Abstract
We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids. Stone duality implies that elements of the free pro-aperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for ω-terms that avoids using McCammond’s normal forms, as well as new proofs and extensions of other structural results concerning free pro-aperiodic monoids.
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J. Almeida, Finite Semigroups and Universal Algebra, Series in Algebra, Vol. 3, World Scientific Publishing, River Edge, NJ, 1994.
J. Almeida, On hyperdecidable pseudovarieties of simple semigroups, International Journal of Algebra and Computation 10 (2000), 261–284.
J. Almeida, Profinite groups associated with weakly primitive substitutions, Fundamentalnaya i Prikladnaya Matematika 11 (2005), 13–48.
J. Almeida and A. Costa, Infinite-vertex free profinite semigroupoids and symbolic dynamics, Journal of Pure and Applied Algebra 213 (2009), 605–631.
J. Almeida, A. Costa, J. C. Costa and M. Zeitoun, The linear nature of pseudowords, Publications Matématiques 63 (2019), 361–422.
J. Almeida, J. C. Costa and M. Zeitoun, Some structural properties of the free profinite aperiodic semigroup, in Automatha Plenary Conference, Liège, 2009, https://hal.archives-ouvertes.fr/hal-00948998.
J. Almeida, J. C. Costa and M. Zeitoun, Iterated periodicity over finite aperiodic semigroups, European Journal of Combinatorics 37 (2014), 115–149.
J. Almeida, J. C. Costa and M. Zeitoun, McCammond’s normal forms for free aperiodic semigroups revisited, LMS Journal of Computation and Mathematics 18 (2015), 130–147.
J. Almeida, O. Klima and M. Kunc, The ω-inequality problem for concatenation hierarchies of star-free languages, Forum Mathematicum 30 (2018), 663–679.
J. Almeida and M. V. Volkov, Subword complexity of profinite words and subgroups of free profinite semigroups, International Journal of Algebra and Computation 16 (2006), 221–258.
S. L. Bloom and Z. Ésik, The equational theory of regular words, Information and Computation 197 (2005), 55–89.
M. Bojańczyk, Recognisable languages over monads, in Developments in Language Theory. DLT 2015, Lecture Notes in Computer Science, Vol. 9168, Springer, Cham, 2015, pp. 1–13.
J. A. Brzozowski, Hierarchies of aperiodic languages, Revue Francaise d’Automatique Informatique Recherche Opérationnelle Série Rouge Informatique Théorique 10 (1976), 33–49.
J. R. Büchi, Weak second-order arithmetic and finite automata, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 6 (1960), 66–92.
O. Carton, T. Colcombet and G. Puppis, Regular languages of words over countable linear orderings, in Automata, Languages and Programming. Part II, Lecture Notes in Computer Science, Vol. 6756, Springer, Heidelberg, 2011, pp. 125–136.
O. Carton, T. Colcombet and G. Puppis, An algebraic approach to mso-definability on countable linear orderings, Journal of Symbolic Logic 83 (2018), 1147–1189.
C.-C. Chang and J. H. Keisler, Model Theory, Studies in Logic and the Foundations of Mathematics, Vol. 73, North-Holland, Amsterdam, 1990.
H. C. Doets, Completeness and Definability: Applications of the Ehrenfeucht game in second-order and intensional logic, Ph.D. thesis, Universiteit van Amsterdam, May 1987.
S. Eilenberg, Automata, Languages, and Machines. Vol. B, Pure and Applied Mathematics, Vol. 59, Academic Press, New York London, 1976.
M. Gehrke, Stone duality, topological algebra, and recognition, Journal of Pure and Applied Algebra 220 (2016), 2711–2747.
M. Gehrke, S. Grigorieff and J.-É. Pin, Duality and equational theory of regular languages, in Automata, Languages and Programming. Part II, Lecture Notes in Computer Science, Vol. 5126, Springer, Berlin, 2008, pp. 246–257.
M. Gehrke, A. Krebs and J.-É. Pin, Ultrafilters on words for a fragment of logic, Theoretical Computer Science 610 (2016), 37–58.
M. Gehrke, D. Petrisan and L. Reggio, The Schützenberger product for syntactic spaces, in 43rd International Colloquium on Automata, Languages and Programming, Leibniz International Proceedings in Informatics, Vol. 55, Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl, 2016, pp. 112:1112:14.
