A group-embeddable non-automatic semigroup whose universal group is automatic
نویسنده
چکیده
The concept of an automatic structure has been generalized from groups [ECH+] to semigroups [CRRT]. Several authors [CRR, Hof, Kam] have asked the following question: Let S be a finitely generated semigroup embeddable in a group and let G be its universal group [CP, Chapter ]. If G is automatic, must S be automatic? Examples in favour of this implication include: free groups and semigroups; braid groups and semigroups [ECH+, Chapter ]; abelian groups and their subsemigroups [CRR, Proposition .]. A similar question asked whether the automatism of a group implied the automatism of its positive subsemigroups. (A positive subsemigroup is a subsemigroup generated by a group generating set.) This question was recently answered negatively [CRR, Section ]. However, the techniques used cannot be adapted to answer the original question. The purpose of this paper is to answer the original question in the negative. Section contains an example of a finitely generated semigroup S such that
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