24.3. Greatest and Least Elements in a Poset

So, imagine you are given an arbitrary poset less than equal to relationship. This is not again a numerical less than equal to, this is an arbitrary relation, R, which is reflexive, anti symmetric and transitive. Then if I take a pair of elements x, y then the element y is called the cover of element x if the following two conditions hold. The element x should be related to the element y and of course x ≠ y. And there should not exist any intermediate element.

In a partially ordered set (poset), we can define a 'cover' relationship between two elements. For an element y to be called the cover of another element x, two conditions must be satisfied: first, x must be related to y (meaning x appears before y in the ordering), and second, there should be no element z that stands between them in the order. This means that the only connection is directly from x to y, without any elements in between.

Think of a stack of blocks. If you consider block x at the bottom and block y directly on top of it, then y is the cover of x. If there’s no other block in between them, then we can say y covers x. If you placed an additional block z between them, then y can no longer be the cover of x.

Let us next define what we call as the maximal and minimal element in a poset. So, if you are given an arbitrary poset and an element a from the set S. Then the element a is called as the maximal element if it is on the topmost layer. More formally, a is a maximal element if there is no element b such that a is related to b. Similarly, an element is called as a minimal element if it is at the lower level of your Hasse diagram or if it covers no element.

In a poset, a maximal element is defined as an element that has no element above it in the order. This means that there is no relationship that shows an element greater than a. Conversely, a minimal element has no elements below it, indicating it is the lowest in the hierarchy. For instance, in a Hasse diagram representing a poset, a maximal element is at the topmost layer, while a minimal element is at the bottom.

Imagine the hierarchy in a company. The CEO is the maximal element because there are no higher-ranking positions above them. On the other hand, entry-level employees are the minimal elements since there are no lower positions under them. Both roles represent the highest and lowest positions respectively in the organizational structure.

Now finally let us define what we call as the greatest element and the least element of a poset. An element a is called the greatest element if every element b is related to a. Similarly, an element a is called the least element if it is related to every other element b.

In the context of a poset, the greatest element is one that is 'greater than' or is related to every other element in the set. This means you can reach every other element by traversing the relation. Conversely, the least element is one that every other element is related to, making it the starting point in terms of 'less than' relationship. Not all posets will have a greatest or least element.

Think of the strongest player in a sports league as the greatest element; this player can defeat every other player. In contrast, the weakest player can be seen as the least element, as they lose to everyone else. However, in some leagues, there isn’t a definitive strongest or weakest player that dominates all others, similar to how a poset might have multiple elements without a clear greatest or least.

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