Abstract
Let ℙ be a cycle-free presentation of a semigroup. Adian showed that the semigroup S of ℙ is embedded in the group G of ℙ. Let Q = S•S-1 ⊂ G. In this paper we give a geometric argument, using diagrams, to prove a result, due basically to O. A. Sarkisian, which says: If n > 1, and q1 ∈ Q for i = 1,…,n, and if 1 = q1•q2…qn is true in G, the there exists i such that q1•q1+1 ∈ Q. This gives a set of examples of a concept invented by Baer. related to pregroups, about which little is known. A number of questions and conjectures are discussed, about the relations between these “S-pregroups,” cycle-free groups, left-ordered groups, and vector fields.
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References
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© 1987 Springer-Verlag New York Inc.
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Stallings, J.R. (1987). Adian Groups and Pregroups. In: Gersten, S.M. (eds) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9586-7_5
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DOI: https://doi.org/10.1007/978-1-4613-9586-7_5
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