The Constructions of the Square Complex of a Diagram Group from a Graphical Presentation

• Kalthom M. Alaswed

  Department of Mathematical, Faculty of Science, Gharyan University, Libya

  Author

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

Cohen, D. E. (1989). Combinatorial group theory: a topological approach. CUP Archive.

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.

Guba, V., & Sapir, M. (2006a). Diagram groups and directed 2-complexes: homotopy and homology. Journal of Pure and Applied Algebra, 205(1), 1-47.

Guba, V., & Sapir, M. (2006b). Diagram groups are totally orderable. Journal of Pure and Applied Algebra, 205(1), 48-73.

Guba, V. S. (2002). Some properties of periodic words. Mathematical Notes, 72, 301-307.

Guba, V. S., & Sapir, M. V. (1999). On subgroups of R. Thompson's group and other diagram groups. Sbornik: Mathematics, 190(8), 1077.

Guba, V. S., & Sapir, M. V. (2002). Rigidity properties of diagram groups. International Journal of Algebra and Computation, 12(01n02), 9-17.

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.

Meakin, J., & Sapir, M. (1993). Congruences on free monoids and submonoids of polycyclic monoids. Journal of the Australian Mathematical Society, 54(2), 236-253.

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.

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.

• PDF

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 the original publisher. Also, they accept the article remains published by the MJSc website (except in the occasion of a retraction of the article).