B. Banaschewski and A. Pultr, Booleanization. Cahiers de Topologie et Géométrie Différentielle Catégoriques, vol.37, pp.41-60, 1996.

D. Barrington, K. Compton, H. Straubing, and D. Thérien, Regular languages in N C 1, Journal of Computer and System Sciences, vol.44, issue.3, pp.478-499, 1992.

C. Borlido and M. Gehrke, A note on powers of Boolean spaces with internal semigroups, 2018.

O. Carton, D. Perrin, and J. Pin, A survey on difference hierarchies of regular languages, Logical Methods in Computer Science, vol.14, issue.1, 2017.
URL : https://hal.archives-ouvertes.fr/hal-02104436

E. Casanovas and R. Farré, Omitting Types in Incomplete Theories, The Journal of Symbolic Logic, vol.61, issue.1, pp.236-245, 1996.

C. C. Chen, Free Boolean extensions of distributive lattices, Nanta Math, vol.1, pp.1-14, 1966.

B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, 2002.

L. L. Esakia, Topological Kripke models, Dokl. Akad. Nauk SSSR, vol.214, pp.298-301, 1974.

L. L. Esakia, Heyting algebras: Duality theory, Metsniereba, 1985.

M. Gehrke, Canonical extensions, Esakia spaces, and universal models, Leo Esakia on duality in modal and intuitionistic logics, vol.4, pp.9-41, 2014.

M. Gehrke, S. Grigorieff, and J. Pin, Duality and equational theory of regular languages, Automata, languages and programming. Part II, vol.5126, pp.246-257, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00340803

M. Gehrke and B. Jónsson, Bounded distributive lattices expansions, Mathematica Scandinavica, vol.94, issue.2, pp.13-45, 2004.

M. Gehrke and H. A. Priestley, Canonical extensions and completions of posets and lattices, Rep. Math. Logic, vol.43, pp.133-152, 2008.

S. Van-gool and B. Steinberg, Pro-aperiodic monoids and model theory, Israel Journal of Mathematics

C. Glaßer, H. Schmitz, and V. Selivanov, Efficient algorithms for membership in Boolean hierarchies of regular languages, Theoret. Comput. Sci, vol.646, pp.86-108, 2016.

G. Grätzer and E. T. Schmidt, On the generalized Boolean algebra generated by a distributive lattice, Indag. Math, vol.20, pp.547-553, 1959.

F. Hausdorff-;-john and R. Aumann, Set theory, 1957.

B. Jónsson and A. Tarski, Boolean algebras with operators I, Amer. J. Math, vol.73, pp.891-939, 1951.

L. Libkin, Elements of finite model theory, 2004.

A. Maciel, P. Péladeau, and D. Thérien, Programs over semigroups of dot-depth one, Semigroups and algebraic engineering, vol.245, pp.135-148, 1997.

H. M. Macneille, Extension of a distributive lattice to a Boolean ring, Bull. Amer. Math. Soc, vol.45, issue.6, pp.452-455, 1939.

A. Nerode, Some Stone spaces and recursion theory, Duke Math. J, vol.26, pp.397-406, 1959.

W. Peremans, Embedding of a distributive lattice into a Boolean algebra, Indag. Math, vol.60, pp.73-81, 1957.

T. Place and M. Zeitoun, Going higher in first-order quantifier alternation hierarchies on words, Journal of the ACM, 2018.

H. A. Priestley, Representation of distributive lattices by means of ordered stone spaces, Bull. London Math. Soc, vol.2, pp.186-190, 1970.

M. Sipser, Introduction to the theory of computation, 2006.

M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc, vol.41, issue.3, pp.375-481, 1937.

H. Straubing, Recognizable sets and power sets of finite semigroups, Semigroup Forum, vol.18, issue.4, pp.331-340, 1979.

H. Straubing, Finite automata, formal logic, and circuit complexity, 1994.

H. Straubing, Languages defined with modular counting quantifiers, Information and Computation, vol.166, issue.2, pp.112-132, 2001.