3 Lattice In this section, we will illustrate lattice in two approaches. One is to define it on the base of a partial order. And another is to define it in traditionally algebraic style 3.1 Orders Perspective Definition 6. A lattice (structure) is a poset (A, s) in which any two elements a, b have a LUB(a, b)and a GLB(a, b) From now on, we define a Ub=LU B(a, b) and anb=GLB(a, b) in brief. We also call them join and meet respectively. With the following example, we show you why the notations are taken. Example 2.(P(A), S), with two operations U(union) and n(intersection), is a lattice The verification is simple. You just follow the definition of lattice There are several means to represent lattice. For lattice is generally a partial order, it can be described by Hasse diagram. Another way is to use table, for a lattice is also determined by two Example 3.(P(a, b), S), with two operations U(defined as union) and n(defined as intersection) It is a lattice according to previous Example 2. It can be described by a Hasse diagram as shown Figure 2 {a,b} b} Figure 2: Hasse diagram of (P(a, b9),s) It can aslo be described by joint and meet table as shown in the following Table 1. Obviously, 0{a}{b}{a,b} falfa fa fa, b] fa, bj {a}0{a}0 {b}{b}{a,b}{b}{a,b} {b}00{b}{b} {a,b}|{a,b}{a,b}{a,b}{a,b {a,b}0{a}{b}{a,b} (a)Joint table (b) Meet table Table 1:(P(a, b9),S) joint/meet table is symmetric. Sometimes, they could be merged into a table by taken upper/lower3 Lattice In this section, we will illustrate lattice in two approaches. One is to define it on the base of a partial order. And another is to define it in traditionally algebraic style. 3.1 Order’s Perspective Definition 6. A lattice (structure) is a poset hA, ≤i in which any two elements a, b have a LUB(a, b) and a GLB(a, b). From now on, we define a ∪ b = LUB(a, b) and a ∩ b = GLB(a, b) in brief. We also call them join and meet respectively. With the following example, we show you why the notations are taken. Example 2. hP(A), ⊆i, with two operations ∪(union) and ∩(intersection), is a lattice. The verification is simple. You just follow the definition of lattice. There are several means to represent lattice. For lattice is generally a partial order, it can be described by Hasse diagram. Another way is to use table, for a lattice is also determined by two operations. Example 3. hP(a, b), ⊆i, with two operations ∪(defined as union) and ∩(defined as intersection). It is a lattice according to previous Example 2. It can be described by a Hasse diagram as shown in Figure 2. ∅ {a} {b} {a, b} Figure 2: Hasse diagram of hP({a, b}), ⊆i It can aslo be described by joint and meet table as shown in the following Table 1. Obviously, ∪ ∅ {a} {b} {a, b} ∅ ∅ {a} {b} {a, b} {a} {a} {a} {a, b} {a, b} {b} {b} {a, b} {b} {a, b} {a, b} {a, b} {a, b} {a, b} {a, b} (a) Joint table ∩ ∅ {a} {b} {a, b} ∅ ∅ ∅ ∅ ∅ {a} ∅ {a} ∅ {a} {b} ∅ ∅ {b} {b} {a, b} ∅ {a} {b} {a, b} (b) Meet table Table 1: hP({a, b}), ⊆i in table joint/meet table is symmetric. Sometimes, they could be merged into a table by taken upper/lower 3