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Degrees of Word Problem for Algebras Without Finitely Presented Expansions

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Theory and Applications of Models of Computation (TAMC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10185))

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Abstract

Bergstra and Tucker [1, 2] proved that computable universal algebras have finitely presented expansions. Bergstra and Tucker, and Goncharov, independently, asked whether all finitely generated computably enumerable algebras have finitely presented expansions. Khoussainov and Hirschfeldt [3] constructed finitely generated, infinite c.e. semigroups without finitely presented expansions; furthermore, Khoussainov and Miasnikov [6] found such examples in class of groups and algebras over finite fields. In this paper, we consider Turing degrees of the word problem for semigroups constructed in [3] and for algebras over finite fields constructed in [6], and prove that the word problem for such semigroups and algebras appears in all nonzero c.e. degrees respectively.

Both authors are partially supported by MOE2011-T2-1-071 (ARC 17/11, M45110030) from Ministry of Education of Singapore, and by AcRF grants RG29/14, M4011274 from Nanyang Technological University.

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References

  1. Bergstra, J.A., Tucker, J.V.: Initial and final algebra semantics for data type specifications: two characterization theorems. SIAM J. Comput. 12, 366–387 (1983)

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  2. Bergstra, J.A., Tucker, J.V.: Algebraic specifications of computable and semicomputable data types. Theor. Comput. Sci. 50, 137–181 (1987)

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  3. Hirschfeldt, D.R., Khoussainov, B.: Finitely presented expansions of computably enumerable semigroups. Algebra Logic 51, 435–444 (2012)

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  4. Kasymov, N.Kh.: Algebras with finitely approximable positively representable enrichments. Algebra Logic 26, 441–450 (1987)

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  5. Khoussainov, B.: Randomness, computability, and algebraic specifications. Ann. Pure Appl. Logic 91, 1–15 (1998)

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  6. Khoussainov, B., Miasnikov, A.: Finitely presented expansions of semigroups, groups, and algebras. Trans. Am. Math. Soc. 366, 1455–1474 (2014)

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  7. Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)

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Acknowledgement

G. Wu is partially supported by AcRF grants MOE2016-T2-1-083 from Ministry of Education of Singapore, RG29/14, M4011274 and RG32/16, M4011672 from Nanyang Technological University and Ministry of Education of Singapore.

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Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Guohua Wu & Huishan Wu

Authors
  1. Guohua Wu
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  2. Huishan Wu
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Correspondence to Guohua Wu .

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Editors and Affiliations

  1. Anna University, Chennai, India

    T.V. Gopal

  2. Universität Bern , Bern, Switzerland

    Gerhard Jäger

  3. Universität Bern , Bern, Switzerland

    Silvia Steila

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Wu, G., Wu, H. (2017). Degrees of Word Problem for Algebras Without Finitely Presented Expansions. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_46

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  • DOI: https://doi.org/10.1007/978-3-319-55911-7_46

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-55910-0

  • Online ISBN: 978-3-319-55911-7

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