The Maximal Subgroups and the Complexity of the Flow Semigroup of Finite (Di)graphs

Preprint of an article first published online in International Journal of Algebra and Computation, September 2017, doi: https://doi.org/10.1142/S0218196717500412. © 2017 Copyright World Scientific Publishing Company. http://www.worldscientific.com/worldscinet/ijac. Accepted Manuscript version is under embargo. Embargo end date: 26 September 2018.The flow semigroup, introduced by John Rhodes, is an invariant for digraphs and a complete invariant for graphs. We refine and prove Rhodes's conjecture on the structure of the maximal groups in the flow semigroup for finite, antisymmetric, strongly connected graphs. Building on this result, we investigate and fully describe the structure and actions of the maximal subgroups of the flow semigroup acting on all but k points for all finite digraphs and graphs for all k >=1. A linear algorithm is presented to determine these so-called 'defect k groups' for any finite (di)graph. Finally, we prove that the complexity of the flow semigroup of a 2-vertex connected (and strongly connected di)graph with n vertices is n- 2, completely confirming Rhodes's conjecture for such (di)graphs.Peer reviewe

Set Theory

This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces

Algebraic hierarchical decomposition of finite state automata : a computational approach

The theory of algebraic hierarchical decomposition of finite state automata is an important and well developed branch of theoretical computer science (Krohn-Rhodes Theory). Beyond this it gives a general model for some important aspects of our cognitive capabilities and also provides possible means for constructing artificial cognitive systems: a Krohn-Rhodes decomposition may serve as a formal model of understanding since we comprehend the world around us in terms of hierarchical representations. In order to investigate formal models of understanding using this approach, we need efficient tools but despite the significance of the theory there has been no computational implementation until this work. Here the main aim was to open up the vast space of these decompositions by developing a computational toolkit and to make the initial steps of the exploration. Two different decomposition methods were implemented: the VuT and the holonomy decomposition. Since the holonomy method, unlike the VUT method, gives decompositions of reasonable lengths, it was chosen for a more detailed study. In studying the holonomy decomposition our main focus is to develop techniques which enable us to calculate the decompositions efficiently, since eventually we would like to apply the decompositions for real-world problems. As the most crucial part is finding the the group components we present several different ways for solving this problem. Then we investigate actual decompositions generated by the holonomy method: automata with some spatial structure illustrating the core structure of the holonomy decomposition, cases for showing interesting properties of the decomposition (length of the decomposition, number of states of a component), and the decomposition of finite residue class rings of integers modulo n. Finally we analyse the applicability of the holonomy decompositions as formal theories of understanding, and delineate the directions for further research