Definition:Generator of Monoid

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\left({M, \circ}\right)$ be a monoid

Let $S \subseteq M$.

Let $H$ be the smallest submonoid of $M$ such that $S \subseteq H$.


Then:

  • $S$ is a generator of $\left({H, \circ}\right)$
  • $S$ generates $\left({H, \circ}\right)$
  • $\left({H, \circ}\right)$ is the submonoid of $\left({M, \circ}\right)$ generated by $S$.


This is written $H = \left\langle {S} \right\rangle$.


If $S$ is a singleton, i.e. $S = \left\{{x}\right\}$, then we can (and usually do) write $H = \left\langle {x}\right\rangle$ for $H = \left\langle {\left\{{x}\right\}}\right\rangle$.


Also known as

Some sources refer to such an $S$ as a set of generators of $H$, but this terminology is misleading, as it can be interpreted to mean that each of the elements of $S$ is itself a generator of $H$ independently of the other elements.


Also see