lattice interval

Definition. Let $L$ be a lattice. A subset $I$ of $L$ is called a lattice interval, or simply an if there exist elements $a,b\in L$ such that

$$I=\{t\in L\mid a\le t\le b\}:=[a,b].$$

The elements $a,b$ are called the endpoints of $I$. Clearly $a,b\in I$. Also, the endpoints of a lattice interval are unique: if $[a,b]=[c,d]$, then $a=c$ and $b=d$.

Remarks.

• 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 $I$ of a lattice $L$ is a sublattice of $L$.

• A bounded lattice is itself a lattice interval: $[0,1]$.

• A prime interval is a lattice interval that contains its endpoints and nothing else. In other words, if $[a,b]$ is prime, then any $c\in [a,b]$ implies that either $c=a$ or $c=b$. Simply put, $b$ covers $a$. If a lattice $L$ contains $0$, then for any $a\in L$, $[0,a]$ is a prime interval iff $a$ 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.

• Given a lattice $L$, let $ℬ$ be the collection of all lattice intervals without endpoints, we can form a topolgy on $L$ with $ℬ$ as the subbasis. This does not insure that $\wedge$ and $\vee$ are continuous, so that $L$ with this topological structure may not be a topological lattice.

• Locally Finite Lattice. A lattice that is derived based on the concept of lattice interval is that of a locally finite lattice. A lattice $L$ is locally finite iff every one of its interval is finite. Unless the lattice is finite, a locally finite lattice, if infinite, is either topless or bottomless.