Given the Hasse diagram for the Poset find the following min

Given the Hasse diagram, for the Poset, find the following. minimal elements minimum elements maximal elements maximum elements glb(a, f) lub(g, f) Does h relate to a? Does g relate to e? Given the Hasse diagram, for the Poset, find the following: minimal elements minimum elements maximal elements maximum elements glb c, g) lub(e, b) Does h relate to c? Does d relate to a?

Solution

Maximal: An element a of a poset (S, )is maximal if there is no element b in S, such that a   b. • Similarly, we also have a minimal element in the poset. • They are respectively, the “top” and the “bottom” elements in the diagram.

3) From the given Hasse diagram,

i ) only k is minimal element.

j ) only k is minimum element.

k ) a and b are maximal elements.

l ) No maximum element

m ) Since a f and f f hence f is glb(a , f)

n ) Since g b and f b hence b is lulb(g , f)

o) Since h f , f b and b a  hence h a that is h is related to a

p) g does not related to e.

4)  From the given Hasse diagram,

a ) only d is minimal element.

b ) only d is minimum element.

c ) only a is maximal element.

d ) only a is maximum element.

e)  Since e c , b a and e h , h g => e g hence e is glb(c , g)

f) Since e c and c b => e b and b b hence b is lub(e , b)

g) h is not related to c

h) Since d c , c b and b a => d a hence d is related to a