Page "Inverse element" Paragraph 23
from
Wikipedia
A function is the left ( resp.
right ) inverse of a function ( for function composition ), if and only if ( resp.
) is the identity function on the domain ( resp.
codomain ) of.
The inverse of a function is often written, but this notation is sometimes ambiguous.
Only bijections have two-sided inverses, but any function has a quasi-inverse, i. e. the full transformation monoid is regular.
The monoid of partial functions is also regular, whereas the monoid of injective partial transformations is the prototypical inverse semigroup.
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