Poset Definition Given a poset A,<>, we can define o a is maximal if there does not exist b e a such that a< b
Poset . Definition . . Given a poset < A, ≤>, we can define: 1. 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. Yi Li (Fudan University) Discrete Mathematics February 24, 2014 5 / 15
Poset Definition Given a poset A,<>, we can define o a is maximal if there does not exist b e a such that a< b e a is minimal if there does not exist b e a such that
Poset . Definition . . Given a poset < A, ≤>, we can define: 1. 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. Yi Li (Fudan University) Discrete Mathematics February 24, 2014 5 / 15
Poset Definition Given a poset A,<>, we can define o a is maximal if there does not exist b e a such that a< b e a is minimal if there does not exist b E A such that o a is greatest if for every b∈A, we have b≤a
Poset . Definition . . Given a poset < A, ≤>, we can define: 1. 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. Yi Li (Fudan University) Discrete Mathematics February 24, 2014 5 / 15
Poset Definition Given a poset A,<>, we can define o a is maximal if there does not exist b e a such that a< b e a is minimal if there does not exist b e a such that a is greatest if for every b∈A, we have b≤ a is least if for every b∈A, we have a≤b
Poset . Definition . . Given a poset < A, ≤>, we can define: 1. 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. Yi Li (Fudan University) Discrete Mathematics February 24, 2014 5 / 15
Poset Definition Given a poset A, < and a set Sca
Poset . Definition . . 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. Yi Li (Fudan University) Discrete Mathematics February 24, 2014 6 / 15