Yong Ho Yon and Kyung Ho Kim Mokwon University, Korea National University of Transpotation

Abstract : In this paper, we introduce the notion of $f$-derivations from a semilattice $S$ to a lattice $L$, as a generalization of derivation and $f$-derivation of lattices. Also, we define the simple $f$-derivation from $S$ to $L$, and research the properties of them and the conditions for a lattice $L$ to be distributive. Finally, we prove that a distributive lattice $L$ is isomorphic to the class $SD_f(S,L)$ of all simple $f$-derivations on $S$ to $L$ for every $\wedge$-homomorphism $f : S\to L$ such that $f(x_0)\vee f(y_0) = 1$ for some $x_0, y_0\in S$, in particular, $L\cong SD_f(S,L)$ for every $\wedge$-homomorphism $f : S\to L$ such that $f(x_0) = 1$ for some $x_0\in S$.