24.3.1. Definitions

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.

This explanation introduces the concept of a 'cover' in a partially ordered set (poset). In simpler terms, if you have two elements, x and y, in a poset, y is a cover of x if y is directly above x, with no elements in between them. This means y immediately follows x in the order, which can be visualized in a Hasse diagram. The conditions to be a cover are that x must relate to y, and there can't be any element z that fits in between them.

Think of a stack of boxes where each box is a different height. If you have a box labeled '1' and a taller box labeled '2' directly on top of it without any other boxes in between, then box '2' covers box '1'. If there’s another box, say box '3', between them, then box '2' does not cover box '1'.

Let us next define what we call maximal and minimal elements in a poset. An element a is called a maximal element if it is on the topmost layer, or it has no cover. Formally, b is called maximal if ∃c in the set such that c < b, indicating no element on top of a. Similarly, an element is called a minimal element if it occurs at the lower level and covers no element.

In a poset, maximal elements are those that cannot be covered by any other element, much like the tallest item in a collection. On the contrary, minimal elements are at the bottom - they do not cover any other elements. This idea allows us to categorize elements based on their position in the hierarchy of relationships defined by the poset.

Imagine a series of shelves. The top shelf holds the tallest books (maximal elements), while the bottom shelf holds the shortest ones that no other books are taller than (minimal elements). Each shelf has its unique position in the hierarchy of the book collection.

Now, let us define the greatest and least element of a poset. If you have an element a called the greatest element, every element b is related to a as per the relation. Similarly, a is called a least element if it is related to every other element b.

A greatest element in a poset is one that all other elements relate to, meaning every element leads to it. Conversely, a least element is one that every other element relates back to it. Thus, if both exist, it creates a definitive point that encapsulates the entire poset.

Think of an organizational hierarchy in a company. The CEO is the greatest element because every employee reports to them, while a new intern might be considered the least if they report to everyone else. The CEO is the culminating point of the hierarchy, while the intern starts at the very foundation of it.

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