On Several Types Of Lattice Equivalent Definitions

The definition of lattices is the most important in lattice theory, this paper redefine the modular lattice with unit,the distribution lattice,the distribution lattice with unit,Boolean lattice on the foundation of original work, which can make the definition of the lattice simpler in form. And mail contents are following:1,this paper gave the original definitions of the types of lattices, mainly the definition of algebraic structure, and finally introduced several equivalent definitions of lattices, we got that the lattices can be defined in different forms.2,this paper gave three conditions and two conditions of the equivalent definitions, which is one of the core of this paper. The paper simplified some definitions and proofs on the foundation of original definitions and other definitions, which can make the definition of a modular lattice with unit simpler in form more perfect in structure.3,this paper gave the three conditions and two conditions equivalent definitions of the distribution lattices and the distribution lattices with unit, and the paper simplified three conditions equivalent definitions of the distribution lattices and re-defined the three conditions equivalent definitions of distribution lattices with unit.4,this paper gave equivalent definitions of Boolean lattice. As Boolean lattice's applications are wide and the original definition of the algebraic structure is very complicated, the paper improved Boolean lattice's definition and gave the three equivalent conditions of Boolean lattice.