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
Full Text:
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