## Abstract

This paper is about sets equipped with two operations, one of which distributes over the other (a type of structure that we call a “distributive magma” to emphasize that often no other properties are assumed for the two operations involved). Upon closer scrutiny, one should notice that the paper actually centers on a set with a single associative binary operation (usually a group) with the intention to determine whether a suitable companion operation may be found so that the resulting structure is a distributive magma. The motivation here comes, for example, from the opposite roles that semigroups and abelian groups play in the theories of rings and semirings (such as the max-plus algebra). As there are two underlying operations in a distributive magma, the distributed operation and the distributor, the problem we pose here necessarily comes in two flavors depending on the role one expects from the operation with which one begins. Additional layers arise as one endows the beginning operation with specific algebraic structures or demands additional structure from the companion sought. The type of question explored includes whether there are group-theoretic differences between the two abelian group structures associated with a field and, if so, what are the inherent properties of addition and multiplication that entitles one of them to be a distributor and the other one to be the distributed one?.

Original language | English |
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Title of host publication | Contemporary Mathematics |

Publisher | American Mathematical Society |

Pages | 225-242 |

Number of pages | 18 |

DOIs | |

State | Published - 2018 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 715 |

ISSN (Print) | 0271-4132 |

ISSN (Electronic) | 1098-3627 |

### Bibliographical note

Publisher Copyright:© 2018 Amerian Mathematial Soiety.

### Funding

The second author would like to thank the Ohio University Center of Ring Theory and its applications in Athens, Ohio, for their hospitality. We thank D. Saltman for a helpful conversation.

Funders | Funder number |
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Ohio University |

## Keywords

- Distributivity
- Max-plus algebra
- Semiring