  Lattice induced threshold function is a Boolean function determined by a
particular linear combination of lattice elements. We prove that every isotone
Boolean function is a lattice induced threshold function and vice versa. We
also represent lattice valued up-sets on a finite Boolean lattice in the
framework of cuts and lattice induced threshold functions. In terms of closure
systems we present necessary and sufficient conditions for a representation of
lattice valued up-sets on a finite Boolean lattice by linear combinations of
elements of the co-domain lattice.
