Monzo, Wiesław A. Dudek|arXiv (Cornell University)|Aug 27, 2017

graph theory and CDMA systems2 references3 citations

TL;DR

This paper introduces the concept of $k$-translatable groupoids and proves that quadratical quasigroups induced by the additive group $\mathbb{Z}_m$ are always $k$-translatable for some $k$. It systematically lists $k$-translatable quadratical quasigroups of order $m < 1200$, identifies structural properties such as self-duality and translatable behavior, and establishes that all such quasigroups over $\mathbb{Z}_m$ are $k$-translatable, with explicit constructions and dualities for orders up to 500.

ABSTRACT

The concept of a k-translatable groupoid is introduced. Those k-translatable quadratical quasigroups induced by the additive group of integers modulo m, where k&lt;40, are listed for m&lt;1200. The fine structure of quadratical quasigroups is explored in detail and the Cayley tables of quadratical quasigroups of orders 5, 9, 13 and 17 are produced. All but those of order 9 are k-translatable, for some k. Open questions and thoughts about future research in this area are given.

Motivation & Objective

• To investigate the fine algebraic structure of quadratical quasigroups, particularly their translatable properties.
• To determine which quadratical quasigroups induced by $\mathbb{Z}_m$ are $k$-translatable for $k < 40$ and $m \leq 1200$.
• To characterize the conditions under which a quasigroup induced by $\mathbb{Z}_m$ is $k$-translatable, and to explore duality and self-duality.
• To provide explicit Cayley tables for quadratical quasigroups of orders 5, 9, 13, and 17, and analyze their translatable structure.
• To address open problems regarding the existence and generation of quadratical quasigroups, especially self-dual and non-$\mathbb{Z}_m$-induced ones.

Proposed method

• Introduces the concept of a $k$-translatable groupoid and applies it to classify quadratical quasigroups based on their algebraic behavior.
• Uses the four-cycle structure from prior work to analyze and construct Cayley tables for quadratical quasigroups of orders 5, 9, 13, and 17.
• Employs the characterization of quadratical quasigroups via commutative groups and automorphisms: $xy = \varphi(x) + \psi(y)$, with $\varphi(x) + \psi(x) = x$ and $2\varphi\psi(x) = x$.
• Derives necessary and sufficient conditions for a quasigroup induced by $\mathbb{Z}_m$ to be $k$-translatable, based on modular arithmetic and parameter $a, b$ in $x \cdot y = ax + by \pmod{m}$.
• Constructs and lists $k$-translatable quadratical quasigroups for $m < 500$ and $k < 40$, including duals, using systematic modular parameter search.
• Analyzes duality by comparing $x \cdot y = ax + by \pmod{m}$ with its dual $x \circ y = bx + ay \pmod{m}$, and checks translatable behavior for both.

Experimental results

Research questions

• RQ1 For which integers $k < 40$ and $m \leq 1200$ are quadratical quasigroups induced by $\mathbb{Z}_m$ $k$-translatable?
• RQ2 Is every quadratical quasigroup induced by $\mathbb{Z}_m$ $k$-translatable for some $k$, and can this be characterized algebraically?
• RQ3 Are there self-dual quadratical quasigroups of order greater than 9, and if so, can they be constructed outside $\mathbb{Z}_m$?
• RQ4 If a quadratical quasigroup of order $m$ is $k$-translatable, is its dual $(m-k)$-translatable?
• RQ5 Can every quadratical quasigroup of order $m$ be generated by two distinct elements, especially for $m > 9$?

Key findings

• All quadratical quasigroups induced by $\mathbb{Z}_m$ are $k$-translatable for some $k$, and this is proven for all $m$ under consideration.
• The quadratical quasigroup of order 9 is the only one up to isomorphism and is self-dual, while all others of orders 5, 13, 17, 25, and 29 are $k$-translatable for some $k$.
• For $m = 25$, the quasigroup $x \cdot y = 4x + 22y \pmod{25}$ is 18-translatable, and its dual is 7-translatable.
• For $m = 29$, the quasigroup $x \cdot y = 9x + 21y \pmod{29}$ is 12-translatable, and its dual is 17-translatable.
• For $k < 11$, $k$-translatable quadratical quasigroups induced by $\mathbb{Z}_m$ exist for each $k$, and values of $m$ such that the quasigroup is $(m-k)$-translatable are determined.
• A comprehensive list of $k$-translatable quadratical quasigroups over $\mathbb{Z}_m$ is provided for $k < 40$ and $m < 1200$, including duals, with explicit parameters $a, b$ for each.