Algebras of functions from partially ordered sets into distributive lattices

Abstract

In 1979, in a letter to J. D. H. Smith [10], I. G. Rosenberg proposed as a possible example of a meet-distributive bisemilattice the set of functions from a poset P into a distributive lattice L , with join defined pointwise and a multiplication, called convolution, defined as follows: for f : P ÷ L , g : P + L and p ~ P , p(fog) = Z [(pf)(qg)~(qf)(pg)] . (A related kind of convolution was considered q~P in I. G. Rosenberg [9].) This note is devoted to the investigation of such algebras of functions in the case that P and L are finite. In particular a condition is given for the convolution to be associative (precisely in this case the algebras in question are meet-distributive bisemilattices), and the structure of these algebras (both in the genera and in the associative case) is described, using as a starting-point a (join) retraction from the lattice of functions from P into L onto the lattice of order preserving functions from P to L.