Abstract
The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of sub-pseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal’cev products with the pseudovarieties of definite and reverse definite semigroups.
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References
Almeida, J.: Finite Semigroups and Universal Algebra. Singapore, World Scientific (1994)
Birjukov, P.A.: Varieties of idempotent semigroups. Algebra Log. 9, 255–273 (1970)
Fennemore, C.F.: All varieties of bands I. Math. Nach. 48, 237–252 (1971)
Fennemore, C.F.: All varieties of bands II. Math. Nach. 48, 253–262 (1971)
Fich, F., Brzozowski, J.: A characterization of a dot-depth two analogue of generalized definite languages. In: Proc. 6th ICALP. LNCS, vol. 71, pp. 230–244. Springer, Berlin (1979)
Gerhard, J.A.: The lattice of equational classes of idempotent semigroups. J. Algebra 15, 195–224 (1970)
Gerhard, J.A., Petrich, M.: Varieties of bands revisited. Proc. Lond. Math. Soc. 58(3), 323–350 (1989)
Hall, T.E., Weil, P.: On radical congruence systems. Semigroup Forum 59, 56–73 (1999)
Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, Oxford (1995)
Krohn, K., Rhodes, J., Tilson, B.: Homomorphisms and semilocal theory. In: Arbib, M. (ed.) The Algebraic Theory of Machines, Languages and Semigroups. Academic Press, San Diego (1965)
Kufleitner, M., Weil, P.: On FO 2 quantifier alternation over words. In: Proceedings of the 34th Symposium on Mathematical Foundations of Computer Science (MFCS 2009). LNCS, vol. 5734, pp. 513–524. Springer, Berlin (2009)
Lodaya, K., Pandya, P.K., Shah, S.S.: Marking the chops: an unambiguous temporal logic. In: IFIP TCS 2008, pp. 461–476 (2008)
Pin, J.-E.: Varieties of Formal Languages. North Oxford Academic, London (1986)
Pin, J.-E., Weil, P.: Profinite semigroups, Mal’cev products and identities. J. Algebra 182, 604–626 (1996)
Reilly, N.R., Zhang, S.: Complete endomorphisms of the lattice of pseudovarieties of finite semigroups. Bull. Aust. Math. Soc. 55, 207–218 (1997)
Schützenberger, M.-P.: Sur le produit de concaténation non ambigu. Semigroup Forum 13, 47–75 (1976)
Tesson, P., Thérien, D.: Diamonds are forever: the variety DA. In: Dos Gomes Moreira Da Cunha, G.M., Ventura Alves Da Silva, P., Pin, J.-E. (eds.) Semigroups, Algorithms, Automata and Languages, Coimbra (Portugal) 2001, pp. 475–500. World Scientific, Singapore (2002)
Tesson, P., Thérien, D.: Logic meets algebra: the case of regular languages. Log. Methods Comput. Sci. 3(1), 1–37 (2007)
Thérien, D., Wilke, Th.: Over words, two variables are as powerful as one quantifier alternation. In: STOC, pp. 234–240 (1998)
Trotter, P., Weil, P.: The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue. Algebra Univers. 37, 491–526 (1997)
Weil, P.: Some results on the dot-depth hierarchy. Semigroup Forum 46, 352–370 (1993)
Weil, P.: Profinite methods in semigroup theory. Int. J. Algebra Comput. 12, 137–178 (2002)
Weis, Ph., Immerman, N.: Structure theorem and strict alternation hierarchy for FO2 on words. Log. Methods Comput. Sci. 5(3), 1–23 (2009)
Wismath, S.L.: The lattices of varieties and pseudovarieties of band monoids. Semigroup Forum 33(2), 187–198 (1986)
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Communicated by Jean-Eric Pin.
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Kufleitner, M., Weil, P. On the lattice of sub-pseudovarieties of DA . Semigroup Forum 81, 243–254 (2010). https://doi.org/10.1007/s00233-010-9258-6
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DOI: https://doi.org/10.1007/s00233-010-9258-6