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.
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