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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2017: 0.67

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Volume 30, Issue 3

# The ω-inequality problem for concatenation hierarchies of star-free languages

Jorge Almeida
/ Ondřej Klíma
/ Michal Kunc
Published Online: 2017-09-06 | DOI: https://doi.org/10.1515/forum-2016-0028

## Abstract

The problem considered in this paper is whether an inequality of ω-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing–Thérien hierarchy.

MSC 2010: 20M05; 20M07; 20M35; 68Q70

## References

• [1]

J. Almeida, Implicit operations on finite $\mathcal{𝒥}$-trivial semigroups and a conjecture of I. Simon, J. Pure Appl. Algebra 69 (1991), no. 3, 205–218. Google Scholar

• [2]

J. Almeida, Finite Semigroups and Universal Algebra, Ser. Algebra 3, World Scientific, River Edge, 1994. Google Scholar

• [3]

J. Almeida, Hyperdecidable pseudovarieties and the calculation of semidirect products, Internat. J. Algebra Comput. 9 (1999), no. 3–4, 241–261,

• [4]

J. Almeida, Profinite semigroups and applications, Structural Theory of Automata, Semigroups and Universal Algebra (Montreal 2003), Kluwer Academic Publishers, Dordrecht (2005), 1–45. Google Scholar

• [5]

J. Almeida, J. Bartoňová, O. Klíma and M. Kunc, On decidability of intermediate levels of concatenation hierarchies, Developments in Language Theory, Lecture Notes in Comput. Sci. 9168, Springer, Cham (2015) 58–70. Google Scholar

• [6]

J. Almeida, A. Cano, O. Klíma and J.-E. Pin, On fixed points of the lower set operator, Internat. J. Algebra Comput. 25 (2015), no. 1–2, 259–292.

• [7]

J. Almeida and A. Costa, Infinite-vertex free profinite semigroupoids and symbolic dynamics, J. Pure Appl. Algebra 213 (2009), no. 5, 605–631.

• [8]

J. Almeida, J. C. Costa and M. Zeitoun, Iterated periodicity over finite aperiodic semigroups, European J. Combin. 37 (2014), 115–149.

• [9]

J. Almeida, J. C. Costa and M. Zeitoun, McCammond’s normal forms for free aperiodic semigroups revisited, LMS J. Comput. Math. 18 (2015), no. 1, 130–147.

• [10]

J. Almeida, J. C. Costa and M. Zeitoun, Factoriality and the Pin–Reutenauer procedure, Discrete Math. Theor. Comput. Sci. 18 (2016), no. 3, Paper No. 1. Google Scholar

• [11]

J. Almeida and B. Steinberg, On the decidability of iterated semidirect products with applications to complexity, Proc. Lond. Math. Soc. (3) 80 (2000), no. 1, 50–74.

• [12]

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Grad. Texts in Math. 78, Springer, New York, 1981. Google Scholar

• [13]

R. S. Cohen and J. A. Brzozowski, Dot-depth of star-free events, J. Comput. System Sci. 5 (1971), 1–16.

• [14]

S. Eilenberg, Automata, Languages, and Machines. Vol. B, Academic Press, New York, 1976. Google Scholar

• [15]

H. J. Keisler, Fundamentals of model theory, Handbook of Mathematical Logic, Stud. Logic Found. Math. 90, North Holland, Amsterdam (1977), 47–104. Google Scholar

• [16]

J. P. McCammond, Normal forms for free aperiodic semigroups, Internat. J. Algebra Comput. 11 (2001), no. 5, 581–625.

• [17]

J. D. McKnight, Jr. and A. J. Storey, Equidivisible semigroups, J. Algebra 12 (1969), 24–48.

• [18]

V. Molchanov, Nonstandard characterization of pseudovarieties, Algebra Universalis 33 (1995), no. 4, 533–547.

• [19]

Z.-E. Pèn, Eilenberg’s theorem for positive varieties of languages, Izv. Vyssh. Uchebn. Zaved. Mat. (1995), no. 1, 80–90. Google Scholar

• [20]

J.-E. Pin, Syntactic semigroups, Handbook of Formal Languages, Vol. 1, Springer, Berlin (1997), 679–746. Google Scholar

• [21]

J.-E. Pin and P. Weil, A Reiterman theorem for pseudovarieties of finite first-order structures, Algebra Universalis 35 (1996), no. 4, 577–595.

• [22]

J.-E. Pin and P. Weil, Profinite semigroups, Mal’cev products, and identities, J. Algebra 182 (1996), no. 3, 604–626.

• [23]

J.-E. Pin and P. Weil, Polynomial closure and unambiguous product, Theory Comput. Syst. 30 (1997), no. 4, 383–422.

• [24]

T. Place, and M. Zeitoun, Going higher in the first-order quantifier alternation hierarchy on words, Automata, Languages, and Programming. Part II (Copenhagen 2014), Lecture Notes in Comput. Sci. 8573, Springer, Heidelberg (2014), 342–353. Google Scholar

• [25]

T. Place and M. Zeitoun, Separating regular languages with first-order logic, Log. Methods Comput. Sci. 12 (2016), no. 1, Paper No. 5.

• [26]

J. Reiterman, The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982), no. 1, 1–10.

• [27]

J. Rhodes and B. Steinberg, The q-Theory of Finite Semigroups, Springer Monogr. Math., Springer, New York, 2009. Google Scholar

• [28]

H. Straubing, A generalization of the Schützenberger product of finite monoids, Theoret. Comput. Sci. 13 (1981), no. 2, 137–150.

• [29]

H. Straubing, Finite semigroup varieties of the form $V\ast D$, J. Pure Appl. Algebra 36 (1985), no. 1, 53–94. Google Scholar

• [30]

D. Thérien, Classification of finite monoids: The language approach, Theoret. Comput. Sci. 14 (1981), no. 2, 195–208.

• [31]

W. Thomas, Classifying regular events in symbolic logic, J. Comput. System Sci. 25 (1982), no. 3, 360–376.

• [32]

S. J. van Gool and B. Steinberg, Pro-aperiodic monoids via saturated models, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), LIPIcs. Leibniz Int. Proc. Inform. 66, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2017), 39:1–39:14. Google Scholar

• [33]

Revised: 2017-05-23

Published Online: 2017-09-06

Published in Print: 2018-05-01

Funding Source: Centro de Matemática Universidade do Porto

Award identifier / Grant number: UID/MAT/00144/2013

Funding Source: Fundação para a Ciência e a Tecnologia

Award identifier / Grant number: UID/MAT/00144/2013

Funding Source: Ministério da Ciência, Tecnologia e Ensino Superior

Award identifier / Grant number: UID/MAT/00144/2013

Funding Source: European Regional Development Fund

Award identifier / Grant number: UID/MAT/00144/2013

The first author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MCTES) and European structural funds (FEDER), under the partnership agreement PT2020. The second and third authors were partially supported by the grant 15-02862S of the Grant Agency of the Czech Republic.

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 663–679, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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