Can someone help me to understand how the free regular semigroups are defined? I've been looking this for hours, but I just can't get it!
Any help is greatly appreciated.
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Two hints: 1) the free regular semigroup with k generators doesn't exist. 2) You need to define the allowed homomorphisms first, then you can spell out the universal property: The generators can be assigned to arbitrary elements in the target regular semigroup, and it must be possible to extend this to a homomorphism which is unique. Can you construct an example where uniqueness is bound to fail? – Thomas Klimpel Jun 13 '17 at 00:08
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See also https://math.stackexchange.com/questions/104893/existence-of-universal-enveloping-inverse-semigroup-similar-to-grothendieck-gr/105265#105265 for a concrete proof that no unique regular semigroup satisfying ... can exist. – Thomas Klimpel Jun 13 '17 at 00:10