Ivan Shishkin, Birch Grove

Monoid

Concept history

A revision trail for this concept page.

5 revisions

Back

Revision 812

8/1/2026, 5:42:03 PM · Sequoia

Concept edited

Recorded titleMonoid
Recorded typeDefinition
Compare with revision 8082 changed lines
1A monoid $(M,*)$ is a [[unital|unital magma]] such that the law $*$ is associative, which means that for all $x,y,z\in M$, one has the equality:
1A monoid $(M,*)$ is a [[unital|unital magma]] such that the [[Operation|binary operation]] $*$ is associative, which means that for all $x,y,z\in M$, one has the equality:
2$$
3(x*y)*z=x*(y*z)
4$$
5**Examples:**
6
7$\bullet$ The magma $(\Z,-)$ is not a monoid since $(1-2)-3=-4\neq 2=1-(2-3)$.
8
9$\bullet$ The magma $(\Z,+)$ is a monoid since $(a+b)+c=a+(b+c)$ for all $a,b,c\in\Z$ and $0$ is an identity element.

Revision 808

8/1/2026, 5:33:01 PM · Sequoia

Concept edited

Compare with revision 7868 changed lines
1A monoid $(M,*)$ is a magma such that the law $*$ is associative, which means that for all $x,y,z\in M$, one has the equality:
1A monoid $(M,*)$ is a [[unital|unital magma]] such that the law $*$ is associative, which means that for all $x,y,z\in M$, one has the equality:
2$$
3(x*y)*z=x*(y*z)
4$$
5Examples:
5**Examples:**
6
6$\bullet$ The magma $(\Z,-)$ is not a monoid since $(1-2)-3=-4\neq 2=1-(2-3)$.
7$\bullet$ The magma $(\Z,+)$ is a monoid since $(a+b)+c=a+(b+c)$ for all $a,b,c\in\Z$.
8
9$\bullet$ The magma $(\Z,+)$ is a monoid since $(a+b)+c=a+(b+c)$ for all $a,b,c\in\Z$ and $0$ is an identity element.

Revision 786

8/1/2026, 9:40:00 AM · Sequoia

Concept edited

This older revision predates detailed metadata tracking.

Revision 785

8/1/2026, 9:39:50 AM · Sequoia

Concept edited

This older revision predates detailed metadata tracking.

Revision 750

7/31/2026, 10:05:18 AM · Sequoia

Concept created

A monoid $(M,*)$ is a magma such that the law $*$ is associative, which means that for all $x,y,z\in M$, one has the equality:
$$
(x*y)*z=x*(y*z)
$$
Examples:
$\bullet$ The magma $(\Z,-)$ is not a monoid since $(1-2)-3=-4\neq 2=1-(2-3)$.
$\bullet$ The magma $(\Z,+)$ is a monoid since $(a+b)+c=a+(b+c)$ for all $a,b,c\in\Z$.