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Is there a procedure to construct all semigroups over a given set without backtracking?

(Edit: see also how many associative binary operations are there on a finite set and ratio of semigroups over a set N to magmas over N goes to 0 as the cardinality of N gets bigger.)

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math wannabe
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    You mean that you want to construct every associative binary operation possible from $S\times S\rightarrow S$? – rschwieb Apr 29 '12 at 18:44
  • Yes :) something besides constructing them blindly and removing non-associative ones. – math wannabe Apr 29 '12 at 18:59
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    Considering how difficult it is to constructs all the groups of order, say, 2048, I'd be surprised if there were any useful procedure for constructing all the semigroups. By the way, the number of semigroups on a set of 7 elements is already 7743056064, according to http://oeis.org/A023814. – Gerry Myerson Apr 30 '12 at 03:05
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    We still don't know how many semigroups of order 10 there are I believe. I think $9$ was cracked a couple of years ago. –  Apr 30 '12 at 12:29

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