Semigroups form a big bigger chunk than you might think. Basically you call a symbol 0 and declare xyz=0 for all elements (making associativity trivial). You still have a huge flexibility on how to define the remaining products. This is the content of the paper Michael links. In fact 99% of all semigroups up to isomorphism and anti-isomorphism satisfy xyz=0. A recent paper of Distler and Mitchell count the exact number of these guys up to isomorphism http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p51. I think they also count the number of such multiplication tables.
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Semigroups form a big chunk. Basically you call a symbol 0 and declare xyz=0 for all elements (making associativity trivial). You still have a huge flexibility on how to define the remaining products. This is the content of the paper Michael links. In fact 99% of all semigroups up to isomorphism and anti-isomorphism satisfy xyz=0. A recent paper of Distler and Mitchell count the exact number of these guys up to isomorphism http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p51. I think they also count the number of such multiplication tables. |
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