The kernel of monoid morphisms (Q912218)

This article is a continuation of the work of \textit{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.