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Is there a common presentation of the semigroup of functions from a given (finite) set to itself?

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    $\begingroup$ Are you asking about the monoid of maps from a finite set to itself? $\endgroup$
    – S. Carnahan ♦
    Commented Jan 26, 2013 at 15:17

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There are several well known presentations. One has to first choose your favorite presentation of the symmetric group. Then one usually adds in a rank n-1 idempotent but some people add all of them or other elements. Then one figures out the extra relations. See page 161 of Classical Finite Semigroups by Ganyushkin and Mazorchuk. You should know the monoid of all self-maps of a set is usually called the full transformation monoid or $T_n$ when googling.

Pages 42-43 of http://research-repository.st-andrews.ac.uk/bitstream/10023/2821/3/NikolaRuskucPhDThesis.pdf give two presentations.

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  • $\begingroup$ I found the answer on page 161 in: Olexandr Ganyushkin • Volodymyr Mazorchuk, Classical Finite Transformation Semigroups:An Introduction, Springer 2009 $\endgroup$
    – Hercules
    Commented Feb 5, 2013 at 17:42

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