Demonic composition explained

In mathematics, demonic composition is an operation on binary relations that is similar to the ordinary composition of relations but is robust to refinement of the relations into (partial) functions or injective relations.

Unlike ordinary composition of relations, demonic composition is not associative.

Definition

Suppose

is a binary relation between

and

is a relation between

and

Their

Rbf{;}^\toS

is a relation between

and

Its graph is defined as

\.

Conversely, their

Rbf{;}^\leftarrowS

is defined by

\.