(F,G)-derivations on lattices

Abstract

In this thesis we have generalized the notion of (^, _)-derivation to (F,G)- derivation on a lattice, and investigated its properties in detail. This generalization is based on two arbitrary binary operations F and G instead of the lattice meet (^) and join (_) operations. To that end, a lot of preparatory work was required. In particular, several properties and characterizations of binary operations on an arbitrary lattice were investigated, and two representation theorems of a lattice based on a binary operation were provided. Furthermore, we have studied the isotone and principal f-derivations on a lattice and investigated their properties. We have studied the lattice structure of isotone f-derivations on a lattice, and the ideal structures of the sets of their f-fixed points. Finally, future work is anticipated in multiple directions. We intend to extend the different notions of derivation to other useful algebraic structures and investigate their fundamental properties. Moreover, we believe that the notion of (F,G)-derivations on a lattice is worthy of further investigations. 65