::: {.Definition #magma title="magma"}
A magma is a set M with an operation ⋅ that sends any two elements a, b ∈ M to another element, a⋅b∈M. The symbol ⋅ is a general placeholder for a properly defined operation. This requirement that for all a, b in M, the result of the operation a⋅b also be in M, is known as the magma or closure property. 
:::


::: {.Definition #semigroup title="semigroup"}

To understand semigroup, you need to know magma...

:::: {#incmagma include="magma"}
:::::

... Now you know magma, let's see the definition of semigroup.

(S, ⋅) is a semigroup if it is an associative magma.
:::


::: {.Definition #monoid title="monoid"}
To understand monoid, you need to know semigroup...

:::: {#incsemigroup include="semigroup"}
:::::

... Now you know semigroup, let's see the definition of monoid.

A monoid is a semigroup with an identity element.
:::


::: {.Definition #group title="group"}
To understand group, you need to know monoid...

:::: {#incmonoid include="monoid"}
:::::

... Now you know monoid, let's see the definition of group.

(S, ⋅) is a group is it is a monoid such that every element has an unique inverse.
:::