Distributivity relations on the binary operations over a fixed set

Abstract

In recent years, the word magma has been used to designate a pair of the form $(S,\ast )$ where * is a binary operation on the set S. Inspired by that terminology, we use the notation and terminology $\mathcal{M}(S)$ (the magma of S) to denote the set of all binary operations on the set S (i.e. the set of all magmas with underlying set S.) We study distributivity relations among magmas in the context of a hierarchy graph having $\mathcal{M}(S)$ as vertices and edges occurring precisely when an operation distributes over another one. The graph theoretic imagery and terminology serve to motivate questions in an intuitive way and to express their solutions in a reasonable fashion. While most considerations are done in general for arbitrary sets, particular emphasis is placed frequently on the case when $S=\mathbf{n},$ the archetypal set with n elements. In that case, parameters such as cardinalities of outsets and insets, fully connected subsets, and longest cycle-free paths are explored.

Keywords:

• Complete subgraphs
• distributive hierarchy graph
• isolated operators
• longest paths
• magma monoid
• universal distributors

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 05EXX
• 05C25