A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

TL;DR

This paper explores a novel decomposition of the Fano-Snowflake configuration using the Jacobson radical of the smallest ring of ternions, revealing a fundamental difference from the classical Galois field-based factorization.

Contribution

It introduces a Jacobson radical-based partitioning of the Fano-Snowflake configuration, highlighting a new structural perspective in projective lattice configurations over ternion rings.

Findings

Partition of 21 submodules into sets of 9, 9, and 3 based on Jacobson radical entries.

A fundamental difference in line factorization compared to the Galois field case.

New insights into the structure of the Fano-Snowflake via ternion algebra.

Abstract

The Fano-Snowflake, a specific $non$-unimodular projective lattice configuration associated with the smallest ring of ternions $R_{\diamondsuit}$ (arXiv:0803.4436 and 0806.3153), admits an interesting partitioning with respect to the Jacobson radical of $R_{\diamondsuit}$. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left $R_{\diamondsuit}$-module $R_{\diamondsuit}^{3}$ are shown to split into three disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson radical entries in the generating vector is 2, 1 or 0, respectively. The corresponding "ternion-induced" factorization of the lines of the Fano plane sitting in the middle of the Fano-Snowflake (6 -- 7 -- 3) is found to $differ fundamentally$ from the natural one, i. e., from that with respect to the Jacobson radical of the Galois field of two elements (3 -- 3 -- 1).