2016/09/20 by Susan Hermiller, Hermiller, Susan, Zoran Sunic +1
Mathematics · #06F15 #20E06 #20F60 #20M99 #68Q45 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:06F15 #msc:20E06 #msc:20F60 #msc:20M99 #msc:68Q45
paper · pdf · doi:10.48550/arxiv.1609.06288
8 pages
arxiv created 2017/11/15 · arxiv updated 2017/11/16
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.