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::: {#magma .Definition 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. :::
::::: {#semigroup .Definition title="semigroup"} To understand semigroup, you need to know magma...
:::: {#incmagma include="magma"} ::: {.Definition 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. ::: ::::
... Now you know magma, let's see the definition of semigroup.
(S, ⋅) is a semigroup if it is an associative magma. :::::
::::::: {#monoid .Definition title="monoid"} To understand monoid, you need to know semigroup...
:::::: {#incsemigroup include="semigroup"} ::::: {.Definition title="semigroup"} To understand semigroup, you need to know magma...
:::: {#incmagma include="magma"} ::: {.Definition 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. ::: ::::
... Now you know magma, let's see the definition of semigroup.
(S, ⋅) is a semigroup if it is an associative magma. ::::: ::::::
... Now you know semigroup, let's see the definition of monoid.
A monoid is a semigroup with an identity element. :::::::
::::::::: {#group .Definition title="group"} To understand group, you need to know monoid...
:::::::: {#incmonoid include="monoid"} ::::::: {.Definition title="monoid"} To understand monoid, you need to know semigroup...
:::::: {#incsemigroup include="semigroup"} ::::: {.Definition title="semigroup"} To understand semigroup, you need to know magma...
:::: {#incmagma include="magma"} ::: {.Definition 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. ::: ::::
... Now you know magma, let's see the definition of semigroup.
(S, ⋅) is a semigroup if it is an associative magma. ::::: ::::::
... Now you know semigroup, let's see the definition of monoid.
A monoid is a semigroup with an identity element. ::::::: ::::::::
... 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. :::::::::