×

Adjan’s theorem and conjugacy in semigroups. (English) Zbl 0797.20043

S. I. Adyan has shown that under certain circumstances the natural homomorphism from a semigroup into a group is injective [Defining relations and algorithmic problems in semigroups and groups, Tr. Mat. Inst. Steklova 85 (1967; Zbl 0204.01702)]. It is natural to ask whether under the conditions imposed by Adyan, conjugacy is both preserved and reflected by the natural homomorphism. We say that conjugacy is preserved by the natural homomorphism if the images of conjugate elements are conjugate and we say that conjugacy is reflected by the natural homomorphism if two elements are conjugate in the semigroup whenever their images in the group are conjugate. One problem is that there seems to be no standard definition for two elements in an arbitrary semigroup to be conjugate to one another. One definition has been given in J. Dauns [Semigroup Forum 38, No. 3, 355-364 (1989; Zbl 0671.16002)]. Another definition is used by G. Lallement [“Semigroups and combinatorial applications” (Wiley 1979; Zbl 0421.20025)]. Consider a presentation \(P = \{X \mid r_ 1 = s_ 1\), \(r_ 2 = s_ 2,\dots\}\) where \(X\) is an alphabet and \(r_ j\) and \(s_ j\) are nonempty positive words in \(X\). From this presentation it is possible to construct two undirect graphs; the left graph of \(P\) and the right graph of \(P\). If the presentation \(P\) has the property that both the left graph and right graph are cycle free, then we say that \(P\) satisfies Adyan’s condition (AC). The authors modify the definition of conjugacy used by Lallement so that the main result holds: if the presentation \(P\) satisfies (AC), then conjugacy is both reflected and preserved by the natural homomorphism.

MSC:

20M05 Free semigroups, generators and relations, word problems
20M15 Mappings of semigroups
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Adjan, S. I.,Defining relations and algorithmic problems in semigroups and groups, Proc. of the Steklov Inst. of Math.85 (1967); Translated from Trudy Mathem. In-ta AN SSSR im. V. A. Steklova, Amer. Math. Soc.85 (1966).
[2] Dauns, J.,Centers of semigroup rings and conjugacy classes, Semigroup Forum38 (1989), 355–364. · Zbl 0671.16002 · doi:10.1007/BF02573243
[3] Lallement, G., ”Semigroups and Combinatorial Applications”, John Wiley & Sons, New York, 1979. · Zbl 0421.20025
[4] Lyndon, R. C. and P. E. Schupp, ”Combinatorial Group Theory”, Springer-Verlag, Berlin, 1977. · Zbl 0368.20023
[5] Remmers, J. H.,On the geometry of semigroup presentations, Adv. in Math.36 (1980), 283–296. · Zbl 0438.20041 · doi:10.1016/0001-8708(80)90018-3
[6] Remmers, J.H.,A Geometric Approach For Some Algorithmic Problems For Semigroups, Thesis, Univ. of Mich. (1971).
[7] Sarkisjan, O. A.,On the word and divisibility problems in semigroups and groups without cycles Math. USSR Izvestija 19, Amer. Math. Soc. (1982), 634–656; Translated from Izv. Akad. Nauk SSSR Ser. Matem.45 (1981).
[8] Schupp, P. E.,On Dehn’s algorithm and the conjugacy problem, Math. Ann.178 (1968), 119–130. · Zbl 0164.01901 · doi:10.1007/BF01350654
[9] Teymouri, J.,Geometric Methods In Group Theory, Thesis, SUNY Albany (1988).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.