A Classification of the Projective Lines over Small Rings

Abstract

A compact classification of the projective lines defined over (commutative) rings (with unity) of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of points on the line, the number of points represented by coordinates with at least one entry being a unit, the cardinality of the neighbourhood of a generic point of the line as well as those of the intersections between the neighbourhoods of two and three mutually distant points, the number of `Jacobson' points per a neighbourhood, the maximum number of pairwise distant points and, finally, a list of representative/base rings. The classification is presented in form of a table in order to see readily not only the fine traits of the hierarchy, but also the changes in the structure of the lines as one goes from one type to the other. We hope this study will serve as an impetus to a search for possible applications of these remarkable geometries in physics, chemistry, biology and other natural sciences as well.

7 pages, 1 figure; Version 2: classification extended up to order 20, references updated; Version 3: classification extended up to order 31, two more references added; Version 4: references updated, minor correction

References in corpus (7)

• Projective Representations I. Projective lines over rings
• On distant-isomorphisms of projective lines
• Divisible Designs, Laguerre Geometry, and Beyond
• The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$<x^3-x>$ and Quantum Entanglement II. The Mermin "Magic" Square/Pentagram
• A Classification of the Projective Lines over Small Rings II. Non-Commutative Case
• The Projective Line Over the Finite Quotient Ring GF(2)[$x$]/$<x^3-x>$ and Quantum Entanglement I. Theoretical Background
• On the Fine Structure of the Projective Line Over GF(2) x GF(2) x GF(2)

Cited by in corpus (10)

• Projective Ring Line of an Arbitrary Single Qudit
• Projective Ring Line Encompassing Two-Qubits
• Projective Ring Line of a Specific Qudit
• Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators
• Divisible Designs, Laguerre Geometry, and Beyond
• Qudits of composite dimension, mutually unbiased bases and projective ring geometry
• Weak mutually unbiased bases
• Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions
• From pentacyclic coordinates to chain geometries, and back
• Pauli graph and finite projective lines/geometries