In a preordered set or partially ordered set then the meet or infimum of a subset of elements is, if it exists, the largest element in the set which is smaller or equal to all the elements in the subset. If this element is member of the original subset, 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.
If $x$ and $y$ are elements of a poset, then their meet, or infimum, is an element $x \wedge y$ of the poset such that:
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.
A meet of no elements is a top element.
Any element $a$ is a meet of that one element.
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 $b \leq inf(S)$.
As a poset is a special kind of category, a meet is simply a product in that category.
meet
Last revised on June 2, 2019 at 07:35:47. See the history of this page for a list of all contributions to it.