Abstract
We show that the order type of the simplest version of a hammock for string algebras lies in the class of finite description linear orders–the smallest class of linear orders containing \(\textbf{0}\), \(\textbf{1}\), and that is closed under isomorphisms, finite order sum, anti-lexicographic product with \(\omega \) and \(\omega ^*\), and shuffle of finite subsets–using condensation (localization) of linear orders as a tool. We also introduce two finite subsets of the set of bands and use them to describe the location of left \(\mathbb {N}\)-strings in the completion of hammocks.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Availability of data and materials
Not applicable.
References
Agrawal, S., Kuber, A., Gupta, E.: Euclidean algorithm for a class of linear orders. Discrete Mathematics 346(12), 113639 (2023)
Brenner, S.: A combinatorial characterisation of finite Auslander-Reiten quivers. In: Proc. ICRA 4, Ottawa 1984, SLNM 1177, pages 13–49 (1986)
Gupta, E., Kuber, A., Sardar, S.: On the stable radical of some non-domestic string algebras. Algebras and Representation Theory 25(5), 1207–1230 (2022)
Knuth, D.E., Morris Jr, J.H., Pratt, V.R.: Fast pattern matching in strings. SIAM J. Comput. 6(2):323–350 (1977)
Läuchli, H., Leonard, J.: On the elementary theory of linear order. Fundamenta Mathematicae 59(1), 109–116 (1966)
Prest, M.: Morphisms between finitely presented modules and infinite-dimensional representations. In Canad. Math. Soc. Conf. Proc 24, 447–455 (1998)
Ringel, C.M.: Some algebraically compact modules I. In: Abelian groups and modules, pages 419–439. Springer (1995)
Rosenstein, J.G.: Linear orderings. Academic press (1982)
Schröer, J.: Hammocks for string algebras. PhD thesis, Bielefeld University (Germany) (1998)
Schröer, J.: On the infinite radical of a module category. Proceedings of the London Mathematical Society 81(3), 651–674 (2000)
Sardar, S., Kuber, A.: On the order types of hammocks for domestic string algebras. Journal of Pure and Applied Algebra, 2024. 107763. https://doi.org/10.1016/j.jpaa.2024.107763
Srivastava, S., Kuber, A.: Automating the stable rank computation for special biserial algebras. arXiv:2407.02326 (2024)
Srivastava, S., Sinha, V., Kuber, A.: On the stable radical of the module category for special biserial algebras. arXiv:2311.10178 (2023)
Acknowledgements
All authors are grateful to the anonymous referee for their very careful reading of the manuscript and for suggesting changes to improve the presentation of the paper. They would also like to thank Naivedya Amarnani, Dawood Bin Mansoor and Nupur Jain for discussions on the topic, and Suyash Srivastava for careful reading of the first draft.
Funding
A.S. thanks the Council of Scientific and Industrial Research (CSIR) India - Research Grant No. 09/092(1090)/2021-EMR-I for the financial support.
Author information
Authors and Affiliations
Contributions
V.S. contributed to sections 4-11, A.K. supervised the project and contributed to sections 1, 2, 7, 11 and 12, A.S. contributed to sections 2-4 and 11-12, and B.K. contributed to sections 4-6 and 10. A.S. prepared all the figures. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Ethical approval
Not applicable.
Competing interests
The authors declare no competing interests.
Additional information
Presented by: Henning Krause
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sinha, V., Kuber, A., Sengupta, A. et al. Hammocks for Non-Domestic String Algebras. Algebr Represent Theor (2024). https://doi.org/10.1007/s10468-024-10285-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10468-024-10285-7