grid
Assumption (L, ≼) (L, \preccurlyeq)(L,≼) Is a partially ordered set, if for anya, b ∈ L, a, ba, b\in L ,{a, b}a,b∈L,a,b has both supremum andinfimum, then it is called<L, ≼> <L,\preccurlyeq><L,≼> Is alattice (lattice)（lattice）
Obviously the supremum and the infimum are unique
Supremum L ∩ B (a, b) L \ cap B ({a, b})L∩B(a,b ) is denoted as a∨ \lor∨ b, called the union of a and b(join) (join)（join）
Infimum G ∩ B (a, b) G \ cap B ({a, b})G∩B(a,b ) is denoted as a∧ \land∧ b, called the meeting of a and b(meet)（meet）
Maximum yuan, minimum yuan
The largest element: refers to the element that is not less than everything in the subset of the partially ordered set. The
smallest element: refers to the element that is not greater than everything in the subset of the partially ordered set.
Maximal element, minimal element
Minimal element: Refers to the partial ordering set that has no comparable element larger than it. Minimal element: Refers to the partial ordering set that has no comparable element smaller than it.
Bounded lattice
The lattice with the largest element and the smallest element is called a bounded lattice.
Supplement
设 < L , ≼ > <L,≼> <L,≼> Is a bounded lattice, a and b are two elements in L, ifa ∨ b = 1, a ∧ b = 0 a∨b=1,a∧b=0a∨b=1,a∧b=0 , then it is said that a is the complement of b or b is the complement of a, or a and b are each other's complement.
a ∨ b = 1 a \ lor b = 1 a∨b=1 meansa and ba and ba and b go up, there is only one point in common1 11.
a ∧ b = 1 a \land b=1 a∧b=1 meansa and ba and bThere is only one point in common between a and b going down0 00.
Distribution grid
The distribution grid is the one that satisfies the distribution rate
For any element x, y and z of the lattice, there is x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) x∧(y∨z)=(x∧y)∨(x∧z )x∧( and∨with )=(x∧and )∨(x∧z ) . Due to the symmetry of knot operation and intersection operation in the lattice, the above condition is equivalent tox ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) x∨(y∧z)=(x∨y)∧ (x∨z)x∨( and∧with )=(x∨and )∧(x∨z ) , whenLLWhen L is a distributive lattice, the intersection operation satisfies the distributive law for the knot operation, and vice versa.
Boolean lattices, divisor lattices, ideal lattices, chains, etc. are all distributive lattices.
judgment
There is more than one complement of a certain element in a bounded lattice. Then it is not a distributive lattice √ \surd√