State transition system

Coalgebra formulation

The formal definition can be rephrased as follows. Labelled state transition systems on ${\displaystyle S}$ with labels from ${\displaystyle \Lambda }$ correspond one-to-one with functions ${\displaystyle S\to {\mathcal {P}}(\Lambda \times S)}$, where ${\displaystyle {\mathcal {P}}}$ is the (covariant) powerset functor. Under this bijection ${\displaystyle (S,\Lambda ,T)}$ is sent to ${\displaystyle \xi _{T}:S\to {\mathcal {P}}(\Lambda \times S)}$, defined by