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Neural networks powered research of semigroups

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Open In Colab PyPI version CircleCI Documentation Status codecov

Neural Semigroups

Here we try to model Cayley tables of semigroups using neural networks.

The simplest way to get started is to use Google Colaboratory

More documentation can be found here.

There is also a paper with more maths inside. It can be used to cite the package:

      title={A Neural Network for Semigroups}, 
      author={Edouard Balzin and Boris Shminke},


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:

  1. 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;

  2. 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 amusement. 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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