Definition:Generated Submonoid
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Definition
Let $\struct {M, \circ}$ be a monoid whose identity is $e_M$.
Let $S \subseteq M$
Let $H$ be the smallest (with respect to set inclusion) submonoid of $M$ such that $\paren {S \cup \set {e_M} } \subseteq H$.
Then $\struct {H, \circ}$ is the submonoid of $\struct {M, \circ}$ generated by $S$.
This is written $H = \gen S$.
If $S$ is a singleton, for example $S = \set x$, then we can (and usually do) write $H = \gen x$ for $H = \gen {\set x}$.
Generator
Let $\struct {H, \circ}$ be the submonoid of $\struct {M, \circ}$ generated by $S$.
Then $S$ is known as a generator of $\struct {H, \circ}$.
Also see
Sources
- 1999: J.C. Rosales and P.A. García-Sánchez: Finitely Generated Commutative Monoids ... (previous) ... (next): Chapter $1$: Basic Definitions and Results