Certain elements have special importance in posets. The maximal element(s) is the greatest element(s). The minimal element (s) is the element (s) with the least value. You may have 1 or many maximal or minimal elements. a 1,a 2,a 3 are maximal elements b 1,b 2,b 3 are minimal elements since there is no line between b 2 and b 3 we cannot conclude that one is less than the other or equal to each other. a1a1 b1b1 a2a2 b2b2 a3a3 b3b3

Topological sorting decba ab d e c

To find a topological sorting of poset (A,≤)(linear order) A = {a,b,c,d,e} 1. Choose a minimal element of A. You could choose d or e. We will choose d. 2. Make d the next entry of sort and replace A – {d}. A = {a,b,c,e} Repeat steps 1 and 2 until A = {}

First we note d and e are minimal elements. We choose either d or e and place it in the sort. We select d in this example. Mark off d. Next we see e is the minimal element, so e is the next in our sort. Mark off e. Next we see c is the minimal element. Place c in the sort and mark it off. Next we can choose either a or b as our minimal element. In this example we choose b. b goes in the sort. We are left with a. a is the last entry the sort. a c de b decba

An element is considered a greatest element if it is greater than all the other elements. There can only be one greatest element or none. An element is considered the least element if it is less than all other elements. There can only be one least element or none. S = {a,b,c} A = P(s) (Power set of s) A = {{},{a},{b}, {c},{a,b},{a,c},{b,c},{a,b,c}} The least element in set A is {}. The greatest element in set A is {a,b,c}

d is the greatest element A is the least element There is no greatest element There is no least element d b c a ba e dc

Posets have an upper bound and a lower bound. A = {a,b,c,d,e,f,g,h} Subset B 1 = {a,b} Subset B 2 = {c,d,e} B 1 has no lower bounds. a and b are as low as you can go. Upper bounds = c,d,e,f,g,h B 2 Upperbounds are f,g,h lower bounds are c,a,b. f d a b c e g h

Subset B 2 = {c,d,e} Draw a line at the highest pair (d,e) Anything above the line is considered upper bounds. Anything below the line is considered lower bounds. f d a b c e g h

Least upper bound: LUB Greatest lower bound GLB Subset B = {c,d,e} 1. draw a line at the greatest pair of subset B (d,e). 2. least upper bound of subset B is f. 3. greatest lower bound of subset B is c. g d ba c e f h

If you have no lower bound then, you have no greatest lower bound. a and b are not comparable. This poset has no greatest lower bound. c is the least upper bound. c ba