Consider the poset with the following Hasse diagram.
(a) Find all maximal elements of the poset.
(b) Find all minimal elements of the poset.
(c) Find the least element of the poset if it exists, or show that it does not exist.
(d) Find the greatest element of the poset if it exists, or show that it does not exist.
(e) What is the greatest lower bound of the set {a, b, c}?
(f) What is the least upper bound of the set {a, b, c}?

Question

Consider the poset with the following Hasse diagram.  
(a) Find all maximal elements of the poset. 
(b) Find all minimal elements of the poset. 
(c) Find the least element of the poset if it exists, or show that it does not exist. 
(d) Find the greatest element of the poset if it exists, or show that it does not exist. 
(e) What is the greatest lower bound of the set {a, b, c}? 
(f) What is the least upper bound of the set {a, b, c}?

Quizplus · Accepted Answer

The Answer is (a) The maximal element is

l

.
(b) The minimal elements are

a , b , c

, and

d

.
(c) There is no least element since there is more than one minimal element.
(d) The greatest element is

l

.
(e) There is no lower bound for the set

\{ a , b , c \}

, so there is no greatest lower bound.
(f) The upper bounds for the set

\{ a , b , c \}

are the elements

j

and

l

. Since

j

is less than

l , j

is the least upper bound.

Consider the Poset with the Following Hasse Diagram