# Order Theory/Series-parallel order

**Definition (parallel order)**:

Let be a family of ordered sets, and define . Then the **parallel order** on is defined to be the order

- .

**Definition (parallel composition)**:

Let be a family of ordered sets, and define . Then the pair , where is the parallel order on , is called the **parallel composition** of the sets .

**Definition (series order)**:

Let be a preordered set, and let be a family of ordered sets over . The **series order** on induced by is the order on given by

- .

**Definition (series composition)**:

Let be a preordered set, and let be a family of ordered sets, and define . Then the pair , where is the series order on , is called the **series composition** of the sets over .

**Definition (series-parallel order)**:

A **series-parallel order** is the order of an ordered set that arises from a family of singleton ordered sets (the order being the order that turns into a poset) by applying parallel composition and series composition over a finite number of times.