24.1.1. Definitions and Examples

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 such that x ≤ z ≤ y.

In a poset (partially ordered set), an element y is said to cover another element x if y directly follows x in the ordering with no elements in between. The relationship is defined such that there are no other elements between x and y, meaning you can't jump from x to y without encountering another element. This is crucial in understanding how elements relate to each other in a poset.

Think of a hierarchy in a corporate office. If an employee (x) reports directly to a manager (y) without any supervisors in between, then we can say that the manager covers the employee in the organizational structure.

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 graphical representation of a poset that illustrates how elements are ordered. When you look at a Hasse diagram, if there is a direct line connecting two elements with no other elements in between, the one above covers the one below. For example, if we see direct lines from 1 to 2 in a diagram, 2 is the cover of 1.

Imagine a stack of trays in a cafeteria. If tray 1 is directly below tray 2, then tray 2 covers tray 1. You can't have any tray in-between them that would disrupt that direct relationship.

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 do not have any common. Similarly, an element may cover multiple elements.

In some posets, not every element has a cover. For instance, if an element is the lowest in the hierarchy, it cannot cover anything because there are no elements below it. Conversely, some elements can cover multiple elements, meaning they can have more than one direct relationship where they are the immediate successor to different elements.

Returning to our corporate hierarchy analogy, a team leader may directly oversee several team members (covering multiple elements) but may themselves have no one directly overseeing them if they are at the top level of their department.

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 top most layer

In posets, a maximal element is one that has no element above it, meaning there is no relation to another element that is greater than it. Alternatively, a minimal element is the opposite: it has no element below it. This means no other element relates to it as being smaller or lesser.

Consider a pyramid of blocks. The biggest block on top is a maximal element because nothing is above it, while the smallest block on the ground is a minimal element, as nothing is beneath it.

Now finally let us define what we call as the greatest element and the least element of a poset. 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.

The greatest element in a poset is one that all other elements are related to in a way that they are less than or equal to it. In contrast, the least element is one to which every other element is related, making it lesser in terms of the relationship defined in the poset. This leads to the idea that if these elements exist, the greatest element is unique, and so is the least element.

Think of the class structure in a school. The principal is often considered the greatest element since all teachers report to them, and the students look up to them. Conversely, the students are at the bottom of the hierarchy in this context, making them the least, as they are all subordinate to someone.

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