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For every set M we may construct the monoid of endomorphisms (M → M,◦,id), where ◦ is function composition and id is the identity function.
Theorem 1.1 (Cayley representation for (Set) monoids)
Every monoid (M,⊕,e) is a sub-monoid of the monoid of endomorphisms on M.
Exequiel Rivas, Mauro Jaskelioff. 2014.
https://arxiv.org/abs/1406.4823v1
My hunch about the relationship b/w design and artefact turns out to be true. It also represents a mathematical version of Henry Ford's saying, "You won't get rich through inventions. You get rich through improvements."