Towards a Better Understanding of the Semigroup Tree

TL;DR

This paper investigates the structure and regularities of the semigroup tree, analyzing behaviors of descendant counts, chain properties, and implications for Fibonacci-like conjectures in semigroup enumeration.

Contribution

It characterizes the behaviors of descendant counts in the semigroup tree and explores chain properties, advancing understanding of semigroup structures and their enumeration.

Findings

Identifies two types of regularities in descendant counts.

Characterizes properties of chains in the semigroup tree.

Provides insights towards Fibonacci-like conjecture in semigroup enumeration.

Abstract

In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which of the two types takes place in particular for well-known classes of semigroups. Also we study the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof. We conclude with some thoughts that show how this study of the semigroup tree may help in solving the conjecture of Fibonacci-like behavior of the number of semigroups with given genus.