Free Access

Inverse monoids and immersions of $Δ$ -complexes

John Meakin

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA

E-mail Address: jmeakin@unl.edu

Search for more papers by this author

and

Nóra Szakács

Bolyai Institute, University of Szeged, Aradi vértanúk tere 1. H-6720 Szeged, Hungary

Center of Mathematics, University of Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

E-mail Address: szakacsn@math.u-szeged.hu

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218196721400117Cited by:1 (Source: Crossref)

This article is part of the issue:

Special Issue on Papers From the Conference “Semigroups and Groups, Automata, Logics” SandGAL 2019
Guest Editors: Achille Frigeri, Victoria Gould, Patrizia Longobardi and Emanuele Rodaro

Abstract

An immersion $f : 𝒟 \to 𝒞$ between $Δ$ -complexes is a $Δ$ -map that induces injections from star sets of $𝒟$ to star sets of $𝒞$ . We study immersions between finite-dimensional connected $Δ$ -complexes by replacing the fundamental group of the base space by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex. This extends earlier results of Margolis and Meakin for immersions between graphs and of Meakin and Szakács on immersions into $2$ -dimensional $C W$ -complexes.

Communicated by all the special editors

Keywords:

AMSC: 20M18, 57N35

References

1. A. Hatcher, Algebraic Topology (Cambridge University Press, 2001). Google Scholar
2. J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages and Computation (Addison-Wesley, New York, 1979). Google Scholar
3. M. V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries (World Scientific, 1998). Link, Google Scholar
4. S. Margolis and J. Meakin, Free inverse monoids and graph immersions, Int. J. Algebra Comput. 3(1) (1993) 79–99. Link, Google Scholar
5. J. Meakin and N. Szakács, Inverse monoids and immersions of $2$ -complexes, Int. J. Algebra Comput. 25(1–2) (2015) 301–323. Link, Web of Science, Google Scholar
6. J. Milnor, The geometric realization of a semi-simplicial complex, Ann. Math. 65 (1957) 357–362. Crossref, Web of Science, Google Scholar
7. W. D. Munn, Free inverse semigroups, Proc. London Math. Soc. 30 (1974) 384–404. Google Scholar
8. M. Petrich, Inverse Semigroups (Wiley, 1984). Google Scholar
9. B. M. Schein, Representations of generalized groups, Izv. Vyss. Ucebn. Zav. Mat. 3 (1962) 164–176 (in Russian). Google Scholar
10. J. R. Stallings, Topology of finite graphs, Invent. Math. 71(3) (1983) 551–565. Crossref, Web of Science, Google Scholar
11. J. B. Stephen, Applications of automata theory to presentations of monoids and inverse monoids, PhD. Thesis, University of Nebraska-Lincoln (1987). Google Scholar
12. J. B. Stephen, Presentations of inverse monoids, J. Pure Appl. Algebra 63 (1990) 81–112. Crossref, Web of Science, Google Scholar