Monoid Congruence

An Equivalence Relation $R\subseteq M\times M$ on a Monoid $(M,\cdot,1)$ is a

• Left congruence if $R(x,y)\Rightarrow R(ax,ay)$
• *Right congruence if $R(x,y)\Rightarrow R(xa,ya)$
• Two-sided congruence if it is a left congruence and a right congruence.

Properties

• A monoid congruence is a two-sided congruence if and only if

  $R(a,b)\land R(x,y)\Rightarrow R(ax,by)$

• Every Two-Sided Monoid Ideal $I\subseteq M\times M$ induces a congruence

  $\par\rho(x,y):=(x=y)\lor(x\in I\land y\in I)\par$

  known as the Rees Congruence of $I$.

• If $M$ posesses a Malcev Operation $t$ (for instance, if $M$ is a Group), then every Internal Reflexive Relation is a congruence (see Mal’cev, Protomodular, Homological and Semi-Abelian Categories), but we will only recap the basic proof.

  Symmetry follows from the fact that

  	$\displaystyle x\sim y$	$\displaystyle=x\sim x\land x\sim y\land y\sim y$
  		$\displaystyle\Rightarrow t(x,x,y)\sim t(x,y,y)$
  		$\displaystyle\Rightarrow y\sim x$

  Transitivity follows from

  	$\displaystyle x\sim y\land y\sim z$	$\displaystyle=x\sim y\land y\sim y\land y\sim z$
  		$\displaystyle=t(x,y,y)\sim t(y,y,z)$
  		$\displaystyle=x\sim z$

  Moreover, if $t$ is Para-associative Operation (EG, a Groud), then congruence relations are entirely determined by what they relate to the identity element, as $x\sim y$ if and only if $t(x,y,e)\sim e$.

  This is why we can get away with using Normal Subgroups and Ring Ideals forming Quotient Groups and Quotient Rings instead of congruences.