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

In a partially ordered set (poset), we define a cover relation that describes how one element can be immediately above another without any elements in between. An arbitrary relation R is given, which fulfills three properties: reflexivity (every element is related to itself), anti-symmetry (if x is related to y, and y is related to x, then x must be equal to y), and transitivity (if x is related to y and y is related to z, then x must be related to z). The element y is a cover of x if y is directly above x and there are no other elements between them in the poset.

Imagine a stack of books where each book is placed directly on top of another without anything in between. If book A is directly below book B and there are no other books in between them, then we can say that B 'covers' A. If there is another book C between A and B, then B cannot cover A since C is in the way.

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.

The Hasse diagram is a visual representation of a poset where elements are represented as points, and an upward line or arrow indicates that one element covers another. For example, if you look at the Hasse diagram and see that element 2 is directly above element 1 with no other elements in between, it illustrates the cover relation, meaning 2 covers 1. This visual representation helps easily identify relationships between elements in the poset.

Think of a pyramid with distinct layers. The blocks on the layers represent elements in the poset. If a block (element) on the second layer has no block directly above it in the third layer, it covers that block below. If there is an additional block between them, then the upper block does not cover the lower one; instead, it indirectly connects through the intermediate layer.

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 the element 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, we have more than one cover. So, as I said earlier both 2 and 3 covers 1.

Not every element in a poset has to have a cover. For example, in a poset illustrated by a Hasse diagram, if there are elements like 8 and 12 at the top with no elements above them, they do not have covers since no elements are related to them in the relations defined. Additionally, an element can have more than one cover; for instance, both elements 2 and 3 may cover element 1 in the same poset.

Consider a leadership hierarchy in an organization. In some cases, you might have a department head without any supervisor above them (like elements 8 and 12). In contrast, a manager could have multiple team leaders reporting directly to them, similar to how one element can have multiple covers.

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. 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, if there is no element b on top of a that means there is no element b such that a is related to b where a is different from b.

Maximal elements in a poset are those elements that do not have any elements above them in the structure; they are at the highest position or lack a cover. Conversely, minimal elements are at the lowest end, meaning there are no elements below them. For example, in a Hasse Diagram, if element 8 is not covered by any element, then it is a maximal element. In contrast, if element 1 is at the bottom with nothing related below it, it is a minimal element.

Imagine a ladder. The top rung represents a maximal element since there is nothing above it, while the bottom rung is the minimal element — no rung exists below it. An employee without a supervisor is at a maximal position; an intern without any tasks assigned is at a minimal position.

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 and an arbitrary relation less than equal to and if you have an element a then the element a of the set is called as the greatest element if every element b is related to the element a as per the relation.

In a poset, the greatest element is one that is related to every other element in the set. This means that for every other element, you can say it is less than or equal to the greatest element. A least element, on the other hand, is the one that relates to all other elements meaning it is less than or equal to every other element. In some cases, posets might not have a greatest or least element, depending on the relationships defined within that poset.

Think of a class where the teacher is the greatest element. Every student is 'below' the teacher in terms of authority and knowledge and thus related to them. Conversely, consider a situation where a beginner student starting out with no prior knowledge is the least element; they are at the very beginning of the learning curve with no one below them.

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