Abstract
This paper is a survey of some recent results on the word problem for amalgams of inverse semigroups. Some decidability results for special types of amalgams are summarized pointing out where and how the conditions posed on amalgams are used to guarantee the decidability of the word problem. Then a recent result on undecidability is shortly illustrated to show how small is the room between decidability and undecidability of the word problem in amalgams of inverse semigroups.
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References
Bennett, C.: Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973)
Bennett, P.: Amalgamated free product of inverse semigroups. J. Algebra 198, 499–537 (1997)
Bennett, P.: On the structure of inverse semigroup amalgams. Int. J. Algebra Comput. 7(5), 577–604 (1997)
Cherubini, A., Mazzucchelli, M.: On the decidability of the word problem for amalgamated free products of inverse semigroups. Semigroup Forum 76(2), 309–329 (2008)
Cherubini, A., Rodaro, E.: Amalgams vs Yamamura’s HNN-extensions of inverse semigroups. Algebra Colloq. 18(04), 647–657 (2011)
Cherubini, A., Jajcayová, T., Rodaro, E.: Maximal subgroups of amalgams of finite inverse semigroups. Semigroup Forum 9(2), 401–424 (2015)
Cherubini, A., Meakin, J., Piochi, B.: Amalgams of free inverse semigroups. Semigroup Forum 54, 199–220 (1997)
Cherubini, A., Meakin, J., Piochi, B.: Amalgams of finite inverse semigroups. J. Algebra 285, 706–725 (2005)
Cherubini, A., Nuccio, C., Rodaro, E.: Multilinear equations in amalgams of finite inverse semigroups. Int. J. Algebra Comput. 21(01n02), 35–59 (2011)
Cherubini, A., Nuccio, C., Rodaro, E.: Amalgams of finite inverse semigroups and deterministic context-free languages. Semigroup Forum 85(1), 129–146 (2012)
Haataja, S., Margolis, S., Meakin, J.: Bass-Serre theory for groupoids and the structure of full regular semigroup amalgams. J. Algebra 183, 38–54 (1996)
Hall, T.E.: Finite inverse semigroups and amalgamation. In: Goberstein, S.M., Higgins, P.M. (eds.) Semigroups and Their Applications, pp. 51–56. Reidel, Dordrecht (1987)
Jajcayová, T.: HNN-extensions of inverse semigroups. Ph.D. thesis at University of Nebraska-Lincoln Department of Mathematics and Statistics (1997)
Janin, D.: Toward a higher-dimensional string string theory for the modeling of computerized systems. Technical report RR-1477-13, LaBRI, IPB, Université de Bordeaux (2013)
Jones, P.R., Margolis, S.W., Meakin, J.C., Stephen, J.B.: Free products of inverse semigroups. Glasg. Math. J. 33, 373–387 (1991)
Kellendonk, J.: The local structure of tiling and their integer group of coinvariance. Commun. Math. Phys. 187, 115–157 (1997)
Kellendonk, J.: Topological equivalence of tilings. J. Math. Phys. 38, 1823–1842 (1997)
Kimura, N.: On semigroups. Ph.D. thesis at Tulane University of Louisiana (1957)
Lawson, M.V.: Inverse Semigroups. The Theory of Partial Symmetries. World Scientific, River Edge (1998)
Margolis, S., Meakin, J., Sapir, M.: Algorithmic problems in groups, semigroups and inverse semigroups. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups, pp. 147–214 (1995)
Meakin, J.: Inverse semigroups: some open questions (2012)
Morita, K.: Universality of a reversible two-counter machine. Theor. Comput. Sci. 168, 303–320 (1996)
Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Trudy Matematicheskogo Instituta imeni VA Steklova 3(29), 44–143 (1955)
Paterson, A.L.T.: Grupoids, Inverse Semigroups, and Their Operator Algebras. Birkhauser Boston, Boston (1999)
Petrich, M.: Inverse Semigroups. Wiley, New York (1984)
Rodaro, E.: Bicyclic subsemigroups in amalgams of finite inverse semigroups. Int. J. Algebra Comput. 20(1), 89–113 (2010)
Rodaro, E., Cherubini, A.: Decidability of the word problem in Yamamura’s HNN-extensions of finite inverse semigroups. Semigroup Forum 77(2), 163–186 (2008)
Rodaro, E., Silva, P.V.: Amalgams of inverse semigroups and reversible two-counter machines. J. Pure Appl. Algebra 217(4), 585–597 (2013)
Sapir, M.V.: Algorithmic problems for amalgams of finite semigroups. J. Algebra 229(2), 514–531 (2000)
Steinberg, B.: A topological approach to inverse and regular semigroups. Pac. J. Math. 208(2), 367–396 (2003)
Stephen, J.B.: Presentation of inverse monoids. J. Pure Appl. Algebra 198, 81–112 (1990)
Stephen, J.B.: Amalgamated free products of inverse semigroups. J. Algebra 208, 339–424 (1998)
Acknowledgments
The first author is grateful to the organizers of ICSAOT 2014 for inviting her to present these results and acknowledges support from PRIN project 2011 “Automi e Linguaggi Formali: Aspetti Matematici e Applicativi.” The last author acknowledges support from the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT—Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2013 as well as support from the FCT project SFRH/BPD/65428/2009.
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Cherubini, A., Rodaro, E. (2015). Decidability Versus Undecidability of the Word Problem in Amalgams of Inverse Semigroups. In: Romeo, P., Meakin, J., Rajan, A. (eds) Semigroups, Algebras and Operator Theory. Springer Proceedings in Mathematics & Statistics, vol 142. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2488-4_1
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