1.1.7. Question 5: Minimum Element in Poset

You are given an arbitrary poset. And for any subset T of that set S, an element x from that set T will be called as a minimum element, if that element x is related to all other elements y of depth subset T.

In a partially ordered set (poset), a minimum element is defined concerning a subset of that set. This means if we have a subset T, the minimum element x must relate to every other element y (where y is another element from T). In simpler terms, for x to be considered a minimum, it should be comparable to all other elements in the subset T. We are not talking about a 'global' minimum that applies to the entire poset but a minimum defined only within the context of a specific subset.

Consider a situation where several people are standing in a group (the poset) and there are different subsets of this group (like friends or team members). If one person in the subset is the organizer (the minimum element), they have to communicate with all other friends or team members present in that subset, showing they are responsible for coordinating everything.

The condition is that your poset is such that every non-empty subset T of S has a minimum element.

This condition states that regardless of which subset T you pick from the set S, there will always be at least one element that qualifies as a minimum. In essence, this means that for each and every possible subset you can form from set S, you will find an element that is comparable (less than or equal to) all others in that subset. This property is quite strong and influences the overall structure of the poset.

Imagine a classroom with students (the set S) where every possible group of students (the subsets) always has a team leader (the minimum element). No matter how you group the students, you’ll always have someone designated to make decisions or give directions, making sure there’s always order in how things are done.

Under this condition, you have to show that your poset is actually a total ordering.

A total ordering is a stronger condition than a partial ordering. In total ordering, every pair of distinct elements must be comparable: for any two distinct elements a and b in the set, at least one of the following must hold: a ≤ b or b ≤ a. When every non-empty subset of the poset has a minimum element, it ensures that all elements can be compared with each other, thus indicating that the poset displays a total order rather than just a partial order.

Think of a ranking system in sports. If you have several players, and each specific subset of players can always determine a top player (the minimum), then this implies that all players can be ranked against each other—there is no ambiguity about who ranks higher or lower. Hence, every player can be compared, establishing a clear rank order among all.

Let us take an arbitrary pair of elements a, b which are distinct and we will show they are comparable.

To show that the poset is totally ordered, we select any two distinct elements a and b. By forming a subset T that contains just a and b, and applying the earlier condition (which guarantees every subset has a minimum element), we can conclude that either a or b must be the minimum of T. If a is the minimum, then we conclude that a ≤ b. Conversely, if b is the minimum, then b ≤ a. Hence, any two distinct elements will be comparable, fulfilling the criteria for total ordering.

This is like setting up a tournament where every participant must face off against one another. If each match determines a winner (minimum element), then eventually every participant can be directly compared to one another in terms of who won or lost. Hence, in the end, each participant can be ranked based on their performance against every other participant.

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