Rational series in the free group and the Connes operator
Aaron Lauve, Christophe Reutenauer
TL;DR
This paper characterizes rational series over free groups using Connes' operator, shows that rational Malcev–Neumann series have rational expressions without simplifications, and develops an algorithm for the free skew field's word problem.
Contribution
It introduces a new characterization of rational series via Connes' operator and provides an effective algorithm for the free skew field's word problem.
Findings
Rational series over free groups can be characterized using Connes' operator.
Rational Malcev–Neumann series have simplified rational expressions.
An algorithm is developed for solving the word problem in the free skew field.
Abstract
We characterize rational series over the free group by using an operator introduced by A. Connes. We prove that rational Malcev--Neumann series posses rational expressions without simplifications. Finally, we develop an effective algorithm for solving the word problem in the free skew field.
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Taxonomy
Topicsadvanced mathematical theories · semigroups and automata theory · Quantum chaos and dynamical systems