The Constructions of the Square Complex of a Diagram Group from a Graphical Presentation
DOI:
https://doi.org/10.54172/mjsc.v38i2.1206Keywords:
Diagram groups, Semigroup presentation, Generators, Maximal subtreeAbstract
In this paper, we may obtain diagram groups for any given graphical presentation. These groups can be viewed as the fundamental group of squire complexes. Let 4 be a semigroup presentation. The problems are divided into several cases according to the length of words, with all vertices in 4 being words of the length . The main aim of this article is to construct the connected square complex graph 4 of a diagram group from semigroup presentation 4 . Then we will prove 4 is the covering squire complexes for 4 for all . Then the covering space is identified for all connected square complex graphs by picking normal subgroups from the diagram group that was previously obtained from the semigroup presentation. This research introduces how to associate with the covering space 4 , how to determine the generators for covering space 4 , and what 4 looks like
Downloads
References
Cohen, D. E. (1989). Combinatorial group theory: a topological approach. CUP Archive. DOI: https://doi.org/10.1017/CBO9780511565878
Gheisari, Y., & Ahmad, A. G. (2010a). The class of isomorphic diagram groups over semigroup presentations. International Journal of Contemporary Mathematical Sciences, 5, 2311-2318.
Gheisari, Y., & Ahmad, A. G. (2010b). Component of graphs from diagram groups over the union of two semigroup presentation with three different initial generators by adding new relation. Journal of International Mathematical Forum,
Guba, V., & Sapir, M. (1997). Diagram groups (Vol. 620). American Mathematical Soc. DOI: https://doi.org/10.1090/memo/0620
Guba, V., & Sapir, M. (2006a). Diagram groups and directed 2-complexes: homotopy and homology. Journal of Pure and Applied Algebra, 205(1), 1-47. DOI: https://doi.org/10.1016/j.jpaa.2005.06.012
Guba, V., & Sapir, M. (2006b). Diagram groups are totally orderable. Journal of Pure and Applied Algebra, 205(1), 48-73. DOI: https://doi.org/10.1016/j.jpaa.2005.06.001
Guba, V. S. (2002). Some properties of periodic words. Mathematical Notes, 72, 301-307. DOI: https://doi.org/10.1023/A:1020539118744
Guba, V. S., & Sapir, M. V. (1999). On subgroups of R. Thompson's group and other diagram groups. Sbornik: Mathematics, 190(8), 1077. DOI: https://doi.org/10.1070/SM1999v190n08ABEH000419
Guba, V. S., & Sapir, M. V. (2002). Rigidity properties of diagram groups. International Journal of Algebra and Computation, 12(01n02), 9-17. DOI: https://doi.org/10.1142/S0218196702000882
Kilibarda, V. (1994). On the algebra of semigroup diagrams [PhD thesis, The University of Nebraska-Lincoln].
Kilibarda, V. (1997). On the algebra of semigroup diagrams. International Journal of Algebra and Computation, 7(03), 313-338. DOI: https://doi.org/10.1142/S0218196797000150
Meakin, J., & Sapir, M. (1993). Congruences on free monoids and submonoids of polycyclic monoids. Journal of the Australian Mathematical Society, 54(2), 236-253. DOI: https://doi.org/10.1017/S1446788700037149
Nieveen, S., & Smith, A. (2006). Covering spaces and subgroups of the free group. Oregon State University, Corvallis, Oregon.
Pride, S. (1991). Identities among relations of group presentations, in “Group Theory from a Geometrical Viewpoint”, edited by E. Ghys, A. Haefliger and A. Verjovsky: World Scientific.
Pride, S. (1993). Geometric methods in combinatorial semigroup theory in Semigroups, Formal Languages and Groups, ed. J. Fountain: Kluwer.
Pride, S. (1995). Geometric methods in combinatorial semigroup theory. NATO ASI Series C Mathematical and Physical Sciences-Advanced Study Institute, 466, 215-232. DOI: https://doi.org/10.1007/978-94-011-0149-3_7
Rotman, J. (2002). Advanced Modern Algebra. New Jersey 07458: Pearson Education, Inc. Upper Saddle River.
Rotman, J. J. (1995). An introduction to homological algebra (Vol. 2). Springer.
Serre, J.-P., & Serre, J.-P. (1980). SL 2. Trees, 69-136. DOI: https://doi.org/10.1007/978-3-642-61856-7_2
Downloads
Published
How to Cite
License
Copyright (c) 2023 Kalthom M. Alaswed
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright of the articles Published by Almukhtar Journal of Science (MJSc) is retained by the author(s), who grant MJSc a license to publish the article. Authors also grant any third party the right to use the article freely as long as its integrity is maintained and its original authors and cite MJSc as original publisher. Also they accept the article remains published by MJSc website (except in occasion of a retraction of the article).
