Join and meet

Imagine you have a group of things.

You can find a top point for them. This is called a join. It is the smallest thing that is still above all others.

You can also find a bottom point. This is called a meet. It is the largest thing that is still below all others.

Some groups have both a join and a meet. These special groups are called lattices.

Not every group has these points. Some groups may not have a top or a bottom.

Math helps us find these points in a set.

Imagine you have a group of items. These items follow an order. Some items are "above" others.

We can look for a special top point. This is called a join. The join is the smallest item that stays above the whole group. It is also called a supremum.

We can also look for a bottom point. This is called a meet. The meet is the largest item that stays below the whole group. This is also called an infimum.

Not every group has these points. Sometimes a group has no top or bottom at all. If every pair of items has a join, we call it a join-semilattice. If every pair has a meet, it is a meet-semilattice.

When a group has both, we call it a lattice. Some very special groups are called complete lattices. In these groups, every subset has a join and a meet.

A diagram can show how these points work in a group.

In math, we can organize items into an order. Some items sit above others in a special way. We call these groups partially ordered sets. Within these groups, we look for special meeting points. One point is called the join. This is also known as the supremum. It is the smallest item that is above a whole group.

Another point is called the meet. This is also known as the infimum. The meet is the largest item that stays below a group. These two ideas are opposites of each other. They work in reverse ways to find bounds.

Finding a join or a meet follows specific rules. To find a join, we look for an upper bound. An upper bound is an item that is greater than or equal to everything in our group. The join must be the smallest of these upper bounds. To find a meet, we look for a lower bound. A lower bound is an item that is less than or equal to everything in the group. The meet must be the largest of these lower bounds.

Not every group has a join or a meet. Sometimes there are no bounds at all. Or, there might be many bounds with no single best one.

Mathematicians use special names for groups that follow rules. If every pair of items has a join, it is a join-semilattice. If every pair has a meet, it is a meet-semilattice. When a group has both, we call it a lattice.

Some groups are even more organized. A complete lattice is a group where every subset has a join and a meet. We can also have a partial lattice. In those, not every pair has a join or a meet. However, the ones that do exist still follow certain math rules.

We can use symbols to help us remember these ideas. For joins, we use the symbol for union. This looks like the symbol for a supremum. For meets, we use the symbol for intersection. This looks like the symbol for an infimum.

These symbols act as helpful memory tools. If you have sets of things, joining them makes a union. Meeting them makes an intersection. This helps you remember which symbol goes with which idea. The symbols help us see how the math works.

These ideas connect to things you might already know. Think about a list of numbers in order. In a totally ordered set, the join is just the biggest number. The meet is just the smallest number in that set.

If you have a group of sets, joining them is like putting them all together. Meeting them is like finding only what they share. This is how math describes how things fit together. It helps us find the best way to group or split things.

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.