How to find lower and upper bound of an element from the given hasse diagram?

Question

Consider the poset P given by the following Hasse diagram. We have to find lower and upper bound using the hasse diagram.

i) Is there is at least an element of P? If so provide this.

ii) What are the upper bounds of {b, c)? Is there a least an upper bound for this set? If so provide this.

iii) What are the lower bounds of {c,e,f}? Is there a greatest lower bound for this set? If so provide this.

Summary

In this question, a hasse diagram is given, and based on this hasse diagram there are three questions given. We have been asked that if there is any least element in this hasse diagram. We have been also asked about the upper and lower bound of an element in the hasse diagram.

Explanation

Posets are also called partially ordered sets. They are reflexive, anti-symmetric, and transitive in nature and they are represented using the hasse diagram.

We can get the upper and lower bound of an element in the hasse diagram like this:

(i) In any poset, an element is called the least element if it is less than all the other elements in the poset. In the given hasse diagram, a is the smallest of all the elements. Also, there is no other element that is below the element a, so a will be the least element of this hasse diagram.

(ii) Upper bound of the set {b,c} will be the intersection of the upper bound of b and upper bound of c. The elements which are above the particular element will be its upper bound elements and the elements should be connected to that particular element also. From this hasse diagram, we can see that the upper bound of the element b is {d,e}, and the upper bound of the element c is {d,e,f}, and the intersection of these two sets will be {d,e}, so {d,e} is the upper bound of set {b,c}. We can refer to the below diagram for more understanding.

Also, we can see here that, that the upper bound of d is d and e is e and there is no common element between them, so there will not be any least upper bound.

(iii) Lower bound of the set {c, e, f} will be the intersection of the lower bound of c, e, and f. The elements which are below the particular element will be its lower bound elements and the elements should be connected to that particular element also. From this hasse diagram, we can see that the lower bound of the element c is {c, a}, and the lower bound of the element e is {e, b, c, a}, and the lower bound of the element f is {f, c, a}, and the intersection of these three sets will be {c, a}, so {c, a} is the lower bound of set {c,e,f}. We can refer to the below diagram for more understanding.

Also, we can see here that, the lower bound of a is a and c is c and a. So the greatest lower bound will be c only.

Also read, What is the best way to iterate over a dictionary