Discrete Math with SageMath: Learn math with open-source software

A lattice is a partially ordered set (poset) in which any two elements have a least upper bound (also known as join) and greatest lower bound (also known as meet).

In Sage, a lattice can be represented as a poset using the Poset() function. This function takes a tuple as its argument, where the first element is the set of elements in the poset, and the second element is a list of ordered pairs representing the partial order relations between those elements.

First, let’s define the lists of elements and relations we will use for the following examples:

Create a poset from a tuple of elements and relations.

The function is_lattice() determines whether the poset is a lattice.

We can also use LatticePoset() function to plot the lattice. The function Poset() can be used with any poset, even when the poset is not a lattice. The LatticePoset() function will raise an error if the poset is not a lattice.

The join of two elements in a lattice is the least upper bound of those elements.

To check if a poset is a join semi-lattice (every pair of elements has a least upper bound), we use is_join_semilattice() function.

We can also find the join for individual pairs using the join() function.

The meet of two elements in a lattice is their greatest lower bound.

To check if a poset is a meet semi-lattice (every pair of elements has a greatest lower bound), we use is_meet_semilattice() function.

We can also find the meet for individual pairs using the meet() function.