Abstract
The Krohn-Rhodes theorem describes how an arbitrary finite monoid can be decomposed into a wreath product of groups and aperiodic monoids. New tools have recently been introduced to refine and extend this fundamental result. New theorems can be obtained by considering monoids as a special case of categories, thus allowing more general structures to appear as building blocks in decompositions results. Also, a two-sided version of the wreath product may be used as the connecting operation. This paper combines the two ideas: the new operation, called the block product, is defined directly as acting on categories and basic properties are presented. As an application, an open problem in the theory of regular languages is solved.
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