Join and meet

In the field of order theory, mathematicians study how elements in a set relate to one another. These relationships are often organized into a partially ordered set. Within these sets, we often look for specific points that act as boundaries for subsets. These boundary points are known as the join and the meet. The join is also called the supremum, or the least upper bound. The meet is also called the infimum, or the greatest lower bound. These two concepts are duals of each other. This means they function as opposites through order inversion.

To find a join for a subset, we must first identify the upper bounds. An upper bound is an element that is greater than or equal to every member of that subset. However, a set might have many different upper bounds. The join is the specific element that is the smallest among all those upper bounds. We denote the join of a subset using the symbol $\bigvee$. Similarly, finding a meet requires looking for lower bounds. A lower bound is an element that is less than or equal to every member of the subset. The meet is the greatest of these lower bounds. We denote the meet using the symbol $\bigwedge$.

Not every subset in a partially ordered set will have a join or a meet. A subset might have no lower or upper bounds at all. In other cases, there might be several bounds, but none is the "greatest" or "least." If a meet or join does exist, it is always unique. This uniqueness is important for mathematical consistency. If a set is a meet-semilattice, every pair of elements has a meet. If every pair has a join, it is a join-semilattice. A set that possesses both is called a lattice.

There are even more structured versions of these sets. A complete lattice is a special type where every possible subset has both a meet and a join. This includes infinite subsets, not just pairs. We can also define a partial lattice. In a partial lattice, not every pair of elements is guaranteed to have a meet or a join. However, the operations that do exist must satisfy specific mathematical axioms. If a subset is also an upward directed set, its join is called a directed join. If it is a downward directed set, its join is called a directed meet.

Mathematicians can also look at these ideas through universal algebra. In this approach, the meet and join are treated as binary operations. For these to be valid operations on a set, they must follow three specific rules. First, they must satisfy commutativity, meaning the order of elements does not change the result. Second, they must satisfy associativity, which allows us to group operations differently. Third, they must satisfy idempotency, where the operation on an element with itself results in that same element. These rules allow us to define a partial order directly from the operations.

We can see these concepts in action using power sets. A power set is a collection of all possible subsets of a given set. When a power set is partially ordered by inclusion, the join and meet become very familiar. The join of two sets is their union, which combines all elements. The meet of two sets is their intersection, which finds only the shared elements. The symbols for union ($\cup$) and intersection ($\cap$) serve as helpful mnemonics. The union symbol looks like the supremum symbol, and the intersection symbol looks like the infimum symbol.

These ideas help us understand how complex systems are organized. In a totally ordered set, like a standard number line, the join and meet are simple. The join is just the maximal element, and the meet is the minimal element. If those specific elements do not exist in a subset, then the join or meet does not exist either. By studying joins and meets, we can describe the fundamental structure of how different mathematical objects relate, overlap, or stack upon one another.