The Completion of n-Kernels of the Skeletal Congruences on a Distributive Lattice
Muhammad Amer Latif · JOURNAL OF INTERNATIONAL ACADEMY OF PHYSICAL SCIENCES · 2011
The Skeleton SC(L) = { q I(L) : q =j* for some jI C(L)} = {qIC(L) : q = j **} is a complete Boolean lattice. The meet of the set { qi } I SC(L) is C qI while the join is Uqi =( Uqi)** = (Cqi)* and the complement of q ISC(L) is q*. For any nIL, the set K SC(L) = { Ker n q : q I SC(L)} which is also a complete lattice, where Ker n q = { x : x o n (q)}, is an n-ideal. In this paper, we have studied the n-kernels of the Skeletal congruence K SC(L) on a distributive lattice L, and generalized many results on the completions of special classes of lattices. We have also shown that the set K SC(L) of all n-kernels forms an upper continuous distributive lattice and the map a® n ={ x I L : a U x £ x £ a U n } is a lower join – dense embedding of L into K SC(L).