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
S. I. Adjan, Defining Relations and Algorithmic Problems for Groups and Semigroups, Proc. of the Steklov Inst. of Math. 85, Amer. Math. Soc. (1967). Translated from Trudy Matem. In-ta AN SSSR im. V. A. Steklova 85 (1966).
R. Baer, “Free sums of groups and their generalizations II, III,” Amer. J. of Math. 72 (1950), 625–670.
N. Harrison, “Real length functions in groups,” Transactions of the Amer. Math. Soc. 174 (1972), 77–106.
R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Ergeb. der Math. 89, Springer-Verlag (1977).
J. W. Morgan and P. B. Shalen, “Valuations, trees, and degenerations of hyperbolic structures, I,” Ann. of Math. 120 (1984), 401–476.
J. H. Remmers, “On the geometry of semigroup presentations,” Adv. in Math. 36 (1980), 283–296.
F. S. Rimlinger, “A subgroup theorem for pregroups,” in Combinatorial Group Theory and Topology, Ann. of Math. Studies 111, Princeton Univ. Press (to appear).
F. S. Rimlinger, Pregroups and Bass-Serre Theory, Memoirs of the Amer. Math. Soc. (to appear).
O. A. Sarkisjan, “Some relations between the word and divisibility problems in groups and semigroups,” Math. USSR Izvestija 15, Amer. Math. Soc. (1980), 161–171. Translated from Izv. Akad. Nauk SSSR Ser. Matem. 43 (1979).
O. A. Sarkisjan, “On the word and divisibility problems in semigroups and groups without cycles,” Math. USSR Izvestija 19, Amer. Math. Soc. (1982), 643–656. Translated from Izv. Akad. Nauk SSSR Ser. Matern. 45 (1981).
P. B. Shalen, “Dendrology of Groups: An Introduction,” this volume.
J. R. Stallings, Group Theory and Three-Dimensional Manifolds, Yale Math. Monograph 4, Yale Univ. Press (1971).
J. R. Stallings, “The cohomology of pregroups,” in Conference on Group Theory Univ. of Wisc.-Parkside 1972, Lecture Notes in Math. 319, Springer-Verlag (1973), 169–182.
J. R. Stallings, “A graph-theoretic lemma and group-embeddings,” in Combinatorial Group Theory and Topology, Ann. of Math. Study 111, Princeton Univ. Press (to appear).
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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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