f-Fixed Points of Isotone f-Derivations on a Lattice

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Abstract

In a recent paper, Çeven and Öztürk have generalized the notion of derivation on a lattice to f-derivation, where f is a given function of that lattice into itself. Under some conditions, they have characterized the distributive and modular lattices in terms of their isotone f-derivations. In this paper, we investigate the most important properties of isotone f-derivations on a lattice, paying particular attention to the lattice (resp. ideal) structures of isotone f-derivations and the sets of their f-fixed points. As applications, we provide characterizations of distributive lattices and principal ideals of a lattice in terms of principal f-derivations.

[1] M. Ashraf, S. Ali and C. Haetinger, On derivations in rings and their applications, Aligarh Bull. Math. 25 (2006) 79–107.

[2] G. Birkhoff, Lattice Theory, 3rd edition, Amer. Math. Soc. (Providence, RI, 1967).

[3] Y. Çeven and M. Öztürk, On f-derivations of lattices, Bull. Korean Math. Soc. 45 (2008) 701–707.

[4] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, 2nd edition (Cambridge University Press, 2002).

[5] L. Ferrari, On derivations of lattices, Pure Math. and Appl. 12 (2001) 365–382.

[6] P. He, X. Xin and J. Zhan, On derivations and their fixed point sets in residuated lattices, Fuzzy Sets and Systems 303 (2016) 97–113. doi:10.1016/j.fss.2016.01.006

[7] K.H. Kim and B. Davvaz, On f-derivations of BE-algebras, J. Chungcheong Math. Soc. 28 (2015) 127–138. doi:10.14403/jcms.2015.28.1.127

[8] B. Kolman, R.C. Busby and S.C. Ross, Discrete Mathematical Structures, 4th edition (Prentice Hall PTR, 2000).

[9] S.M. Lee and K.H. Kim, A note on f-derivations of BCC-algebras, Pure Math. Sci. 1 (2012) 87–93.

[10] Ş.A. Özbal and A. Firat, On f-derivations of incline algebras, Int. Electronic J. Pure and Appl. Math. 3 (2011) 83–90.

[11] M.S. Rao, Congruences and ideals in a distributive lattice with respect to a derivation, Bulletin of the Section of Logic 42 (2013) 1–10.

[12] G.C. Rao and K.R. Babu, The theory of derivations in almost distributive lattice, Bulletin of the International Mathematical Virtual Institute 7 (2017) 203–216.

[13] S. Roman, Lattices and Ordered Sets (Springer Science+Business Media, New York, 2008).

[14] B.S. Schröder, Ordered Sets (Birkhauser, Boston, 2003).

[15] G. Szász, Translationen der verbände, Acta Fac. Rer. Nat. Univ. Comenianae 5 (1961) 53–57.

[16] G. Szász, Derivations of lattices, Acta Sci. Math. 37 (1975) 149–154.

[17] J. Wang, Y. Jun, X.L. Xin, T.Y. Li and Y. Zou, On derivations of bounded hyperlattices, J. Math. Res. Appl. 36 (2016) 151–161. doi:10.3770/j.issn:2095-2651.2016.02.003

[18] Q. Xiao and W. Liu, On derivations of quantales, Open Mathematics 14 (2016) 338–346. doi:10.1515/math-2016-0030

[19] X.L. Xin, T.Y. Li and J.H. Lu, On derivations of lattices, Information Sciences 178 (2008) 307–316. doi:10.1016/j.ins.2007.08.018

[20] X.L. Xin, The fixed set of a derivation in lattices, Fixed Point Theory and Applications 218 (2012) 1–12. doi:10.1186/1687-1812-2012-218

[21] Y.H. Yon and K.H. Kim, On f-derivations from semilattices to lattices, Commun. Korean Math. Soc. 29 (2014) 27–36. doi:10.4134/CKMS.2014.29.1.027

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