nLab meet

Idea

In a preordered set or partially ordered set $P$, the meet (or infimum, abbreviated inf, or greatest lower bound, abbreviated glb) of a subset $S$ of $P$ is, if it exists, the largest element of $P$ which is smaller or equal to all the elements in $S$. If this element is itself member of $S$, then it is also called the minimum of that subset.

If we think of the pre-ordered set as a category (a (0,1)-category) then the meet is the limit over the given subset, if it exists, regarded as a diagram. Thus in a partially ordered set this is unique if it exists, otherwise it is unique up to isomorphism.

Definition

If $x$ and $y$ are elements of a poset, then their meet is an element $x \wedge y$ of the poset such that:

• $x \wedge y \leq x$ and $x \wedge y \leq y$;
• if $a \leq x$ and $a \leq y$, then $a \leq x \wedge y$.

Such a meet may not exist; if it does, then it is unique.

In a proset, a meet may be defined similarly, but it need not be unique. (However, it is still unique up to the natural equivalence in the proset.)

The above definition is for the meet of two elements of a poset, but it can easily be generalised to any number of elements. It may be more common to use ‘meet’ for a meet of finitely many elements and ‘infimum’ for a meet of (possibly) infinitely many elements, but they are the same concept. The meet may also be called the minimum if it equals one of the original elements.

A poset that has all finite meets is a meet-semilattice. A poset that has all infima is an inflattice.

A meet of subsets or subobjects is called an intersection.

Examples

General

• A meet of no elements is a top element.

• Any element $a$ is a meet of that one element.

Infimum of real numbers

Often one considers infima of subsets of the real numbers $\mathbb{R}$, regarded with their canonical preordering, which in this case is in fact a total order.

For $S \subset \mathbb{R}$ a subset, say that a lower bound is an element $b \in \mathbb{R}$ such that $\underset{s \in S \subset \mathbb{R}}{\forall}( b \leq s )$.

Then the infimum of $S$ is, if it exists, that lower bound $inf(S)$ of $S$ such that for $b$ any other lower bound of $S$ then $b \leq inf(S)$.

See join#constructive for the case in constructive analysis.

Properties

As a poset is a special kind of category, a meet is simply a product in that category.

Related concepts

• join

• meet