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The kernel of monoid morphisms. (English) Zbl 0698.20056
This 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.
Reviewer: V.Fleischer

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)
Full Text: DOI
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