24.5.1. Partial Ordering

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, that is why the less than symbol. And there should not exist any intermediate element ∃z, x ≤ z ≤ y.

So, pictorially, you can imagine that y is a cover of x if I view the Hasse Diagram then in the when I go from bottom to up y is immediately occurring or y is occurring on top of x layer wise and there is no intermediate element or no element z in the intermediate layer. So, for instance here in this Hasse diagram the element 2 covers the element 1 because in between 2 and 1 there is no intermediate element. You have the element 1 which is related to the element 2 and between 1 and 2 there is no intermediate elements. But the element 6 does not cover the element 1, even though the element 1 is less than 6, because element 1 is indeed related to 6 as per this Hasse diagram.

In a partially ordered set (poset), covering refers to a specific relationship between two elements where one element is directly above another in the ordering. To say that y covers x means two things: first, x must be related to y (x ≤ y), and second, there must be no element z that exists between them (i.e., there cannot be an element z such that x ≤ z ≤ y). This condition is significant because it describes a direct 'step' in the ordering without skipping any elements. For example, if in the Hasse diagram of a poset, we see that element 1 is directly below element 2 with no elements in between, we describe 2 as covering 1.

Consider a hierarchy in a company where the CEO is at the top, followed by managers, then employees. If we think of an employee as element x and a manager as element y, a manager covers their direct employee if there are no other managerial ranks between them. In this analogy, the manager directly oversees the employee, affecting them without any layers of hierarchy in between.

So, it turns out that in a partially partial order set every element need not have a cover. So, for instance, if you take the Hasse diagram on your left-hand side the elements 8 and 12, it does not have any common. There is no element on top of 8, there is no element on top of 12. Similarly, an element may have more than one cover. So, as I said earlier both 2 and 3 cover 1. An element may cover multiple elements. So, for instance here, in this Hasse diagram or in this poset 6 covers 2 as well as 3.

In a poset, not every element is guaranteed to have a cover. For instance, elements like 8 and 12 may not cover any elements because they are at the top of their layers without any elements above them. Conversely, it’s also possible for a single element to cover more than one element, as seen in our example where element 6 covers both elements 2 and 3. This illustrates the flexibility and complexity of relationships in partially ordered sets.

Imagine a group of students in a class. Some students (the top performers) may be at the top of the class, meaning no one performs better than them, while others might have multiple friends who they help (cover) in their studies. Just like a top student has no one above them, certain elements in a poset can represent leaders with no one covering them.

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 the maximal element if it is on the top most layer informally, or in a loose sense or if it has no cover. More formally, a is called maximal element, ∃b ∈ S, b < a; i.e., is no element b on top of a.

Similarly, I can define what we call as a minimal element. So, an element is called a minimal element if it occurs at the lower level of your Hasse diagram or in other words, it covers no element.

Maximal elements in a poset are those that have no elements above them; they represent the highest points in the ordering. In contrast, minimal elements are those that have no elements below them, marking the lowest points in the ordering. For example, if an element is such that there is no other element greater than it in the poset, it is a maximal element. On the opposite end, if an element is the lowest in the ordering, it is termed minimal.

Consider a group of friends who have a hierarchy based on popularity. The most popular friend (the one no one ranks higher than) is like a maximal element, while the least popular one (who no one considers less popular than) resembles a minimal element. Understanding this hierarchy helps make sense of social dynamics in any group.

Now finally let us define what we call as the greatest element and the least element of a poset. So, if you are given a poset S and with an arbitrary relation less than equal to and if you have an element a then the element a of the set S is called as the greatest element if every element b is related to the element a as per the relation R or the relation less than equal to. In the same way the element a is called as the least element if it is related to every other element b as per your relationship less than equal to.

A greatest element in a poset indicates an element that is greater than or equal to every other element. Conversely, the least element is less than or equal to every other element. It's crucial to note that not all posets will contain a unique greatest or least element—some may exist, some may not.

Imagine a group of students scoring on an exam. The student with the highest score represents the greatest element, while the student with the lowest score represents the least element. Depending on how scores are distributed, there might not be a single highest scorer (greatest) if they all scored the same, nor necessarily a lowest if everyone performed well.

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