24.2. Maximal and Minimal Elements 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, and there should not exist any intermediate element ∃z, x ≤ z ≤ y. Pictorially, in the Hasse Diagram, y is a cover of x if y is immediately occurring on top of x and there is no intermediate element.

In a partially ordered set (poset), we use a specific kind of relationship called 'cover' to determine how elements relate to one another. We consider the reflexive, anti-symmetric, and transitive relationship R for any two elements x and y. If y is considered a cover of x, it means there is a direct connection between them (x is related to y), and no other element is situated between them in terms of this relationship. This means there are no intermediate elements z that could fit between x and y.

Imagine a hierarchy in a company where each level of management is a person. If the manager (y) is immediately above an employee (x) without any other layer of management in between, we say the manager covers the employee.

So, it turns out that in a partially ordered set, every element need not have a cover. If you take the Hasse diagram, some elements like 8 and 12 do not have any covers. Similarly, an element may have more than one cover, like 2 and 3 both covering 1. Additionally, an element can cover multiple elements; for instance, 6 covers both 2 and 3.

In a poset, not every element needs to have a cover, which means there can be entities at the top without any other entities above them. This can be visualized with a Hasse diagram where no elements exist at certain levels. Furthermore, some elements can cover multiple items, indicating a more complex relational structure where an element stands directly over more than one item in the order.

Think about animals in an ecosystem: a lion may not have any animals above it in the food chain (no cover) but it may prey on multiple smaller animals like zebras and antelopes (it covers them).

Let us define what we call as the maximal element in a poset. If you are given an arbitrary poset and an element a from the set, then a is called the maximal element if it is in the topmost layer or if it has no cover. More formally, P is called maximal element, ∃Q ∈ S, Q < P; i.e., there is no element b on top of a.

A maximal element in a poset is one that does not have any cover above it; it essentially resides at the 'top' of the ordering. Any element that can be covered by another will disqualify itself from being maximal because it indicates that there exists something above it in the order. For example, elements like 8 and 12 in a diagram with no upper connections are considered maximal.

Think of a mountain; the peak of the mountain represents a maximal element. There is no higher point above the peak, just as there are no coverings above a maximal element.

Similarly, we have the minimal element. An element is called a minimal element if it occurs at the lower level of your Hasse diagram or if it covers no elements, meaning there is no element b which occurs below a. For instance, the element 1 is the minimal element because there is no element related to 1 that is lower than it.

A minimal element functions in the opposite manner from a maximal element: it is situated at the lowest level in the poset without elements below it. Thus, there cannot be a relationship that would designate other elements as lower, making minimal elements critical in understanding the structure of the ordering.

Consider a basement in an apartment building; it is the lowest level of the structure, and there are no rooms or apartments below it, representing a minimal element.

If you have a non-empty poset, then it has at least one maximal element and one minimal element. For example, a singleton element like a would indicate that this element is both maximal and minimal. In sets with multiple elements, there will always be at least one element at the lowest level and one at the highest level.

It’s an important principle that in any non-empty poset, there will be at least one element recognized as maximal and one recognized as minimal. If we take a set consisting of just one item, that item will automatically fit both categories. In a more complex structure, we can always identify elements at both the topmost and bottommost levels.

Consider a classroom setting: there’s at least one top-performing student (maximal) and one student who struggles the most (minimal), reflecting the same hierarchical order.

Now let us define what we call as the greatest and least element of a poset. An element a is called the greatest element if every element b is related to a. In the same way, a is called the least element if it is related to every other element.

The concepts of greatest and least elements are clear extensions of maximal and minimal elements. A greatest element in a poset is one that all other elements directly relate to. Conversely, a least element is one that every other element relates to, meaning it sits at the lowest relational level among all elements. It’s crucial to differentiate that while maximal and minimal can have multiple instances, greatest and least items are unique if they exist.

Think of a hierarchy within a company: the CEO can be seen as the greatest since everyone reports to them, while the intern may be considered the least because they typically do not supervise others.

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