The kernel of monoid morphisms

Abstract

This paper introduces what the authors believe to be the correct definition of the kernel of a monoid morphism a, : M+ N. This kernel is a category, constructed directly from the constituents of 9. In the case of a group morphism, our kernel is a groupoid that is divisionally equivalent to the traditional kernel. This article is a continuation of the work in [8]. The thesis of [8] is that categories, as generalized monoids, are essential ingredients in monoid decomposition theory. The principal development in [S] was the introduction of division, a new ordering for categories, which extend the existing notion for monoids. Since its introduction in [3], division has proved to be the ordering of choice for monoids. This extension of division to categories allows for the useful comparison of monoids and categories. A strong candidate for the title ‘kernel’ was introduced in [8]. This candidate is also a category and is called the derived category of 9. The derived category operation and the wreath product of monoids are shown to have an adjoint-like relationship. This relationship is summed up in the Derived Category Theorem [8, Theorem 5.21. The derived category has its origins in [6], where it appears as the derived semigroup. The kernel construction of this paper is an improvement over the derived category for a variety of reasons. First, it is smaller in the divisional sense. Second, it is a reversal invariant construction. Third, it combines more effectively with classical structure theories. For example, when applied to surmorphisms that cannot be further factored, the kernel has a particularly simple form. This leads to important decomposition theorems for finite monoids.