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arXiv · 1609.06288

No positive cone in a free product is regular

Susan Hermiller·Zoran Sunic
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Abstract

We show that there exists no left order on the free product of two nontrivial, finitely generated, left-orderable groups such that the corresponding positive cone is represented by a regular language. Since there are orders on free groups of rank at least two with positive cone languages that are context-free (in fact, 1-counter languages), our result provides a bound on the language complexity of positive cones in free products that is the best possible within the Chomsky hierarchy. It also provides a strengthening of a result by Cristobal Rivas stating that the positive cone in a free product of nontrivial, finitely generated, left-orderable groups cannot be finitely generated as a semigroup.

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Susan Hermiller, Zoran Sunic. 2016-09-20. No positive cone in a free product is regular. https://arxiv.org/abs/1609.06288

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