Green index in semigroups: generators, presentations and automatic structures - Mathematics > Group TheoryReport as inadecuate




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Abstract: Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S byleft- and by right multiplication. This gives rise to a partition of thecomplement of T in S, and to each equivalence class of this partition wenaturally associate a relative Schutzenberger group. We show how generatingsets for S may be used to obtain generating sets for T and the Schutzenbergergroups, and vice versa. We also give a method for constructing a presentationfor S from given presentations of T and the Schutzenberger groups. Theseresults are then used to show that several important properties are preservedwhen passing to finite Green index subsemigroups or extensions, including:finite generation, solubility of the word problem, growth type, automaticity,finite presentability for extensions and finite Malcev presentability in thecase of group-embeddable semigroups. These results provide commongeneralisations of several classical results from group theory and Rees indexresults from semigroup theory.



Author: Alan J. Cain, Robert Gray, Nik Ruskuc

Source: https://arxiv.org/



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