Editing Monoid (section)

== Monoid homomorphisms ==
[[File:Exponentiation as monoid homomorphism svg.svg|thumb|x150px|Example monoid homomorphism {{math|''x'' ↦ 2<sup>''x''</sup>}} from {{math|{{color|#004000|('''N''', +, 0)}}}} to {{math|{{color|#400000|('''N''', ×, 1)}}}}. It is injective, but not surjective.]]
A [[homomorphism]] between two monoids {{math|(''M'', ∗)}} and {{math|(''N'', •)}} is a function {{math|''f'' : ''M'' → ''N''}} such that
* {{math|1=''f''(''x'' ∗ ''y'') = ''f''(''x'') • ''f''(''y'')}} for all {{math|''x''}}, {{math|''y''}} in {{math|''M''}}
* {{math|1=''f''(''e''<sub>''M''</sub>) = ''e''<sub>''N''</sub>}},
where {{math|''e''<sub>''M''</sub>}} and {{math|''e''<sub>''N''</sub>}} are the identities on {{math|''M''}} and {{math|''N''}} respectively. Monoid homomorphisms are sometimes simply called '''monoid morphisms'''.

Not every [[semigroup homomorphism]] between monoids is a monoid homomorphism, since it may not map the identity to the identity of the target monoid, even though the identity is the identity of the image of the homomorphism.{{efn|{{math|1=''f''(''x'') ∗ ''f''(''e''<sub>''M''</sub>) = ''f''(''x'' ∗ ''e''<sub>''M''</sub>) = ''f''(''x'')}} for each {{math|''x''}} in {{math|''M''}}, when {{math|''f''}} is a semigroup homomorphism and {{math|''e''<sub>''M''</sub>}} is the identity of its domain monoid {{math|''M''}}.}} For example, consider {{math|['''Z''']<sub>''n''</sub>}}, the set of [[residue class]]es modulo {{math|''n''}} equipped with multiplication. In particular, {{math|[1]<sub>''n''</sub>}} is the identity element. Function {{math|''f'' : ['''Z''']<sub>3</sub> → ['''Z''']<sub>6</sub>}} given by {{math|[''k'']<sub>3</sub> ↦ [3''k'']<sub>6</sub>}} is a semigroup homomorphism, since {{math|1=[3''k'' ⋅ 3''l'']<sub>6</sub> = [9''kl'']<sub>6</sub> = [3''kl'']<sub>6</sub>}}. However, {{math|1=''f''([1]<sub>3</sub>) = [3]<sub>6</sub> ≠ [1]<sub>6</sub>}}, so a monoid homomorphism is  a semigroup homomorphism between monoids that maps the identity of the first monoid to the identity of the second monoid and the latter condition cannot be omitted.

In contrast, a semigroup homomorphism between groups is always a [[group homomorphism]], as it necessarily preserves the identity (because, in the target group of the homomorphism, the identity element is the only element {{math|''x''}} such that {{math|1=''x'' ⋅ ''x'' = ''x''}}).

A [[bijective]] monoid homomorphism is called a monoid [[isomorphism]]. Two monoids are said to be isomorphic if there is a monoid isomorphism between them.