Lattice
A lattice is any partially ordered set (L, ) where every non-empty finite subset X ⊆L has a greatest lower bound (meet) and a least upper bound (join).
任意非空有限集合也可以换成任意两个元素 a、b，定义等价（对 X 的大小归纳可证）
A complete lattice is any partially ordered set (L, ) where every subset X ⊆L has a greatest lower bound and a least upper bound.
显然，有限非空格总是完全格。完全格也一定是非空的。
Every complete lattice has a greatest element (top, ) and a least element (bottom ).
A lattice that additionally has a top and a bottom is called a bounded lattice. Equivalently, a bounded lattice has glb and lub for every finite subset, including the empty set.
Semilattice
- If for every non-empty finite subset, only lub exists, it is a join semilattice
- If for every non-empty finite subset, only glb exists, it is a meet semilattice
Product
The product of n lattices is also lattice. Product of complete lattices is complete.
构造乘积的方式是，集合算笛卡儿积，偏序关系为按元素偏序关系的合取。乘积格的 glb 和 lub 为按元素依次应用对应格的算子。