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Universal locally finite maximally homogeneous semigroups and inverse semigroups

  • Igor Dolinka ORCID logo und Robert D. Gray EMAIL logo
Veröffentlicht/Copyright: 19. Dezember 2017
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 30 Heft 4

Abstract

In 1959, Philip Hall introduced the locally finite group 𝒰, today known as Hall’s universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in 𝒰. It can explicitly be described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic sense since it is the FraĂŻssĂ© limit of the class of all finite groups. Since its introduction Hall’s group and several natural generalisations have been studied widely. In this article we use a generalisation of FraĂŻssé’s theory to construct a countable, universal, locally finite semigroup 𝒯, that arises as a direct limit of finite full transformation semigroups, and has the highest possible degree of homogeneity. We prove that it is unique up to isomorphism among semigroups satisfying these properties. We prove an analogous result for inverse semigroups, constructing a maximally homogeneous universal locally finite inverse semigroup ℐ which is a direct limit of finite symmetric inverse semigroups (semigroups of partial bijections). The semigroups 𝒯 and ℐ are the natural counterparts of Hall’s universal group for semigroups and inverse semigroups, respectively. While these semigroups are not homogeneous, they still exhibit a great deal of symmetry. We study the structural features of these semigroups and locate several well-known homogeneous structures within them, such as the countable generic semilattice, the countable random bipartite graph, and Hall’s group itself.

Keywords: Hall’s universal countable homogeneous group; homogeneous structures; maximally homogeneous semigroup; amalgamation
MSC 2010: 20M20; 20M10; 03C07; 20F50

Communicated by Manfred Droste

Funding source: Engineering and Physical Sciences Research Council

Award Identifier / Grant number: EP/N033353/1

Funding source: London Mathematical Society

Award Identifier / Grant number: Ref: 41530

Funding source: Ministarstvo Prosvete, Nauke i TehnoloĆĄkog Razvoja

Award Identifier / Grant number: 174019

Funding statement: This work was supported by the London Mathematical Society Research in Pairs (Scheme 4) grant “Universal locally finite partially homogeneous semigroups and inverse semigroups” (Ref: 41530), to fund a 9-day research visit of the first named author to the University of East Anglia (Summer 2016). The research of I. Dolinka was supported by the Ministry of Education, Science, and Technological Development of the Republic of Serbia through the grant No. 174019. This research of R. D. Gray was supported by the EPSRC grant EP/N033353/1 “Special inverse monoids: subgroups, structure, geometry, rewriting systems and the word problem”.

Acknowledgements

The authors would like to thank the anonymous referee for their helpful comments, and also to thank the handling editor of the paper Manfred Droste whose comments led to the questions that are now posed in Section 8.

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Received: 2017-04-05
Revised: 2017-10-06
Published Online: 2017-12-19
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Distinct distances between points and lines in đ”œq2
  3. A global perspective to connections on principal 2-bundles
  4. Integral group rings of solvable groups with trivial central units
  5. Lefschetz property and powers of linear forms in 𝕂[x,y,z]
  6. On quasi-convex null sequences in infinite cyclic groups
  7. On autocommutators and the autocommutator subgroup in infinite abelian groups
  8. Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
  9. Finite-dimensional representations of Leavitt path algebras
  10. Orlicz dual affine quermassintegrals
  11. Universal locally finite maximally homogeneous semigroups and inverse semigroups
  12. Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups
  13. Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
  14. A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)
  15. Equivariant Gauss sum of finite quadratic forms
  16. Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras
  17. Solution of Brauer’s k⁹(B)-conjecture for π-blocks of π-separable groups
https://doi.org/10.1515/forum-2017-0074
Schlagwörter fĂŒr diesen Artikel
Hall’s universal countable homogeneous group; homogeneous structures; maximally homogeneous semigroup; amalgamation

Artikel in diesem Heft

  1. Frontmatter
  2. Distinct distances between points and lines in đ”œq2
  3. A global perspective to connections on principal 2-bundles
  4. Integral group rings of solvable groups with trivial central units
  5. Lefschetz property and powers of linear forms in 𝕂[x,y,z]
  6. On quasi-convex null sequences in infinite cyclic groups
  7. On autocommutators and the autocommutator subgroup in infinite abelian groups
  8. Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
  9. Finite-dimensional representations of Leavitt path algebras
  10. Orlicz dual affine quermassintegrals
  11. Universal locally finite maximally homogeneous semigroups and inverse semigroups
  12. Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups
  13. Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
  14. A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)
  15. Equivariant Gauss sum of finite quadratic forms
  16. Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras
  17. Solution of Brauer’s k⁹(B)-conjecture for π-blocks of π-separable groups
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