In this paper we define a weak auxiliary relation<on completely distributive lattice L as follows:x<y iff for any subset X of L with y ≤supX there exists x*∈X such that x≤x*.Then,we prove the follow~ ing results:
Theorem. Suppose that L₁,L₂are completely distributive lattices,f:L₁ →L₂is a function,then f is a generalized order-homomorphism iff f pre serves arbitrary sups and <,and iff f preserves the minimal sets.
Theorem. If a∈L,BcL,then B is a minimal set of a iff sup B=a and B C↓a.Moreover,if there exists a minimal set of a,then the greatest minimal set of a is β(a)=↓a,where ↓a={x∈L|x<a}.
In the second section of this paper,we establish two extention theo- rems of generalized order-homomorphsms.Fanally,we prove a duality theo- rem of the categories CD and COMP ,whose objects are completely dis- tributive lattices(in both cases)and whose morphisms are generalized or- der-homomorphisms(respectively,functions preserving arbitrary sups and infs),and introduce an important and interesting functor of category CD×CD to CD,
