On locally order-polynomially complete modular lattices

Abstract

Let L be a lattice. F L L (L') , ( ) = is the set of all k-place functions on L. If we define pointwise meet and join operation on Fk(L), then Fk(L) becomes a lattice. The elements of the sublattice Pk(L) of Fg(L) generated by the projections and the constant functions will be called k-place polynomial functions on L. If fEFk(L) has the property that for every finite subset MC=L k there exists a pEPk(L) such that f and p coincide on M, then p is called a local polynomial function, fEFk(L) is called order-preserving if a~.<b,,= i = 1 . . . . . k implies f (a l , ..., ak)<--f(bl . . . . 9 bk). L is called (locally) order-polynomially complete iff every order-preserving function on L is a (iocal) polynomial function. The first characterization of finite order-polynomially complete lattices was given in Wille [6]. For finite modular lattices he proved the following