1.1 KiB
::: {.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. :::