Classification of Connected Shelves

TL;DR

This paper classifies small connected shelves, explores their group structures, and introduces a new polynomial invariant, contributing to the understanding of finite algebraic structures called shelves.

Contribution

It provides a complete classification of connected shelves with fewer than six elements and proposes a new polynomial invariant for shelves.

Findings

Classified all connected shelves of order less than six.

Analyzed the group structure generated by latin shelves.

Introduced a two-variable shelf polynomial and conjecture for idempotent shelves.

Abstract

We investigate finite right-distributive binary algebraic structures called shelves. We first use symbolic computations with Python to classify (up to isomorphism) all connected shelves with order less than six. We explore the group structure generated by the rows of \textit{latin} shelves. We also define two-variable shelf polynomial by analogy with the quandle polynomial and then state a conjecture about connected idempotent shelves.