The kernel of monoid morphisms.

*(English)*Zbl 0698.20056This article is a continuation of the work of B. Tilson [J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)]. In the paper the kernel of a relation \(\phi\) : \(M\to N\) of monoids is introduced. A relation of monoids is a relation whose graph #\(\phi\) \(=\{(m,n)|\) \(n\in m\phi \}\) is a submonoid of \(M\times N\). This concept includes morphism and division. The kernel of a relation \(\phi\) is a category, constructed directly from the constituents of \(\phi\). The kernel provides the foundation for a prime decomposition theorem of finite relations of monoids. It is shown that every relation may be written as a composition of “primitive” relations. A relation is primitive if its kernel bears a certain relationship to a simple monoid, that is a monoid with non non- trivial congruences.

A new product, the block product, is introduced to complement the kernel construction. The block product is a specific form of the two-sided semidirect product, called a double semidirect product. It is shown that there is a deep connection between the kernel and the block product. An adjoint-like relationship between these two concepts is established.

A new product, the block product, is introduced to complement the kernel construction. The block product is a specific form of the two-sided semidirect product, called a double semidirect product. It is shown that there is a deep connection between the kernel and the block product. An adjoint-like relationship between these two concepts is established.

Reviewer: V.Fleischer

##### MSC:

20M50 | Connections of semigroups with homological algebra and category theory |

20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |

18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |

##### Keywords:

kernel; relation of monoids; prime decomposition theorem; finite relations of monoids; simple monoid; block product; semidirect product
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\textit{J. Rhodes} and \textit{B. Tilson}, J. Pure Appl. Algebra 62, No. 3, 227--268 (1989; Zbl 0698.20056)

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##### References:

[1] | Clifford, A.; Preston, G., The algebraic theory of semigroups, () · Zbl 0178.01203 |

[2] | Eilenberg, S., Automata, Languages and machines, Vol. B, (1976), Academic Press New York |

[3] | Krohn, K.; Rhodes, J., Algebraic theory of machines, Trans. amer. math. soc., 116, 450-464, (1965) · Zbl 0148.01002 |

[4] | MacLane, S., Categories for the working Mathematician, (1971), Springer Berlin |

[5] | Rhodes, J., A homomorphism theorem for finite semigroups, Math. systems theory, 1, 289-304, (1967) · Zbl 0204.03303 |

[6] | Tilson, B., On the complexity of finite semigroups, J. pure appl. algebra, 5, 187-208, (1974) · Zbl 0293.20049 |

[7] | Tilson, B., Depth decomposition theorem, () |

[8] | Tilson, B., Categories as algebra: an essential ingredient in the theory of monoids, J. pure appl. algebra, 48, 83-198, (1987) · Zbl 0627.20031 |

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