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Abstract. A proof of the Generalized Road Coloring Problem, independent of the recent work by Beal and Perrin, is presented, using both semigroup methods and Trakhtman's algorithm. Algebraic properties of periodic, strongly connected digraphs are studied in the semigroup context. An algebraic condition which characterizes periodic, strongly connected digraphs is determined in the context of periodic Markov chains.

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Consider a semigroup generated by matrices associated with an edge-coloring of a strongly connected, aperiodic digraph. We call the semigroup Lie-solvable if the Lie algebra generated by its elements is solvable. We show that if the semigroup is Lie-solvable then its kernel is a right group. Next, we analyze the Lie algebras generated by the kernel. The Lie structure of a subalgebra generated by two idempotents is completely described. Finally, we discuss an infinite class of examples that are shown to always produce strongly connected aperiodic digraphs. ICALP Workshop on Semigroups and AutomataLisboa,

The structure of transformation semigroups on a finite set is analyzed by introducing a hierarchy of functions mapping subsets to subsets. The resulting hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or kernels. This kernel hierarchy produces a set of tools that provides direct access to computations of interest in probability limit theorems; in particular, finding certain factors of idempotent limit measures. In addition, when considering transformation semigroups that arise naturally from edge colorings of directed graphs, as in the road-coloring problem, the hierarchy produces simple techniques to determine the rank of the kernel and to decide when a given kernel is a right group. In particular, it is shown that all kernels of rank one less than the number of vertices must be right groups and their structure for the case of two generators is described.