Definition:Generator of Subsemigroup
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This page is about generators of subsemigroups. For other uses, see Generator.
Definition
Let $\struct {S, \circ}$ be a semigroup.
Let $\O \subset X \subseteq S$.
Let $\struct {T, \circ}$ be the smallest subsemigroup of $\struct {S, \circ}$ such that $X \subseteq T$.
Then:
- $X$ is a generator of $\struct {T, \circ}$
- $X$ generates $\struct {T, \circ}$
- $\struct {T, \circ}$ is the subsemigroup of $\struct {S, \circ}$ generated by $X$.
This is written:
- $T = \gen X$
Also known as
Some sources refer to such an $X$ as a set of generators of $T$, but this terminology is misleading, as it can be interpreted to mean that each of the elements of $X$ is itself a generator of $T$ independently of the other elements.
Examples
Positive Odd Numbers
Let $\struct {\Z, +}$ be the additive group of integers.
Let $A$ be the set of positive odd integers.
The subsemigroup of $\struct {\Z, +}$ generated by $A$ is the semigroup of strictly positive integers under addition.
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings