Compatible functions on BDLs - on a result of G. Gratzer

By an affine complete algebra is meant an algebra whose polynomial functions are the only compatible (i.e. congruence-preserving) functions. Hence one can imagine affine complete algebras as algebras having "many" congruences. G. Gratzer in 1962 showed that every Boolean algebra is affine complete and in 1964 he characterized affine complete bounded distributive lattices. In general, the problem of characterizing algebras which are affine complete was firstly formulated in his 1968 monograph "Universal algebra". Our aim is to initiate a project of describing the compatible functions of an algebra in a given (favourite) variety no matter whether the algebra is affine complete or not. We start with describing the compatible functions of an arbitrary bounded distributive lattice. G. Gratzer's characterization of affine complete bounded distributive lattices is then coming as an immediate corollary of our result.