Neural networks powered research of semigroups
Here we try to model Cayley tables of semigroups using neural networks.
More documentation can be found here.
This work was inspired by a sudoku solver. A solved Sudoku puzzle is nothing more than a Cayley table of a quasigroup from 9 items with some well-known additional properties. So, one can imagine a puzzle made from a Cayley table of any other magma, e. g. a semigroup, by hiding part of its cells.
There are two major differences between sudoku and puzzles based on semigroups:
it's easy to take a glance on a table to understand whether it is a sudoku or not. That's why it was possible to encode numbers in a table cells as colour intensities. Sudoku is a picture, and a semigroup is not. It's difficult to check a Cayley table's associativity with a naked eye;
sudoku puzzles are solved by humans for fun and thus catalogued. When solving a sudoku one knows for sure that there is a unique solution. On the contrary, nobody guesses values in a partially filled Cayley table of a semigroup as a form of amuzement. As a result, one can create a puzzle from a full Cayley table of a semigroup but there may be many distinct solutions.
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