Higher Derivations of Finitary Incidence Algebras

  • Ivan Kaygorodov
  • Mykola Khrypchenko
  • Feng Wei


Let P be a partially ordered set, R a commutative unital ring and FI(P,R) the finitary incidence algebra of P over R. We prove that each R-linear higher derivation of FI(P,R) decomposes into the product of an inner higher derivation of FI(P,R) and the higher derivation of FI(P,R) induced by a higher transitive map on the set of segments of P.


Finitary incidence algebra Higher derivation Inner higher derivation Higher transitive map 

Mathematics Subject Classification (2010)

Primary 16S50, 16W25 Secondary 16G20, 06A11 


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The authors are grateful to the reviewer whose suggestions helped them to improve the readability of the paper.


  1. 1.
    Baclawski, K.: Automorphisms and derivations of incidence algebras. Proc. Amer. Math. Soc. 36(2), 351–356 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brusamarello, R., Fornaroli, É.Z., Khrypchenko, M.: Jordan isomorphisms of finitary incidence algebras. Linear Multilinear Algebra 66(3), 565–579 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brusamarello, R., Fornaroli, É.Z., Santulo, E.A.: Anti-automorphisms and involutions on (finitary) incidence algebras. Linear Multilinear Algebra 60(2), 181–188 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brusamarello, R., Fornaroli, É.Z., Santulo, E.A.: Classification of involutions on finitary incidence algebras. Int. J. Algebra Comput. 24(8), 1085–1098 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Coelho, S.P.: The automorphism group of a structural matrix algebra. Linear Algebra Appl. 195, 35–58 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Coelho, S.P.: Automorphism groups of certain structural matrix rings. Comm. Algebra 22(14), 5567–5586 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Coelho, S.P., Polcino Milies, C.: Derivations of upper triangular matrix rings. Linear Algebra Appl. 187, 263–267 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Courtemanche, J., Dugas, M., Herden, D.: Local automorphisms of finitary incidence algebras. Linear Algebra Appl. 541, 221–257 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Doubilet, P., Rota, G.-C., Stanley, R.P.: On the foundations of combinatorial theory. VI. The idea of generating function. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. II: Probability theory. Univ. California Press, pp. 267–318 (1972)Google Scholar
  10. 10.
    Dugas, M.: Homomorphisms of finitary incidence algebras. Comm. Algebra 40 (7), 2373–2384 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ferrero, M., Haetinger, C.: Higher derivations and a theorem by Herstein. Quaest. Math 25(2), 249–257 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ferrero, M., Haetinger, C.: Higher derivations of semiprime rings. Comm. Algebra 30(5), 2321–2333 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Heerema, N.: Convergent higher derivations on local rings. Trans. Amer. Math. Soc. 132, 31–44 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Heerema, N.: Higher derivations and automorphisms of complete local rings. Bull. Amer. Math. Soc. 76, 1212–1225 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaygorodov, I., Popov, Yu.: A characterization of non-associative nilpotent algebras by invertible Leibniz-derivations. J. Algebra 456, 1086–1106 (2016)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kaygorodov, I., Popov, Yu.: Generalized derivations of (color) n-ary algebras. Linear Multilinear Algebra 64(6), 1086–1106 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Khripchenko, N.S.: Automorphisms of finitary incidence rings. Algebra Discrete Math. 9(2), 78–97 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Khripchenko, N.S.: Derivations of finitary incidence rings. Comm. Algebra 40 (7), 2503–2522 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Khripchenko, N.S., Novikov, B.V.: Finitary incidence algebras. Comm. Algebra 37(5), 1670–1676 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Khrypchenko, M.: Jordan derivations of finitary incidence rings. Linear Multilinear Algebra 64(10), 2104–2118 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Khrypchenko, M.: Local derivations of finitary incidence algebras. Acta Math. Hungar 154(1), 48–55 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Koppinen, M.: Automorphisms and higher derivations of incidence algebras. J. Algebra 174, 698–723 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mirzavaziri, M.: Characterization of higher derivations on algebras. Comm. Algebra 38(3), 981–987 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nowicki, A.: Higher R-derivations of special subrings of matrix rings. Tsukuba J. Math. 8(2), 227–253 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nowicki, A.: Inner derivations of higher orders. Tsukuba J. Math. 8(2), 219–225 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nowicki, A., Nowosad, I.: Local derivations of subrings of matrix rings. Acta Math. Hungar. 105, 145–150 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ribenboim, P.: Higher derivations of rings. I. Rev. Roumaine Math. Pures Appl. 16, 77–110 (1971)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Ribenboim, P.: Higher derivations of rings. II. Rev. Roumaine Math. Pures Appl. 16, 245–272 (1971)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Rota, G.-C.: On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2(4), 340–368 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rota, G.-C., Goldman, J.: On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions. Stud. In Appl. Math. 49, 239–258 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rota, G.-C., Mullin, R.: On the foundations of combinatorial theory. III. Theory of binomial enumeration. In: Graph Theory and its Appl., B. Harris, Ed. Acad. Press., pp. 167–213 (1970)Google Scholar
  32. 32.
    Saymeh, S.A.: On Hasse-Schmidt higher derivations. Osaka J. Math. 23(2), 503–508 (1986)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Spiegel, E.: Automorphisms of incidence algebras. Comm. Algebra 21(8), 2973–2981 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Spiegel, E.: On the automorphisms of incidence algebras. J. Algebra 239, 615–623 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Spiegel, E., O’Donnell, C.J.: Incidence algebras. Marcel Dekker, New York (1997)zbMATHGoogle Scholar
  36. 36.
    Stanley, R.: Enumerative combinatorics, vol. 1. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  37. 37.
    Stanley, R.P.: Structure of incidence algebras and their automorphism groups. Bull. Amer. Math. Soc. 76, 1236–1239 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ward, M.: Arithmetic functions on rings. Ann. Math. (2) 38, 725–732 (1937)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wei, F., Xiao, Z.: Higher derivations of triangular algebras and its generalizations. Linear Algebra Appl. 435(5), 1034–1054 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Xiao, Z.: Jordan derivations of incidence algebras. Rocky Mountain J. Math. 45 (4), 1357–1368 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zhang, X., Khrypchenko, M.: Lie derivations of incidence algebras. Linear Algebra Appl. 513, 69–83 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.CMCCUniversidade Federal do ABCSanto AndréBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal de Santa CatarinaFlorianópolisBrazil
  3. 3.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingPeople’s Republic of China

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