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Semilatice Decompositions of Semigroups. Hereditariness and Periodicity—An Overview

  • Melanija MitrovićEmail author
  • Sergei Silvestrov
Conference paper
  • 39 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 317)

Abstract

A semigroup is an algebraic structure consisting of a set with an associative binary operation defined on it. We can say that most of the work within theory is done on semigroups with a finiteness condition, i.e. a semigroups possessing any property which is valid for all finite semigroups—like, for example, completely \(\pi \)-regularity, periodicity are. There are many different techniques for describing various kinds of semigroups. Among the methods with general applications is a semilattice decomposition of semigroups. Here, we are interested, in particular, in the decomposability of a certain type of semigroups with finiteness conditions into a semilattice of archimedean semigroups. Having in mind that the definition of finiteness condition may be given, also, in terms of elements of the semigroup, its subsemigroups, in terms of ideals or congruences of certain types, we choose to characterize them mostly by making connections between their elements and/or their special subsets. We are, also, going to list some of the applications of presented classes of semigroups and their semilattice decompositions in certain types of ring constructions. This overview, which is, by no mean, comprehensive one, is mainly based on the results presented in the book [27], and articles [8, 28, 29].

Keywords

Semigroups Semilattices of archimedean semigroups MBC-semigroups GVS-semigroups Hereditary GVS-semigroups Periodic MBC-semigroups Combinatorial GVS-semigroups 

MSC 2010 Classification

06A12 20M05 20M15 20M17 20M18 16E60 

Notes

Acknowledgements

Melanija Mitrović is financially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, Grant 174026, and by the Faculty of Mechanical Engineering, University of Niš, Serbia, Grant “Research and development of new generation machine systems in the function of the technological development of Serbia”. Melanija Mitrović is grateful to Mathematics and Applied Mathematics Research Environment MAM, Division of Applied Mathematics at the School of Education, Culture and Communication at Mälardalen University for cordial hospitality and excellent environment for research and cooperation during her visit.

