The elements are called the endpoints of . Clearly . Also, the endpoints of a lattice interval are unique: if , then and .
It is easy to see that the name is derived from that of an interval on a number line. From this analogy, one can easily define lattice intervals without one or both endpoints. Whereas an interval on a number line is linearly ordered, a lattice interval in general is not. Nevertheless, a lattice interval of a lattice is a sublattice of .
A bounded lattice is itself a lattice interval: .
A prime interval is a lattice interval that contains its endpoints and nothing else. In other words, if is prime, then any implies that either or . Simply put, covers . If a lattice contains , then for any , is a prime interval iff is an atom.
Since no operations of meet and join are used, all of the above discussion can be generalized to define an interval in a poset.
|Date of creation||2013-03-22 15:44:56|
|Last modified on||2013-03-22 15:44:56|
|Last modified by||CWoo (3771)|
|Defines||locally finite lattice|