1. a is maximal if there does not exist b E A such that a< b 2. a is minimal if there does not exist b E A such that b< a 3. a is greatest if for every b∈A, we have b≤a 4. a is least if for every b∈A, we have a≤b When we consider a subset of a poset A. a bound of it could be found, which is a element in A Definition 3. Given a poset A. < and a set ScA 1.a∈ A is a upper bound of s if s≤ u for every s∈S. 2.l∈ A is a lower bound of s if l≤ s for every s∈S You can observe that there sometimes exist more than one upper/lower bound of a given subset We are specially interested in some special one, such as the least or the greatest. So the following concepts are derived Definition 4. Given a poset A, < and a set SCA 1. u is a least upper bound of S,(LUB(), if u is the upper bound of s and u s u' for any other upper bound uof s 2. I is a greatest lower bound of S,(GLB(S)), if l is the upper bound of s and l'< I for any other lower bound l of s Example 1. Given two poset described in Figure 1 Figure 1: Extreme element of poset They can demonstrate concepts mentioned before Theorem 5. A subset of a poset has at most one LUB Or gLB The proof is left as an exercise1. a is maximal if there does not exist b ∈ A such that a ≤ b. 2. a is minimal if there does not exist b ∈ A such that b ≤ a. 3. a is greatest if for every b ∈ A, we have b ≤ a. 4. a is least if for every b ∈ A, we have a ≤ b. When we consider a subset of a poset A. A bound of it could be found, which is a element in A. Definition 3. Given a poset < A, ≤> and a set S ⊆ A. 1. u ∈ A is a upper bound of S if s ≤ u for every s ∈ S. 2. l ∈ A is a lower bound of S if l ≤ s for every s ∈ S. You can observe that there sometimes exist more than one upper/lower bound of a given subset. We are specially interested in some special one, such as the least or the greatest. So the following concepts are derived. Definition 4. Given a poset < A, ≤> and a set S ⊆ A. 1. u is a least upper bound of S, (LUB(S)), if u is the upper bound of S and u ≤ u 0 for any other upper bound u 0 of S. 2. l is a greatest lower bound of S, (GLB(S)), if l is the upper bound of S and l 0 ≤ l for any other lower bound l 0 of S. Example 1. Given two poset described in Figure 1. a0 a1 a2 a3 a4 a b c 1 0 Figure 1: Extreme element of poset They can demonstrate concepts mentioned before. Theorem 5. A subset of a poset has at most one LUB or GLB. The proof is left as an exercise. 2