S. J. van Gool and B. Steinberg, Pro-aperiodic monoids via saturated models, in 34th Symposium on Theoretical Aspects of Computer Science, Leibniz International Proceedings in Informatics, Vol. 66, Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl, 2017, pp. 39:139:14.
K. Henckell, Idempotent pointlike sets, International Journal of Algebra and Computation 14 (2004), 703–717.
K. Henckell, Stable pairs, International Journal of Algebra and Computation 20 (2010), 241–267.
K. Henckell, J. Rhodes and B. Steinberg, A profinite approach to stable pairs, International Journal of Algebra and Computation 20 (2010), 269–285.
K. Henckell, J. Rhodes and B. Steinberg, An effective lower bound for group complexity of finite semigroups and automata, Transactions of the American Mathematical Society 364 (2012), 1815–1857.
W. Hodges, Model Theory, Encyclopedia of Mathematics and its Applications, Vol. 42, Cambridge University Press, Cambridge, 1993.
M. Huschenbett and M. Kufleitner, Ehrenfeucht Fraïssé games on omega-terms, in 31st International Symposium on Theoretical Aspects of Computer Science, Leibniz International Proceedings in Informatics, Vol. 25, Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl, 2014, pp. 374–385.
K. Krohn and J. Rhodes, Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines, Transactions of the American Mathematical Society 116 (1965), 450–464.
K. Krohn and J. Rhodes, Complexity of finite semigroups, Annals of Mathematics 88 (1968), 128–160.
M. Lohrey and C. Mathissen, Isomorphism of regular trees and words, Information and Computation 224 (2013), 71–105.
D. Marker, Model Theory: An Introduction, Graduate Texts in Mathematics, Vol. 217, Springer, New York, 2002.
J. P. McCammond, The solution to the word problem for the relatively free semigroups satisfying Ta = Ta+b with a = 6, International Journal of Algebra and Computation 1 (1991), 1–32.
J. P. McCammond, Normal forms for free aperiodic semigroups, International Journal of Algebra and Computation 11 (2001), 581–625.
R. McNaughton and S. Papert, Counter-free Automata, M.I.T. Research Monograph, no. 65, MIT Press, Cambridge, MA, 1971.
J.-É. Pin, The dot-depth hierarchy, 45 years later, in The Role of Theory in Computer Science, World Scientific, Hackensack, NJ, 2017, pp. 177–202.
T. Place and M. Zeitoun, Going higher in the first-order quantifier alternation hierarchy on words, in Automata, languages, and programming. Part II, Lecture Notes in Computer Science, Vol. 8573, Springer, Heidelberg, 2014, pp. 342–353.
J. Rhodes and B. Steinberg, Profinite semigroups, varieties, expansions and the structure of relatively free profinite semigroups, International Journal of Algebra and Computation 11 (2001), 627–672.
J. Rhodes and B. Steinberg, The q-theory of Finite Semigroups, Springer Monographs in Mathematics, Springer, New York, 2009.
J. G. Rosenstein, Linear Orderings, Pure and Applied Mathematics, Vol. 98, Academic Press, New York London, 1982.
M.-P. Schützenberger, On finite monoids having only trivial subgroups, Information and Control 8 (1965), 190–194.
S. Shelah, The monadic theory of order, Annals of Mathematics 102 (1975), 379–419.
B. Steinberg, A combinatorial property of ideals in free profinite monoids, Journal of Pure and Applied Algebra 214 (2010), 1693–1695.
M. H. Stone, The theory of representation for Boolean algebras, Transactions of the American Mathematical Society 74 (1936), 37–111.
H. Straubing, Finite Automata, Formal Logic, and Circuit Complexity, Progress in Theoretical Computer Science, Birkhäuser Boston, Boston, MA, 1994.
J. Väänänen, Pseudo-finite model theory, Matemática Contemporânea 24 (2003), 169–183.
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van Gool, S.J., Steinberg, B. Pro-aperiodic monoids via saturated models. Isr. J. Math. 234, 451–498 (2019). https://doi.org/10.1007/s11856-019-1947-6
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DOI: https://doi.org/10.1007/s11856-019-1947-6