Theorem 10. IfL is any set in which there are two operation defined as U and n satisfying the last four properties, then L is a lattice Proof. We first show that L has the following two properties: a∪b= a and a∩b= b are equivalent. if aUb=a, we get anb=(a∪b)∩b= b by absorption 2.<S, < is a poset if b a is defined by anb=b or aUb=a (i)aUa= a imply a≤ i) Suppose a≤ b and b≤a, we have a=aUb=bUa=b. (irfa≤b,b≤c, then au b=b,bUc=c. Hence, aUc=aU(buc=(aubuc=bUc=c We now prove that aUb is the LUB. Since(aUbna=a,asaUb. Similarly b< aUb. Now given any c such that a, b< c, we have aUc=c and bUc=c. Hence b)Uc=aU(bUc ∪c=c and aU b≤c. Similarly, we can prove that a n b is the glB. Totally, L is a lattice with U and n 4 Sublattice Similar to group given some subset, we have the following concep Definition 11. A subset s of a lattice L is called sublattice if it is closed under the operation U and∩ Example 5. Given two posets described in Figure 3 (a) Lattice (b)Sublattice igure 3: Sublattice of (20, 10, 5, 4, 2, 11, It obvious that lattice in Figure 3b is a sublattice of lattice in Figure 3a For sublattice is define relatively. Dually, we can also define a concept ertension as followinTheorem 10. If L is any set in which there are two operation defined as ∪ and ∩ satisfying the last four properties, then L is a lattice. Proof. We first show that L has the following two properties: 1. a ∪ b = a and a ∩ b = b are equivalent. if a ∪ b = a, we get a ∩ b = (a ∪ b) ∩ b = b by absorption. 2. < S, ≤> is a poset if b ≤ a is defined by a ∩ b = b or a ∪ b = a. (i) a ∪ a = a imply a ≤ a. (ii) Suppose a ≤ b and b ≤ a, we have a = a ∪ b = b ∪ a = b. (iii) If a ≤ b, b ≤ c, then a ∪ b = b, b ∪ c = c. Hence, a ∪ c = a ∪ (b ∪ c) = (a ∪ b) ∪ c = b ∪ c = c We now prove that a∪b is the LUB. Since (a∪b)∩a = a, a ≤ a∪b. Similarly b ≤ a∪b. Now given any c such that a, b ≤ c, we have a ∪ c = c and b ∪ c = c. Hence (a ∪ b) ∪ c = a ∪ (b ∪ c) = a ∪ c = c and a ∪ b ≤ c. Similarly, we can prove that a ∩ b is the GLB. Totally, L is a lattice with ∪ and ∩. 4 Sublattice Similar to group, given some subset, we have the following concept. Definition 11. A subset S of a lattice L is called sublattice if it is closed under the operation ∪ and ∩. Example 5. Given two posets described in Figure 3 1 2 5 4 10 20 (a) Lattice 1 2 5 10 (b) Sublattice Figure 3: Sublattice of h{20, 10, 5, 4, 2, 1}, |i It obvious that lattice in Figure 3b is a sublattice of lattice in Figure 3a For sublattice is define relatively. Dually, we can also define a concept extension as following: 5