nLab
meet

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Limits and colimits

limits and colimits

1-Categorical

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(0,1)-Category theory

(0,1)-category theory: logic, order theory

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Definition

If x and y are elements of a poset, then their meet, or infimum, is an element x∧y of the poset such that:

  • x∧y≤x and x∧y≤y;
  • if a≤x and a≤y, then a≤x∧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 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.

Special cases

A meet of zero elements is a top element.

Any element a is a meet of that one element.

Properties

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

Revised on March 15, 2012 15:41:57 by Urs Schreiber (82.172.178.200)
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