Special Elements of Partially Ordered Sets

Solved Problems

Solution.

We first draw the Hasse diagram for the poset \(\left( {A,\mid} \right):\)

Determine the extremal elements:

1. The maximal elements are \(24,30.\)
2. The minimal elements are \(2,3,5.\)
3. The greatest element does not exist.
4. The least element does not exist.
5. There is one upper bound for the subset \(\left\{ {8,12} \right\}:\) \(24.\)
6. The lower bounds of the subset \(\left\{ {8,12} \right\}\) are \(2,4.\)
7. The least upper bound of \(\left\{ {4,8} \right\}\) is \(8\) (the smallest element of the upper bounds \(8\) and \(24\)).
8. The greatest lower bound of \(\left\{ {4,8} \right\}\) is \(4\) (the largest element of the lower bounds \(2\) and \(4\)).

Solution.

The Hasse diagram of this poset is shown in Figure \(7.\)

Find the special elements in \(\left( {B, \mid } \right)\):

1. The maximal element is \(240.\)
2. The minimal element is \(2.\)
3. The greatest element exists and is equal to \(240.\)
4. The least element exists and is equal to \(2.\)
5. The upper bounds of the subset \(\left\{ {30,60} \right\}\) are \(120\) and \(240.\)
6. The lower bounds of \(\left\{ {30,60} \right\}\) are \(2,10.\)
7. The least upper bound of the subset \(\left\{ {8,30,60} \right\}\) is \(120\) (the smallest element among the upper bounds \(120\) and \(240\)).
8. The greatest lower bound of \(\left\{ {8,30,60} \right\}\) is \(2\) (this subset has only one lower bound).

Solution.

1. The upper bounds of \(\left\{ {d,f} \right\}\) are \(f,g,h,i,j,k.\) The lower bounds are \(a,b,d.\)
2. The subset \(\left\{ {e,f,g} \right\}\) does not have an upper bound. Its lower bound is \(b.\)
3. The upper bounds of \(\left\{ {h,i} \right\}\) are the elements \(i,k.\) The lower bounds are \(a,b,d,f.\)

Solution.

1. The upper bounds of the subset \(\left\{ {e,f} \right\}\) are \(g,i,j,k.\) The lower bound is \(c.\)
2. The upper bounds of \(\left\{ {f,h} \right\}\) are the elements \(h,\ell.\) The lower bounds are \(b,c,d,f.\)
3. The subset \(\left\{ {e,h} \right\}\) does not have an upper bound. The lower bound is the element \(c.\)

Solution.

We draw a Hasse diagram of the poset:

1. The greatest lower bound (glb) of \(\left\{ {\left\{ {1,3} \right\},\left\{ {2,4} \right\}} \right\}\) is \(\varnothing.\) The least upper bound (lub) of the subset is \(\left\{ {1,2,3,4} \right\}.\)
2. The glb of \(\left\{ {\left\{ {1,2,3} \right\},\left\{ {2,3,4} \right\}} \right\}\) is the element \(\left\{ {2,3} \right\}\) (the largest element among the lower bounds \(\varnothing,\) \(\left\{ {2} \right\},\) \(\left\{ {3} \right\},\) and \(\left\{ {2,3} \right\}.\) The lub of \(\left\{ {\left\{ {1,2,3} \right\},\left\{ {2,3,4} \right\}} \right\}\) is \(\left\{ {1,2,3,4} \right\}.\)

Solution.

To determine the extremal elements, it is convenient to use a Hasse diagram of the poset:

1. The subset \(\left\{ {\left\{ {1,2} \right\},\left\{ {3,4} \right\}} \right\}\) does not have a greatest lower bound (glb). The least upper bound (lub) of the subset is the top element \(\left\{ {1,2,3,4} \right\}.\)
2. The glb of \(\left\{ {\left\{ {3} \right\},\left\{ {2,3} \right\}} \right\}\) is equal to \(3.\) The upper bounds of the subset are \(\left\{ {1,2,3} \right\},\) \(\left\{ {2,3,4} \right\},\) and \(\left\{ {1,2,3,4} \right\}.\) However, the subset \(\left\{ {\left\{ {3} \right\},\left\{ {2,3} \right\}} \right\}\) does not have a lub since the elements \(\left\{ {1,2,3} \right\}\) and \(\left\{ {2,3,4} \right\}\) are incomparable.