标题： 半群中的共轭和最小交换同余 显示英文标题

标题： Conjugacy and Least Commutative Congruences in Semigroups

摘要： 给定一个半群$S$和$s,t \in S$，如果$s=pr$和$t=rp$，则写成$s \sim_p^1 t$，对于某些$p,r \in S \cup \{1\}$。 这种关系被称为“主要共轭”，以及它的传递闭包$\sim_p$，在代数的许多领域中已被广泛使用和研究。 本文致力于一种自然推广，由$s \sim_s^1 t$在$s=p_1\cdots p_{n}$和$t=p_{f(1)}\cdots p_{f(n)}$时定义，对于某些$p_1, \dots, p_n \in S \cup \{1\}$和排列$f$的$\{1, \dots, n\}$，以及其传递闭包$\sim_s$。 关系$\sim_s$是由$\sim_p^1$或$\sim_p$生成的同余，而且是在任何半群上的最小交换同余。我们探讨了$\sim_s$的一般性质，在群和环的背景下进行讨论，将其与其他半群共轭关系进行比较，并在自由半群、Rees矩阵半群、图逆半群以及各种变换半群中完整描述其等价类。 显示英文摘要

摘要： Given a semigroup $S$ and $s,t \in S$, write $s \sim_p^1 t$ if $s=pr$ and $t=rp$, for some $p,r \in S \cup \{1\}$. This relation, known as "primary conjugacy", along with its transitive closure $\sim_p$, has been extensively used and studied in many fields of algebra. This paper is devoted to a natural generalization, defined by $s \sim_s^1 t$ whenever $s=p_1\cdots p_{n}$ and $t=p_{f(1)}\cdots p_{f(n)}$, for some $p_1, \dots, p_n \in S \cup \{1\}$ and permutation $f$ of $\{1, \dots, n\}$, together with its transitive closure $\sim_s$. The relation $\sim_s$ is the congruence generated by either $\sim_p^1$ or $\sim_p$, and is moreover the least commutative congruence on any semigroup. We explore general properties of $\sim_s$, discuss it in the context of groups and rings, compare it to other semigroup conjugacy relations, and fully describe its equivalence classes in free, Rees matrix, graph inverse, and various transformation semigroups.