When we have two sets, we can define their Cartesian product.

The partition of a given set is a set that contains subsets of the set that are non-overlapping and whose union is the whole set.


Relations can have several properties.

Relations can be represented as a matrix or a digraph. The properties of relations can be easily determined from these representations.

Relations can be composed with each other. When a relation is composed with itself, we call it a power.

There are several types of relations:

Partially Ordered Sets

Partially ordered sets (or poset, for short) are sets with a partial order relation.

There are several kinds of “boundary” elements we can find in a poset:

  • The Minimal and Maximal (and their absolute versions) refer to “least” and “greatest” elements in the poset (i.e. those that aren’t preceded or succeeded by another element)
  • The Upper and Lower Bounds refer to the elements in the poset that bound a subset of the poset (i.e. precede or succeed all the elements of the subset)
    • The Minimum and Maximum (not to be confused with minimal and maximal) refer to the upper/lower bounds of a subset that are also within that same subset
    • The Supremum and Infimum refer to the “least”/“greatest” upper/lower bounds in the whole poset

Posets can be represented with Hasse diagrams, which can be seen as simplified digraphs.


Posets where every subset of two elements has a unique supremum and a unique infimum are called lattices.