Grid Only Complement And Distribution

In the early development of lattice theory,there was a well-publicized debate regarding the basic axiomatic of the subject. This lead Huntington to ask, in1904,whether every uniquely complemented lattice was distributive. This problem advanced the development of lattice theory.By1940,Huntington's conjecture had to be further studied, if we placed some further assumptions on the uniquely complemented lattice,such as its being modular,or being atomic,it was distributive lattice.At that time,they widely believed that Huntington's problem was trueHowerve,In1945,R.P.Dilworth proved that every lattice can be embedded into an uniquely complemented lattice. This result refuted Huntington's conjecture, and it showed the existence of non-distributive uniquely complemented lattice. In1969, Chen and Gratzer gave a simpler proof of R.P.Dilworth's Theorem,but we have not found a suitable example of non-distributive unquely complemented.This articlefirst introduces definitions and related properties of lattice,uniquely complemented lattices and distributive lattices.Mainly to study the relationship between uniquely complemented lattices and distributive lattices, and discuss uniquely complemented lattices are distributive lattices under what additional conditions. On the basis of the original conditions,we get more additional conditions and inferences.At the same time, we consider the existence of non-distributive uniquely complemented lattices, and explore how to describe it.