On derivations of lattices

Authors:
Xiao Long Xin;Ti Yao Li;Jing Hua Lu
Affiliations:
Department of Mathematics, Northwest University, Xi'an 710069, PR China;Department of Mathematics, Northwest University, Xi'an 710069, PR China;Department of Mathematics, Northwest University, Xi'an 710069, PR China and College of Science, Southwest University of Science and Technology, Mianyang 621000, PR China
Venue:
Information Sciences: an International Journal
Year:
2008

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Abstract

In this paper, we introduce the notion of derivation for a lattice and discuss some related properties. We give some equivalent conditions under which a derivation is isotone for lattices with a greatest element, modular lattices, and distributive lattices. We characterize modular lattices and distributive lattices by isotone derivation. Moreover, we prove that if d is an isotone derivation of a lattice L, the fixed set Fix"d(L) is an ideal of L. Finally, we prove that D(L) is isomorphic to L in a distributive lattice L.