23.1.1. Partial Ordering

Hello everyone, welcome to this lecture on partial orderings. So, we will introduce the definition of partial ordering in this lecture. We will discuss Hasse diagram and we will compute with Topological sorting.

This is the introduction to the lecture on partial orderings. The aim is to provide definitions and examples that help understand the concept of partial ordering, as well as how to visualize these relationships using Hasse diagrams.

Think of a library where books are organized by genre. Each genre can have books that might be ordered in a way that some books fall under multiple genres. This complexity resembles partial ordering, where not all books are directly comparable.

So, what is the partial ordering? If you consider a dictionary then the words in a dictionary are arranged alphabetically or we also say that the words are arranged lexicographically. The relationship here is, I say that a word a is related to the word b provided the word a appears before the word b. This alphabetical arrangement satisfies three properties: Reflexive property, Antisymmetric property, and Transitive property.

A partial ordering can be defined using the example of an alphabetical dictionary. Here, words are related based on their position in the dictionary. There are three essential properties that this arrangement satisfies:

1. Reflexive: A word is considered to be before itself, even if this isn't typically true in literal terms.
2. Antisymmetric: Two different words cannot appear before each other simultaneously. If word 'a' precedes 'b', then 'b' cannot precede 'a.'
3. Transitive: If a word 'a' appears before 'b', and 'b' before 'c', then 'a' must appear before 'c'.

Consider managing a set of projects. If Project A must precede Project B and Project B must precede Project C, you can deduce that Project A must precede Project C, illustrating the transitive property.

Imagine a big software project with various modules defined by a dependency relationship R. Module i is related to module j if module j can start only after module i is done. This dependency relationship is reflexive, antisymmetric, and transitive.

In a software project, each module potentially depends on others. The relationship of dependency forms a partial ordering as it respects the three properties:

• Reflexive: Each module has an implicit dependency on itself.
• Antisymmetric: No two different modules can depend on each other, as that would lead to a deadlock scenario.
• Transitive: If Module A is necessary before Module B, and Module B is necessary before Module C, then Module A is inherently necessary before Module C.

Think of a recipe with steps that need to follow a certain order. For example, you must mix ingredients before baking, but you cannot bake before mixing. Each step depends on the previous one, similar to module dependencies.

A relation R over a set S is called a partial ordering if it is reflexive, antisymmetric, and transitive. In that case, the set S along with the relation R is called a poset. The full form of poset is partially ordered set.

Mathematically, a relation is defined over a set S. If this relation satisfies the criteria of being reflexive, antisymmetric, and transitive, then it qualifies as a partial ordering. When we refer to a set and its relation together, we call it a poset.

Imagine organizing an event where tasks must be completed in a specific order. The tasks form a set, and the relationships between them (which tasks depend on others) create a partial ordering or a poset.

A total ordering is a special type of poset where any two elements in the set are comparable. In contrast, a partial ordering allows for the existence of pairs of elements that are not comparable.

In a total ordering, for every pair of elements in a set, one element must relate to the other. This makes it easier to understand relationships within the set. For example, in a total order, every book would have a clear hierarchy based on some criteria. If two books can't be directly compared, then they fall into a partial order.

Consider a competition where each athlete's ranking can be compared against all others for a clear winner; this is similar to total ordering. In contrast, a group project might have various contributions where not all team members can be compared distinctly; this reflects partial ordering.

Hasse diagrams are a graphical representation of partial orders. They help visualize the relationships in a finite poset by encoding the ordering relationships in a clean manner.

Hasse diagrams simplify the representation of posets. They focus on the ordering aspect by omitting redundant elements like self-loops and transitively implied edges, thus making it easier to see the connections between elements at a glance and understand the structure of the partial order.

Imagine organizing a family tree where some relationships (like cousins) may not connect directly but still are related through parents and grandparents. A Hasse diagram would help visualize these relationships clearly without clutter.

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