J. Almeida, A. Costa, R. Kyriakoglou and D. Perrin, Profinite Semigroups and Symbolic Dynamics, Lecture Notes in Mathematics, 2274 (2020), Springer.

Book chapter

J. Almeida and A. Costa, chapter Profinite topologies of Handbook of AutoMathA (edited by Jean-Éric Pin), European Math. Soc. Publ. House (2021).
Preprint of the chapter: arXiv:1804.08004

Papers in refereed journals

  1. A. Costa and B. Steinberg, The Karoubi envelope of the mirage of a subshift, Communications in Algebra, 49 (2021), nº11, 4820-4856; doi:
  2. J. Almeida, A. Costa, R. Kyriakoglou and D. Perrin, On the group of a rational maximal bifix code, Forum Mathematicum, 32 (2020), nº3, 553-576; doi:
  3. A. Costa and A. Escada, Bases for pseudovarieties closed under bideterministic product, Algebra Universalis 80 (2019), nº4, Paper No. 46, 35 pp; doi: 10.1007/s00012-019-0621-5
  4. J. Almeida, A. Costa, J.C. Costa and M. Zeitoun, The linear nature of pseudowords, Publicacions Matemàtiques 63 (2019), nº2, 361-422; doi: 10.5565/PUBLMAT6321901
  5. A. Costa, Symbolic dynamics and semigroup theory, CIM Bulletin 40 (2018), 54-59. 
  6. J. Almeida and A. Costa, Equidivisible pseudovarieties of semigroups, Publicationes Mathematicae Debrecen 90/3-4 (2017), 435-453. 
  7. J. Almeida and A. Costa, A geometric interpretation of the Schützenberger group of a minimal subshift, Arkiv för Matematik 54 (2016), 243-275; doi:10.1007/s11512-016-0233-7
  8. A. Costa and B. Steinberg, A categorical invariant of flow equivalence of shiftsErgodic Theory and Dynamical Systems 36 (2016), 470-513; doi:10.1017/etds.2014.74
    Pdf file of revised preprint version.
  9. A. Costa and B. Steinberg, The Schützenberger category of a semigroup,  Semigroup Forum 91 (2015), 543-559; doi:10.1007/s00233-014-9657-1
    Preprint version: arXiv:1304.3487
  10. J. Almeida and A. Costa, A note on pseudovarieties of completely regular semigroups,  Bulletin of the Australian Mathematical Society 92 (2015), 233-237; doi: 10.1017/S0004972715000532 
    Tech. Report 15-06 DMUC, 2015.
  11. A. Costa and A. Escada, Some operators that preserve the locality of a pseudovariety of semigroups, International Journal of Algebra and Computation 23 (2013), nº 3, 583-610; doi: 10.1142/S0218196713500112
    Pdf file of revised version.
  12. J. Almeida and A. Costa, Presentations of Schützenberger groups of minimal subshifts, Israel Journal of Mathematics 196 (2013), nº1, 1-31; doi: 10.1007/s11856-012-0139-4
    Pdf file of third version.
  13. J. Almeida and A. Costa, On the transition semigroups of centrally labeled Rauzy graphs, International Journal of Algebra and Computation  22 (2012), nº 2.; doi: 10.1142/S021819671250018X
    Tech. Report 11-23 DMUC, 2011.
  14. A. Costa and B. Steinberg, Profinite groups associated to sofic shifts are free, Proceedings of the London Mathematical Society 102 (2011), nº 2, 341-369; doi:10.1112/plms/pdq
    Tech. Report 09-28 DMUC, 2009.
  15. J. Almeida and A. Costa, Infinite-vertex free profinite semigroupoids and symbolic dynamics, Journal of Pure and Applied Algebra 213 (2009), 605–631; doi: 10.1016/j.jpaa.2008.08.009
    Pdf file of revised version of preprint.
  16. L. Chaubard and A. Costa, A new algebraic invariant for weak equivalence of sofic subshifts, RAIRO-Theoretical Informatics and Applications 42 (2008), 481-502. 
    Tech. Report 06-57 DMUC, 2006.
  17. A. Costa, Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts , Journal of Pure and Applied Algebra 209 (2007), 17-530. 
    Tech. Report 05-31 DMUC, 2005.
  18. A. Costa, Conjugacy invariants of subshifts: an approach from profinite semigroup theory, International Journal of Algebra and Computation 16 (2006), nº 4, 629-655. 
    Tech. Report 05-18 DMUC, 2005.


  1. A. Costa, A profinite approach to complete bifix decodings of recurrent languages, arXiv:2208.11768
  2. J. Almeida and A. Costa, Equidivisibility and profinite coproduct, arXiv:2206.13421 (To appear in Quaestiones Mathematicae)


  1. Doctoral thesis, Semigrupos Profinitos e Dinâmica Simbólica,
    Doutoramento em Matemática, Faculdade de Ciências da Universidade do Porto, 2007.    ERRATA
  2. Master's thesis, Relações entre a dinâmica de operadores implícitos e a estrutura de grupos finitos,
    Mestrado em Matemática - Fundamentos e Aplicações, Departamento de Matemática Pura da Faculdade de Ciências da Universidade do Porto, 2003.
Profinite Semigroups and Symbolic Dynamics