24.1.2. Properties of Covers

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 have a specific type of relationship that is defined by three properties: reflexive, anti-symmetric, and transitive. The relationship is said to be reflexive if every element is related to itself, anti-symmetric if no two different elements can be related to each other in both directions, and transitive if the relation can be passed from one element to another. In this context, we refer to 'covers.' An element y is said to cover an element x if y is directly related to x (meaning there’s no other element in between them), and x is not equal to y. This means y is the next immediate element after x in the ordering.

Think of a hierarchy in a company. If employee A is directly above employee B in the hierarchy, A covers B. There’s no one else between them who could be considered as a supervisor—A is the next person above B, just like how y is the next element above x in the poset.

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.

A Hasse diagram is a way to visually represent a poset where elements are drawn as points and their relations are indicated using lines. In this diagram, if you go from bottom to top, you can see which elements cover others without any intermediaries. The example where element 2 covers element 1 shows that there is a direct connection with 2 sitting immediately above 1, reaffirming the absence of any intermediate elements in between.

Imagine climbing a ladder. When you step from one rung to the next, there’s no rung in between you—that’s akin to y covering x: stepping directly from x to y without stopping at any intermediate point.

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, we have more than one cover. So, as I said earlier both 2 and 3 cover 1. And 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, it's possible for some elements not to have a cover. For example, if an element has no elements above it in the diagram, it lacks a cover. Conversely, multiple elements can cover the same element, such as both elements 2 and 3 covering element 1. Additionally, a single element can cover various others. For example, 6 may cover both 2 and 3 in the same structure.

Consider a city hierarchy where certain neighborhoods have no districts above them, meaning they don't have any higher administrative layers—this illustrates elements that don't have a cover. Conversely, two neighborhoods could fall under the same district, just like two elements covering the same element.

Let us next define what we call as the maximal and minimal element in a poset. 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 informally, or in a loose sense or if it has no cover.

Maximal and minimal elements play essential roles in understanding the structure of a poset. A maximal element is one without a cover—there’s no other element above it, which typically places it at the uppermost position in the Hasse diagram. Conversely, a minimal element has no elements below it, existing at the lower end of the hierarchy.

In a sports team, the team captain could be seen as a maximal element; there's no one above them in rank. Meanwhile, the newest team member is like a minimal element, as they have yet to collaborate with any other members—there's no one below them in terms of experience.

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.

The concepts of greatest and least elements further categorize elements in a poset. The greatest element is one to which every other element is related—meaning all other elements are less than or equal to it. Conversely, the least element is one that every other element is related to, indicating it is lower than all others in the ordering.

Consider a family hierarchy; the grandparent could be the greatest element, as they are the older generation—everyone else is younger. Meanwhile, a newborn could be the least element, being the youngest in the family hierarchy.

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