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.

摘要

近年来，岩浆一词已被用来表示一对形式 $(秒,＊)$ 其中 * 是集合 S 上的二元运算。受该术语的启发，我们使用符号和术语 $\mathcal{米}(秒)$（S的岩浆）来表示集合S上所有二元运算的集合（即所有具有潜在集合S 的岩浆的集合。）我们在层次图的上下文中研究岩浆之间的分布关系$\mathcal{米}(秒)$当一个操作分布在另一个操作上时恰好出现的顶点和边。图论图像和术语有助于以直观的方式激发问题并以合理的方式表达他们的解决方案。虽然大多数考虑通常是针对任意集合进行的，但经常特别强调以下情况$秒=\mathbf{n},$具有n 个元素的原型集。在这种情况下，将探索诸如开始和插入的基数、完全连接的子集和最长无环路径等参数。