23.3. Hasse Diagrams

It turns out that we can represent partial ordering or posets by a very specific type of diagrams which are called as Hasse diagrams. And this is possible provided, this makes this is more interesting for posets where the relation is to find over a finite set.

Hasse diagrams are special graphical representations used to illustrate the concept of partial ordering (posets). They are particularly useful when dealing with finite sets. A Hasse diagram provides a visual representation of the relationships between elements in a poset without needing complex notation.

Imagine you have a collection of books arranged in a hierarchy based on their genres. A Hasse diagram can help visualize which genres are related to one another without cluttering the diagram with unnecessary details about all the books themselves.

So, let me demonstrate this Hasse diagram with this less than equal to relationship which is the numerical less than equal to relationship defined over the set S = {1,2,3,4}.

To create a Hasse diagram for the less than equal to relationship over the numbers 1, 2, 3, and 4, we start by identifying the pairs of numbers where one is less than or equal to the other. We can then represent this relationship using arrows pointing from the smaller number to the larger number. Initially, we include self-loops (each number relating to itself) and multiple edges that represent transitive relationships. However, in the final Hasse diagram, we can omit these due to their implicit nature.

Consider a race with four runners. In a Hasse diagram, you would represent who finished ahead of whom, but you wouldn't need to show that each runner finished ahead of themselves, since that's understood in the context of the race. Thus, the diagram would only need the essential relationships between the runners.

I can say that there is no point of explicitly writing down or stating the self loops. Because I can say that since my relationship is reflexive anyhow, I can always say that the self loops are implicitly present in my diagram.

When constructing the Hasse diagram, it is recognized that since the relationship is reflexive (i.e., every element relates to itself), explicitly showing self-loops is unnecessary. Uncluttering the diagram enhances clarity by allowing viewers to focus on essential relationships without redundancy. Additionally, we can also remove any edges that are transitive, meaning if A relates to B and B relates to C, there's no need to draw the edge from A to C because it can be inferred.

Imagine drawing a family tree; you don’t need to indicate that each person is related to themselves—everyone already knows that. Similarly, in a Hasse diagram, we streamline connections so that it focuses only on the key relationships that define the structure.

So, what I am doing is in each stage, I am trying to make my diagram more and more cleaner, tidy and try to remove all unnecessary information or redundant information, which I am not supposed to explicitly state in my graph of the relation of a partial ordering.

The final stage of creating a Hasse diagram involves ensuring that only the essential connections are present. By removing both self-loops and transitively implied edges, the resulting diagram is clear and easily interpretable while still accurately representing the prposed partial ordering. The implied direction of the arrows in the diagram always moves from the bottom up, indicating the nature of the ordering.

Think of an organizational chart. You focus only on the direct reporting lines without showing details about everyone’s personal past experiences or self-evaluations. By keeping it tidy, anyone looking at the chart can quickly understand the hierarchy without being overwhelmed by excess information.

So, let us see another example here. So, you are given the divide (|) relationship. So, your ≤ is the divider relationship. So, again, we can start with our directed graph with the nodes 1, 2, 3, 4, 6, 8, 12.

In this example, we are looking at the divide relationship (|), which indicates that one number divides another. Starting with the numbers 1, 2, 3, 4, 6, 8, and 12, we identify which numbers divide which others. We can create a directed graph that shows these relationships. By removing self-loops and transitive relationships, we create a simplified Hasse diagram representing these divisions.

Think of a team of chefs where each chef specializes in a different dish. A chef who can cook a full three-course meal (e.g., soup, main dish, dessert) can also be seen as being able to prepare a soup only. The Hasse diagram here would show the capacity of chefs to manage larger and larger meals, simplifying the intricate relationships.

Again, I have removed all the self loops. I have removed all the transitively implied edges and I have removed the direction of the edges.

In this final chunk, we discuss representing the subset relationship. The elements are subsets rather than single elements, and an important part of creating the Hasse diagram is recognizing which subsets are contained within other subsets. Like previous examples, we simplify the diagram by removing unnecessary elements while preserving the essential structure of relations.

Consider a set of boxes, where each box can contain smaller boxes (subsets). A Hasse diagram would illustrate not just each box but how they relate to one another in terms of containment—simplifying it by focusing on the parent-child relationships among the boxes without needing to show the trivial relationships of each box holding itself.

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