24.2.1. Maximal Elements

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 relationship called 'covering.' An element 'y' covers 'x' if 'x' is directly related to 'y' without any elements in between. This means when you visualize it with a Hasse diagram (a type of graph used to represent posets), 'y' should be immediately above 'x' with no other elements between them. Essentially, 'covering' denotes a direct connection in this order.

Think of a staircase where each step represents an element in a poset. If you are on the first step (x) and the second step (y) is directly above you, then the second step covers the first. There are no intermediate steps (like z) between them. If, however, there is a third step between the first and second, then the second step does not cover the first anymore.

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 informally, or in a loose sense or if it has no cover. More formally, a is called maximal element, ∄b ∈ S, b > a.

A maximal element in a poset is defined as an element that has no others above it in the ordering. In simpler terms, if you can’t find any element 'b' in your poset that is greater than 'a,' then 'a' is a maximal element. This means that there is no element covering 'a,' further confirming its position at the top.

Imagine you're on the highest floor of a building. If there are no higher floors to go to, then you are at the topmost level, just like a maximal element in a poset. In this case, you cannot step up to any other floor, which highlights your maximal status.

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.

A minimal element is the opposite of a maximal element. It appears at the lowest level of the poset and has no elements beneath it. In formal terms, if there is no element 'b' that is less than 'a,' then 'a' is considered a minimal element, meaning that there are no covers underneath it.

Consider the foundations of a house. The foundation lays at the very bottom, directly on the ground with no blocks underneath it. Just like how a minimal element stands at the lowest position in a poset with no other element below it.

It turns out that in a partially ordered set, every element need not have a cover. For instance, if you take the Hasse diagram, the elements 8 and 12 do not have any common. Also, an element can have more than one cover, and an element may cover multiple elements.

In a poset, not every element is guaranteed to have a maximal element above it or a minimal element below it. Some elements may stand alone without others being related to them in the upper or lower layers. Furthermore, an element can cover more than one other element, indicating that positional relationships can vary and are flexible.

Think of a tree structure. Not every branch has a further branch extending from it, just like not every element has to cover another. Some branches may only exist by themselves, while others branch out to multiple directions, illustrating how a single element can relate to various others in a poset.

It is easy to prove that if you have a poset over a non-empty set, then it has at least one maximal element and one minimal element. For instance, if the poset is defined over a singleton element, then that element is both a maximal and minimal element.

Every non-empty poset will contain at least one maximal and one minimal element due to its structure. In a singleton set (like a single element), that one element serves both roles because there are no other elements to compare it with. This principle remains true regardless of how complex the poset becomes.

Imagine a single piece of fruit, like an apple. It has no other fruits around it; thus, it stands alone at the top and bottom of its own hierarchy—maximal and minimal. You can relate this idea to any basic collection of items: at least one will always be highest and one will always be lowest.

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

A greatest element in a poset is one that all other elements relate to; thus, it stands at the ultimate top of the hierarchy—no other element surpasses it. Conversely, the least element is one that every other element relates to directly, marking its station at the base of the hierarchy.

Think of a family hierarchy. The grandparents might be the greatest elements of the family tree, as everyone relates back to them. In contrast, the youngest child would be the least element since they relate to all the older family members but none below them.

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