Abstract
Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NP-completeness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bekic, H.: Definable operation in general algebras, and the theory of automata and flowcharts. In: Bekic, H. (ed.) Programming Languages and their Definition. LNCS, vol. 177, pp. 30–55. Springer, Heidelberg (1984)
Cori, R., Perrin, D.: Automates et commutations partielles. RAIRO — Inform. Théor. Appl. 19, 21–32 (1985)
Diekert, V.: Combinatorics on Traces. LNCS, vol. 454. Springer, Heidelberg (1990)
Diekert, V., Rozenberg, G. (eds.): The Book of Traces. World Scientific, Singapore (1995)
Droms, C.: Graph groups, coherence and three-manifolds. J. Algebra 106(2), 484–489 (1985)
Kapovich, I., Weidmann, R., Myasnikov, A.: Foldings, graphs of groups and the membership problem. Internat. J. Algebra Comput. 15(1), 95–128 (2005)
Kupferman, O., Vardi, M.Y.: An automata-theoretic approach to reasoning about infinite-state systems. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 36–52. Springer, Heidelberg (2000)
Lohrey, M., Ondrusch, N.: Inverse monoids: decidability and complexity of algebraic questions. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 664–675. Springer, Heidelberg (2005)
Margolis, S., Meakin, J.: E-unitary inverse monoids and the Cayley graph of a group presentation. J. Pure Appl. Algebra 58(1), 45–76 (1989)
Margolis, S., Meakin, J.: Inverse monoids, trees, and context-free languages. Trans. Amer. Math. Soc. 335(1), 259–276 (1993)
Margolis, S., Meakin, J., Sapir, M.: Algorithmic problems in groups, semigroups and inverse semigroups. In: Semigroups, Formal Languages and Groups, pp. 147–214. Kluwer, Dordrecht (1995)
Meakin, J., Sapir, M.: The word problem in the variety of inverse semigroups with Abelian covers. J. London Math. Soc. (2) 53(1), 79–98 (1996)
Mihailova, K.A.: The occurrence problem for direct products of groups. Math. USSR Sbornik 70, 241–251 (1966) (English translation)
Munn, W.: Free inverse semigroups. Proc. London Math. Soc. 30, 385–404 (1974)
Petrich, M.: Inverse semigroups. Wiley, Chichester (1984)
Stephen, J.: Presentations of inverse monoids. J. Pure Appl. Algebra 63, 81–112 (1990)
Veloso da Costa, A.: Γ-Produtos de Monóides e Semigrupos. PhD thesis, Universidade do Porto (2003)
Walukiewicz, I.: Pushdown processes: games and model-checking. Inform. and Comput. 164(2), 234–263 (2001)
Wrathall, C.: The word problem for free partially commutative groups. J. Symbolic Comput. 6(1), 99–104 (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Diekert, V., Lohrey, M., Miller, A. (2006). Partially Commutative Inverse Monoids. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_26
Download citation
DOI: https://doi.org/10.1007/11821069_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37791-7
Online ISBN: 978-3-540-37793-1
eBook Packages: Computer ScienceComputer Science (R0)