Abstract
We consider the well-known problem of deciding the union of decidable equational theories. We focus on monadic theories, i.e., theories over signatures with unary function symbols only. The equivalence of the category of monadic equational theories and the category of monoids is used. This equivalence facilitates a translation of the considered decidability problem into the word problem in the pushout of monoids which themselves have decidable word problems. Using monoids, existing results on the union of theories are then restated and proved in a succint way. The idea is then analyzed of first guaranteeing that the union is a “jointly conservative” extension and then using this property to show decidability of the union. It is shown that “joint conservativity” is equivalent to the corresponding monoid amalgam being embeddable; this allows one to apply results from amalgamation theory to this problem. Then we prove that using this property to show decidability is a more difficult matter: it turns out that even if this property and some additional conditions hold, the problem remains undecidable.
This work has been partially supported by European Union project AGILE IST-2001-32747.
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References
Baader, F., Tinelli, C.: Deciding the word problem in the union of equational theories. Information and Computation 178(2), 346–390 (2002)
Fiorentini, C., Ghilardi, S.: Combining word problems through rewriting in categories with products. Theoretical Computer Science 294, 103–149 (2003)
Hall, T.E.: Amalgamation and inverse and regular semigroups. Transactions of the AMS 246, 395–406 (1978)
Higgins, P.M.: Techniques of Semigroup Theory. Oxford University Press, Oxford (1992)
Howie, J.M.: Fundamentals of Semigroup Theory. London Mathematical Society Monographs, New Series, vol. 12. Oxford University Press, Oxford (1996)
Howie, J.M.: Isbell’s zigzag theorem and its consequences. In: Hofmann, K.H., Mislove, M.W. (eds.) Semigroup Theory and its Applications, Proceedings of the 1994 Conference Commemorating the Work of Alfred H. Clifford, New Orleans, LA, London. Mathematical Society Lecture Notes Series, vol. 231, pp. 81–91. Cambridge University Press, Cambridge (1996)
Isbell, J.R.: Epimorphisms and dominions. In: Proceedings of Conference on Categorical Algebra, La Jolla, California, pp. 232–246. Springer, Heidelberg (1966)
Leinster, T.: Higher Operads, Higher Categories. In: Main, M.G., Mislove, M.W., Melton, A.C., Schmidt, D. (eds.) MFPS 1987. LNCS, vol. 298, Springer, Heidelberg (1988)
Pigozzi, D.: The join of equational theories. Colloquium Mathematicum 30(1), 15–25 (1974)
Renshaw, J.: Extension and amalgamation in monoids and semigroups. Proceedings of the London Mathematical Society 52(3), 119–141 (1986)
Renshaw, J.: Flatness and amalgamation in monoids. Journal of the London Mathematical Society 33(2), 73–88 (1986)
Sapir, M.V.: Algorithmic problems for amalgams of finite semigroups. Journal of Algebra 229(2), 514–531 (2000)
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Hoffman, P. (2005). Union of Equational Theories: An Algebraic Approach. In: Giesl, J. (eds) Term Rewriting and Applications. RTA 2005. Lecture Notes in Computer Science, vol 3467. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32033-3_6
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DOI: https://doi.org/10.1007/978-3-540-32033-3_6
Publisher Name: Springer, Berlin, Heidelberg
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