24.4.1. Definition and Algorithm Overview

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: (1) the element x should be related to the element y and (2) there should not exist any intermediate element z such that x ≤ z ≤ y.

In a partial order set (poset), certain elements can be considered as 'covers' of other elements. For y to be a cover for x, two conditions must be satisfied: first, x must relate to y (meaning x precedes y in the poset), and second, there cannot be any other element z that is in between x and y, as defined by the ordering. This ensures that y is the immediate successor of x, adhering to the definition of coverage in a poset.

Think of a hierarchy in a company. If employee x is the boss of employee y and there are no other employees between them in terms of authority, then we can say that y covers x. For example, if John is directly in charge of Sarah and no one else is involved, then Sarah is said to be covered by John.

So, it turns out that in a partially ordered set, every element need not have a cover. Similarly, an element may have more than one cover, and an element may cover multiple elements.

A significant takeaway is that not every element in a poset has to have a cover. For instance, in some diagrams, there may be elements at the top of the hierarchy (like '8' and '12' in the text) that do not have anything above them, hence they cannot have a cover. Additionally, elements can be connected in ways where one element can serve as a cover for several others, emphasizing the versatility and complexity of relationships within posets.

Consider a family tree: some people (like grandparents) may not have parents above them; hence, they don’t have covers. On the other hand, an aunt can have many nieces and nephews, illustrating that one can cover multiple others.

Let us next define what we call as the maximal and minimal element in a poset. An element a is called as the maximal element if it is on the topmost layer or has no cover. An element is called as a minimal element if it occurs at the lower level of your Hasse diagram or has no one covering it.

In a poset, a maximal element is one that does not have any elements above it, meaning there is no cover for this element. Conversely, a minimal element is at the bottom of the ordering; there are no elements beneath it to cover it. These definitions help further categorize elements based on their position within the structure and help to understand the boundaries of the ordering in more detail.

Think of a football league: the team at the top of the standings (a maximal element) has no team above it; while the team at the bottom (a minimal element) is the last in the ranking, having no team or rank below it.

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

The distinction between greatest and least elements is crucial: the greatest element is characterized by having relationships with all other elements in the set; similarly, a least element is related universally to all others but in the opposite direction. If one exists, the greatest or least element will be unique within the poset, though they may not exist at all in some configurations.

Imagine a school setting where the 'greatest student' could be the valedictorian, acknowledged by all for their achievements; meanwhile, the 'least student' could refer to someone who has credits for all their academic classes. Both terms fit because each represents a singular position based on comparisons to fellow students.

--