Congruence Relations Of Lattice Theory

Firstly, in this paper according to the characterization of two elements covered principal congruence relations of a maximal chain and the distributivity of congruence lattices we proved that the congruence lattice of a chain with finite element is a Boole lattice, and we further proved that the congruence lattice of a chain with countable element is a Boole lattice. Secondly, we supplied a new iff which characterize distribute lattices, and we proved that the congruence lattice of distributive lattice with finite element is Boole lattice based on the method of a chains congruence lattices, but the converse isn't true, the congruence lattice of distributive lattice with countable element is Boole lattice, but the converse isn't true. Thirdly, according to the characterization of two elements covered principal congruence relations of modular lattices we proved the congruence lattice of modular lattice with finite element is Boole lattice, but the converse is not true, and we further proved that the congruence lattice of modular lattice with countable element is Boole lattice, but the converse is not true. For a finite lattices, at last according to the research of the number of two elements covered principal congruence relations of congruence lattices we provided the maximum number of congruence relations which generate from it.It also proved a series of theories about congruence relation on δ-lattice and ortho modular lattice.