Divisibility in a Semigroup

Let $(S,\cdot)$ be a Semigroup. An element $x:S$ is said to left divide an element $y:S$ when there Merely exists some $r:S$ such that $x\cdot r=y$. Likewise $x$ is said to right divide $y$ if there Merely exists some $r:S$ such that $r\cdot x=y$. Finally, $x$ is a two-sided divisor of $y$ if it is both a left and right divisor.

If $S$ is Commutative Semigroup, then these three notions coincide, and we speak merely of divisibility, and write $x\mid y$.

Properties

• Associativity ensures that the divisibility relation is Transitive: if $r_{1}\cdot x=y$ and $r_{2}\cdot y=z$, then we have

  	$\displaystyle(r_{2}\cdot r_{1})\cdot x$	$\displaystyle=r_{2}\cdot(r_{1}\cdot x)$
  		$\displaystyle=r_{2}\cdot y$
  		$\displaystyle=z$

• If $S$ is a Monoid, then the divisibility relation is Reflexive, as $\varepsilon\cdot x=x$.

• In general, the divisibility relation is not Antisymmetric. A nice example of this is divisibility in a Group, where every element divides every other element, as $(h\cdot g^{-1})\cdot g=h$.

• Most statements about divisibility can be translated into statements about Principal Ideals, as $a\mid b$ if and only if $\langle b\rangle\subseteq\langle a\rangle$.