References

  1. 1.
    Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1994)Google Scholar
  2. 2.
    Anderson, D.F.: Robert Gilmer’s work on semigroup rings. In: Brewer, J.W., Glaz, S., Heinzer, W.J., Olberding, B.M., (eds.) Multiplicative Ideal Theory in Commutative Algebra - A Tribute to the Work of Robert Gilmer. Springer, Berlin (2006)Google Scholar
  3. 3.
    Baird, B.B., Magil, K.D.: Green’s \(\cal{R}\)-relations and climbing mountains. Semigroup Forum 18, 347–370 (1979)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bell, A.D., Stadler, S.S., Teply, M.L.: Prime ideals and radicals in semigroup-graded rings. Proc. Edinb. Math. Soc. 39, 1–25 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bogdanović, S.: Semigroups of Galbiati-Veronesi. In: Proceedings of the Conference “Algebra and Logic”, Zagreb 1984, Novi Sad 1984, 9–20 (1984)Google Scholar
  6. 6.
    Bogdanović, S.: Semigroups of Galbiati-Veronesi II. Facta Univ. Niš, Ser. Math. Inform. 2, 61–66 (1987)Google Scholar
  7. 7.
    Bogdanović, S., Milić, S.: A nil-extension of a completely simple semigroup. Publ. Inst. Math. 36(50), 45–50 (1984)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bogdanović, S., Ćirić, M., Mitrović, M.: Semilattices of hereditary Archimedean semigroups 9(3), 611–617 (1995); In: International Conference on Algebra, Logic and Discrete Math. Niš, April 14-16, 1995, ed. Bogdanović, S., Ćirić, M., Perović, Ž. FilomatGoogle Scholar
  9. 9.
    Chrislock, J.L.: On a medial semigroups. J. Algebra 12, 1–9 (1969)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Clifford, A.H.: Semigroups admitting relative inverses. Ann. Math. 42, 1037–1049 (1941)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Clifford, A.H.: Bands of semigroups. Proc. Am. Math. Soc. 5, 499–504 (1954)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ćirić, M., Bogdanović, S.: Decompositions of semigroups induced by identities. Semigroup Forum 46, 329–346 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Drazin, M.P.: Pseudoinverses in associative rings and semigroups. Am. Math. Mon. 65, 506–514 (1958)CrossRefGoogle Scholar
  14. 14.
    Galbiati, J.L., Veronesi, M.L.: Semigruppi quasi regolari, Atti del convegno: Teoria dei semigruppi, Siena. F. Migliorini (ed.) (1982)Google Scholar
  15. 15.
    Galbiati, J.L., Veronesi, M.L.: On quasi completely regular semigroups. Semigroup Forum 29, 271–275 (1984)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gilmer, R.: Commutative Semigroup Rings. The University of Chicago Press, Chicago (1984)Google Scholar
  17. 17.
    Gopalakrishnan, H.: \(\pi \)-regularity of semigroup graded rings. Commun. Algebra 30(2), 100–977 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Grillet, P.A.: Semigroups - An Introduction to the Structure Theory. Marcel Dekker, Inc., New York (1995)Google Scholar
  19. 19.
    Grillet, P.A.: Commutative Semigroups. Advances in Mathematics. Kluwer Academic Publishers, Boston (2001)Google Scholar
  20. 20.
    Jaspers, E., Okniński, J.: Noetherian Semigroup Algebras. Springer, Berlin (2007)Google Scholar
  21. 21.
    Kelarev, A.V.: Applications of epigroups to graded ring theory. Semigroup Forum 50, 327–350 (1995)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kelarev, A.V.: Ring Constructions and Applications. World Scientific, Singapore (2002)Google Scholar
  23. 23.
    Lallement, G.: Semigroups and Combinatorial Applications. Wiley, New York (1979)zbMATHGoogle Scholar
  24. 24.
    McCoy, N.: Generalized regular rings. Bull. Am. Math. Soc. 45, 175–178 (1939)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mikhalev, A.V., Pilz, G.F.: The Concise Handbook of Algebra. Springer Science + Business Media B. V., Berlin (2002)CrossRefGoogle Scholar
  26. 26.
    Miller, D.W.: Some aspects on Green’s relations on periodic semigroups. Czechoslov. Math. J. 33(4), 537–544 (1983)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mitrović, M.: Semilattices of Archimedean Semigroups. University of Niš - Faculty of Mechanical Engineering, Niš (2003)zbMATHGoogle Scholar
  28. 28.
    Mitrović, M.: On semilattices of archimedean semigroups - a survey. In: Proceedings of Workshop on Semigroups and Languages, 2002, 163–196. World Scientific, Lisbon, Portugal (2004)Google Scholar
  29. 29.
    Mitrović, M.: Regular subsets of semigroups related to their idempotents. Semigroup Forum 70(3), 356–360 (2005)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Munn, W.D.: Pseudoinverses in semigroups. Proc. Camb. Philos. Soc. 57, 247–250 (1961)CrossRefGoogle Scholar
  31. 31.
    Nagy, A.: Special Classes of Semigroups. Springer-Science+Business Media, B. V, Berlin (2001)CrossRefGoogle Scholar
  32. 32.
    Nystedt, P., Öinert, J.: Simple semigroup graded rings. J. Algebra Appl. 14(07) (2015)Google Scholar
  33. 33.
    Okniński, J.: Smigroup Algebras. Marcel Dekker, New York (1991)Google Scholar
  34. 34.
    Petrich, M.: The maximal semilattice decompositions of a semigroup. Math. Zeitschr. 85, 68–82 (1964)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Petrich, M.: Introduction to Semigroups. Merill, Ohio (1973)zbMATHGoogle Scholar
  36. 36.
    Petrich, M.: Lectures in Semigroups. Wiley, New York (1977)Google Scholar
  37. 37.
    Prosvirov, A.S.: On periodic in which no torsion class is a subsemigroup. In: II All-Union Symposium on the Theory of Semigroups. Abstracts of Reports, Sverdlovsk, 72 (1988), (in Russian)Google Scholar
  38. 38.
    Putcha, M.S.: Semilattice decompositions of semigroups. Semigroup Forum 6, 12–34 (1973)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Putcha, M.S.: Viewing results on \(\cal{S}\)-indecomposable semigroups as solutions to mathematical puzzles. Semigroup Forum 9, 181–183 (1974)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Putcha, M.S., Weissglass, J.: A semilattice decompositions into semigroups with at most one idempotent. Pac. J. Math. 39, 225–228 (1971)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Schein, B.M.: Book review - social semigroups a unified theory of scaling and block modelling as applied to social networks. Semigroup Forum 54, 264–268 (1997)Google Scholar
  42. 42.
    Schwarz, Š.: Contribution to the theory of periodic semigroups. Czechoslov. Math. J. 3, 7–21 (1953). (in Russian)CrossRefGoogle Scholar
  43. 43.
    Sedlock, J.T.: Green’s relations on a periodic semigroup. Czechoslov. Math. J. 19(2), 318–323 (1969)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Shevrin, L.N.: The theory of epigroups, I, II, Mat. Sb. 185 (8), 1994, 129–160; 185(9), 1994, 153–176, (in Russian; English translation: Russ. Acad. Sci. Sb. Math. 82, 1995, 485–512; 83, 1995, 133–154Google Scholar
  45. 45.
    Shevrin, L.N.: Epigroups. In: Kudryavtsev, V.B., Rozenberg, I.G. (eds.) Structural Theory of Automata, Semigroups and Universal Algebra, 331–380. Springer, Berlin (2005)Google Scholar
  46. 46.
    Tamura, T.: The theory of construction of finite semigroups I. Osaka Math. J. 8, 243–261 (1956)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Tamura, T.: Note on the greatest semilattice decomposition of semigroups. Semigroup Forum 4, 255–261 (1972)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Tamura, T.: Quasi-orders, generalized archimedeaness, semilattice decompositions. Math. Nachr. 68, 201–220 (1975)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Tamura, T.: Semilattice indecomposable semigroups with a unique idempotent. Semigroup Forum 24, 77–82 (1982)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Tamura, T., Kimura, N.: On decomposition of a commutative semigroup. Kodai Math. Sem. Rep. 4, 109–112 (1954)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Tamura, T., Kimura, N.: Existence of greatest decomposition of semigroup. Kodai Math. Sem. Rep. 7, 83–84 (1955)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Thierrin, G.: Sur une condition necessarie et suffisante pour qu un semigroupe soit un groupe. C. R. Acad. Sci. Paris 232, 376–378 (1951)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Thierrin, G.: Sur queiques propriétiés de certaines classes de demi-groupes. C. R. Acad. Sci. Paris 239, 33–34 (1954)Google Scholar
  54. 54.
    Veronesi, M.L.: Sui semigruppi quasi fortemente regolari. Riv. Mat. Univ. Parma 10(4) 319–329 (1984)Google Scholar
  55. 55.
    Yamada, M.: On the greatest semilattice decomposition of a semigroup. Kodai Mat. Sem. Rep. 7, 59–62 (1955)MathSciNetCrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsFaculty of Mechanical Engineering, University of NišNišSerbia
  2. 2.Division of Applied Mathematics, School of EducationCulture and Communication, Mälardalen UniversityVästeråsSweden

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