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The word problem for \(\kappa \)-terms over the pseudovariety of local groups

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In this paper we study the \(\kappa \)-word problem for the pseudovariety \(\mathbf{LG}\) of local groups, where \(\kappa \) is the canonical signature consisting of the multiplication and the pseudoinversion. We solve this problem by transforming each arbitrary \(\kappa \)-term \(\alpha \) into another one \(\alpha ^*\) called the \(\mathbf{LG}\)-canonical form of \(\alpha \) and by showing that different canonical forms have different interpretations over \(\mathbf{LG}\). The procedure of construction of these canonical forms consists in applying reductions determined by a set \(\Sigma \) of \(\kappa \)-identities. As a consequence, \(\Sigma \) is a basis of \(\kappa \)-identities for the \(\kappa \)-variety generated by \(\mathbf{LG}\).

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Acknowledgements

This work was supported by the European Regional Development Fund, through the programme COMPETE, and by the Portuguese Government through FCT (Fundação para a Ciência e a Tecnologia) within the Projects UIDB/00013/2020, UIDP/00013/2020 and PEst-C/MAT/UI0013/2014.

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Authors and Affiliations

  1. CMAT, Departamento de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    J. C. Costa & M. L. Teixeira

  2. CMAT, Escola Superior de Tecnologia e Gestão, Instituto Politécnico de Leiria, Campus 2, Morro do Lena, Alto Vieiro, 2411-901, Leiria, Portugal

    C. Nogueira

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  1. J. C. Costa
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  2. C. Nogueira
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  3. M. L. Teixeira
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Correspondence to J. C. Costa.

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Communicated by Jorge Almeida.

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Costa, J.C., Nogueira, C. & Teixeira, M.L. The word problem for \(\kappa \)-terms over the pseudovariety of local groups. Semigroup Forum 103, 439–468 (2021). https://doi.org/10.1007/s00233-021-10207-9

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