Abstract
We systematically develop a theory of graded semigroups, that is, semigroups S partitioned by groups Γ, in a manner compatible with the multiplication on S. We define a smash product S#Γ, and show that when S has local units, the category S#Γ-Mod of sets admitting an S#Γ-action is isomorphic to the category S-Gr of graded sets admitting an appropriate S-action. We also show that when S is an inverse semigroup, it is strongly graded if and only if S-Gr is naturally equivalent to Sε-Mod, where Sε is the partition of S corresponding to the identity element ε of Γ. These results are analogous to well-known theorems of Cohen/Montgomery and Dade for graded rings. Moreover, we show that graded Morita equivalence implies Morita equivalence for semigroups with local units, evincing the wealth of information encoded by the grading of a semigroup. We also give a graded Vagner—Preston theorem, provide numerous examples of naturally-occurring graded semigroups, and explore connections between graded semigroups, graded rings, and graded groupoids. In particular, we introduce graded Rees matrix semigroups, and relate them to smash product semigroups. We pay special attention to graded graph inverse semigroups, and characterise those that produce strongly graded Leavitt path algebras.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Abrams, P. Ara and M. Siles Molina, Leavitt Path Algebras, Lecture Notes in Mathematics, Vol. 2191, Springer, London, 2017.
P. N Ánh and L. Márki, Rees matrix rings, Journal of Algebra 81 (1983), 340–369.
P. Ara, J. Bosa, R. Hazrat and A. Sims, Reconstruction of graded groupoids from graded Steinberg algebras, Forum Mathematicum 29 (2017), 1023–1037.
P. Ara, R. Hazrat, H. Li and A. Sims, Graded Steinberg algebras and their representations, Algebra & Number Theory 12 (2018), 131–172.
C. J. Ash and T. E. Hall, Inverse semigroups on graphs, Semigroup Forum 11 (1975), 140–145.
G. M. Bergman, An Invitation to General Algebra and Universal Constructions, Universitext, Springer, Cham, 2015.
L. O. Clark and R. Hazrat, Étale groupoids and Steinberg algebras, a concise introduction, In Leavitt Path Algebras and Classical K-Theory, Indian Statistical Institute Series, Springer, Singapore, 2020, pp. 73–101.
L. O. Clark, R. Hazrat and S. W. Rigby, Strongly graded groupoids and strongly graded Steinberg algebras, Journal of Algebra 530 (2019), 34–68.
L. O. Clark and A. Sims, Equivalent groupoids have Morita equivalent Steinberg algebras, Journal of Pure and Applied Algebra 219 (2015), 2062–2075.
M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Transactions of the American Mathematical Society 282 (1984), 237–258; Addendum, Transactions of the American Mathematical Society 300 (1987), 810-811.
L. G. Cordeiro, D. Gonçalves and R. Hazrat, The talented monoid of a directed graph with applications to graph algebras, Revista Matemática Iberoamericana 38 (2022), 223–256.
E. Dade, Group-graded rings and modules, Mathematische Zeitschrift 174 (1980), 241–262.
I. Dolinka and J. East, Semigroups of rectangular matrices under a sandwich operation, Semigroup Forum 96 (2018), 253–300.
R. Exel, Inverse semigroups and combinatorial C*-algebras, Bulletin of the Brazilian Mathematical Society 39 (2008), 191–313.
J. Funk, M. V. Lawson and B. Steinberg, Characterizations of Morita equivalent inverse semigroups, Journal of Pure and Applied Algebra 215 (2011), 2262–2279.
R. Gordon and E. L. Green, Graded Artin algebras, Journal of Algebra 76 (1982), 111–137.
V. Gould and C. Hollings, Partial actions of inverse and weakly E-ample semigroups, Journal of the Australian Mathematical Society 86 (2009), 355–377.
E. L. Green, Graphs with relations, coverings and group-graded algebras, Transactions of the American Mathematical Society 279 (1983), 297–310.
R. Hazrat, Graded Rings and Graded Grothendieck Groups, London Mathematical Society Lecture Note Series, Vol. 435, Cambridge University Press, Cambridge, 2016.
J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series, Vol. 12, Oxford University Press, New York, 1995.
T. Hüttemann, The “fundamental theorem” for the higher algebraic K-theory of strongly ℤ-graded rings, Documenta Mathematica 26 (2021), 1557–1600.
E. Ilić-Georgijević, On the Jacobson and simple radicals of semigroups, Filomat 32 (2018), 2577–2582.
E. Ilić-Georgijević, A description of the Cayley graphs of homogeneous semigroups, Communications in Algebra 48 (2020), 5203–5214.
D. G. Jones and M. V. Lawson, Graph inverse semigroups: their characterization and completion, Journal of Algebra 409 (2014), 444–473.
A. Kumjian and D. Pask, C*-algebras of directed graphs and group actions, Ergodic Theory and Dynamical Systems 19 (1999), 1503–1519.
M. V. Lawson, Inverse Semigroups, World Scientific, River Edge, NJ, 1998.
M. V. Lawson, Morita equivalence of semigroups with local units, Journal of Pure and Applied Algebra 215 (2011), 455–470.
M. V. Lawson, Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras International Journal of Algebra and Computation 22 (2012), Article no. 6.
Z. Mesyan and J. D. Mitchell, The structure of a graph inverse semigroup, Semigroup Forum 93 (2016), 111–130.
C. Nӑstӑsescu and N. Rodinó, Group graded rings and smash products, Rendiconti del Seminario Matematico della Università di Padova 74 (1985), 129–137.
C. Nӑstӑsescu and F. van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, Vol. 1836, Springer, Berlin, 2004.
A. L. T. Paterson, Groupoids, Inverse Semigroups, and Their Operator Algebras, Progress in Mathematics, Vol. 170, Birkhaüser, Boston, MA, 1999.
A. L. T. Paterson, Graph inverse semigroups, groupoids and their C*-Algebras, Journal of Operator Theory 48 (2002), 645–662.
D. Quillen, Higher algebraic K-theory. I. in Algebraic K-Theory, I: Higher K-Theories (Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972), pp. 85–147, Lecture Notes in Mathematics, Vol. 341, Springer, Berlin, 1973, pp. 85-147.
S. W. Rigby, The groupoid approach to Leavitt path algebras, In Leavitt Path Algebras and Classical K-Theory, Indian Statistical Institute Series, Springer, Singapore, 2020, pp. 23–71.
B. Steinberg, A groupoid approach to discrete inverse semigroup algebras, Advances in Mathematics 223 (2010), 689–727.
B. Steinberg, Diagonal-preserving isomorphisms of étale groupoid algebras, Journal of Algebra 518 (2019), 412–439.
S. Talwar, Morita equivalence for semigroups, Journal of the Australian Mathematical Society 59 (1995), 81–111.
F. Wehrung, Refinement Monoids, Equidecomposability Types, and Boolean Inverse Semigroups, Lecture Notes in Mathematics, Vol. 2188, Springer, Cham, 2017.
Acknowledgment
We are grateful to the referee for a very thoughtful review, and suggestions that have led to improvements in the exposition.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hazrat, R., Mesyan, Z. Graded semigroups. Isr. J. Math. 253, 249–319 (2023). https://doi.org/10.1007/s11856-022-2361-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-022-2361-z