We define our semantics generically in terms of a monad (\mathcal {M},\mu ,\eta ), using the Kleisli composition operator for \mathcal {M}:
\begin {array}{l}
\mathsf {ite} : (\Sigma \rightarrow \mathbb {B}) \times (\Sigma \rightarrow \mathcal {M}(\Sigma )) \times (\Sigma \rightarrow \mathcal {M}(\Sigma )) \rightarrow (\Sigma \rightarrow \mathcal {M}(\Sigma ))\\
\mathsf {ite}(b,t,e)(\sigma ) = \begin {cases}t(\sigma ) & \text {if}\ b(\sigma ) = T \\
e(\sigma ) & \text {otherwise}
\end {cases}
\end {array}
\begin {array}{lcl}
\llbracket \cdot \rrbracket _\mathcal {C} : \mathcal {C} \rightarrow \Sigma \rightarrow \mathcal {M}(\Sigma )\\[0.4em]
\llbracket \mathsf {skip} \rrbracket _\mathcal {C} & = & \eta _\Sigma \\
\llbracket x := e \rrbracket _\mathcal {C} & = & \boldsymbol {\lambda } \sigma .\ \eta _\Sigma (\sigma \left (x \mapsto \llbracket e \rrbracket _\mathcal {E}\right )) \\
\llbracket c_1 ; c_2 \rrbracket _\mathcal {C} & = & \llbracket c_2 \rrbracket _\mathcal {C} \circ _K \llbracket c_1 \rrbracket _\mathcal {C} \\[0.3em]
\llbracket \textsf {if}\ b\ \textsf {then}\ c_1\ \textsf {else}\ c_2 \rrbracket _\mathcal {C} & = & \mathsf {ite}(\llbracket b \rrbracket _\mathcal {B},\llbracket c_1 \rrbracket _\mathcal {C},\llbracket c_2 \rrbracket _\mathcal {C})\\
\llbracket \textsf {while}\ b\ \textsf {do}\ c\ \textsf {od} \rrbracket _\mathcal {C} & = & \mathbf {\mathbf {fix}}(\boldsymbol {\lambda } f.\ \mathsf {ite}(\llbracket b \rrbracket _\mathcal {B}, f \circ _K \llbracket c \rrbracket _\mathcal {C}, \eta _\Sigma ) )
\end {array}