Categories as algebra: An essential ingredient in the theory of monoids.

*(English)*Zbl 0627.20031The purpose of this extensive paper is to present an elaborate algebraic theory of finite categories (with finitely many objects and morphisms). Finite categories are finite partial semigroups satisfying some natural axioms. The paper shows convincingly that for theoretical purposes it is much better to keep them partial, instead of yielding to the easy temptation of the completion by a garbage zero. Among various topics the paper deals with in detail we find a generalization to finite categories of the notion “monoid K divides monoid M”, or of the notion of a pseudovariety of monoids, including a Birkhoff-type theorem for varieties of finite categories and a description of the smallest no-trivial variety. For a homomorphism \(f: M\to N\) of finite monoids, a certain finite category derived from M is suggested as its kernel, on ground that in case of groups it coincides with the ordinary kernel of a group homomorphism. Completely described are the finite categories which are locally trivial, i.e. such that for each object the endomorphism monoid is trivial. A remarkable application of the notion of division of categories is given to the membership problem for products of varieties.

Reviewer: V.Koubek

##### MSC:

20M07 | Varieties and pseudovarieties of semigroups |

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

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

08A55 | Partial algebras |

08B25 | Products, amalgamated products, and other kinds of limits and colimits |

08C15 | Quasivarieties |

##### Keywords:

algebraic theory of finite categories; finite partial semigroups; pseudovariety of monoids; Birkhoff-type theorem; varieties of finite categories; finite monoids; kernel; endomorphism monoid; division of categories; membership problem; products of varieties
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##### References:

[1] | Birkhoff, G., On the structure of abstract algebras, (), 433-454 · Zbl 0013.00105 |

[2] | Brzozowski, J.A.; Simon, I., Characterizations of locally testable events, Discrete math., 4, 243-271, (1973) · Zbl 0255.94032 |

[3] | Eilenberg, S., () |

[4] | Eilenberg, S.; Schützenberger, M.P., On pseudovarieties, Adv. in math., 19, 413-418, (1976) · Zbl 0351.20035 |

[5] | Higgins, P.J., Notes on categories and groupoids, (1971), Van Nostrand Reinhold London · Zbl 0226.20054 |

[6] | Knast, R., Some theorems on graph congruences, RAIRO inform. théor., 17, 331-342, (1983) |

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

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

[9] | S.W. Margolis and J.E. Pin, Inverse semigroups and extensions of groups by semi-lattices, J. Algebra, to appear. · Zbl 0625.20043 |

[10] | Nico, W., Wreath products and extensions, Houston J. math., 9, 71-99, (1983) · Zbl 0516.18002 |

[11] | J.E. Pin, H. Straubing and D. Thérien, Locally trivial categories and unambiguous concatenation, J. Pure Appl. Algebra, to appear. · Zbl 0645.20046 |

[12] | Rhodes, J., Kernel systems — a global study of homomorphisms on finite semigroups, J. algebra, 49, 1-45, (1977) · Zbl 0379.20054 |

[13] | Rhodes, J., Infinite iteration of matrix semigroups, part II, J. algebra, 100, 25-137, (1986) · Zbl 0626.20050 |

[14] | Rhodes, J., On the Cantor-Dedekind property of the tilson order on categories and graphs, J. pure appl. algebra, 48, 55-82, (1987), this issue · Zbl 0628.18001 |

[15] | J. Rhodes and B. Tilson, The kernel of monoid morphisms, Preprint. · Zbl 0698.20056 |

[16] | Straubing, H., Finite semigroup varieties of the form V ∗ D, J. pure appl. algebra, 36, 53-94, (1985) · Zbl 0561.20042 |

[17] | H. Straubing and D. Thérien, A new proof of Knast’s Theorem, Preprint. |

[18] | D. Thérien, Union of groups is a local variety, private communication. (A published version with a different title will appear in Semigroup Forum.) |

[19] | Thérien, D.; Weiss, A., Graph congruences and wreath products, J. pure appl. algebra, 36, 205-215, (1985) · Zbl 0559.20042 |

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

[21] | Tilson, B., Complexity of semigroups and morphisms, () · Zbl 0293.20049 |

[22] | B. Tilson, Languages and the block product, Preprint. · Zbl 0226.20060 |

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