### Abstract

Symmetry is one of the most fundamental concepts of all nature. We discuss the general theory of symmetry based on theory of semigroups. This includes and extends the theory of groups, which is by now broadly used as an algebraic tool for studying many types of symmetries observed by people. However, symmetry groups are only a particular case of more general, symmetry semigroups. The latter are in turn an instance of an even more general groupoids of symmetry, which may further be a topic of special consideration.

Original language | English (US) |
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Pages (from-to) | 1758-1776 |

Number of pages | 19 |

Journal | Journal of Mathematical Chemistry |

Volume | 54 |

Issue number | 9 |

DOIs | |

State | Published - Oct 1 2016 |

### Keywords

- Automorphism (symmetry) group
- Color symmetry
- Endomorphism monoid
- Fractal
- Hidden regularities
- Hidden symmetry
- Inverse semigroup
- Monoid
- Orbit
- Quasicrystal
- Semifractal
- Semigroup

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

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## Cite this

Rosenfeld, V. R., & Nordahl, T. E. (2016). Semigroup theory of symmetry.

*Journal of Mathematical Chemistry*,*54*(9), 1758-1776. https://doi.org/10.1007/s10910-016-0653-4