Using notation similar to Backus-Naur Form (BNF): \begin {array}{lcl} \mathcal {E} & \Coloneqq & n \mid x \mid \mathcal {E}_1 + \mathcal {E}_2 \mid \mathcal {E}_1 - \mathcal {E}_2\\ \mathcal {B} & \Coloneqq & \mathsf {false} \mid \mathsf {true} \mid \neg \mathcal {B} \mid \mathcal {E}_1 = \mathcal {E}_2\\ \mathcal {C} & \Coloneqq & \mathsf {skip} \mid x := \mathcal {E} \mid \mathcal {C}_1 ; \mathcal {C}_2 \mid \mathsf {if}\ \mathcal {B}\ \mathsf {then}\ \mathcal {C}_1\ \mathsf {else}\ \mathcal {C}_2 \mid \mathsf {while}\ \mathcal {B}\ \mathsf {do}\ \mathcal {C}\ \mathsf {od}\\ x & \in & \mathcal {V}\; \textit {(variables)}\\ n & \in & \mathbb {N} \end {array}

A monad on the category \textbf {C} consists of an endofunctor \mathcal {M} : \textbf {C} \rightarrow \textbf {C} and two operations (natural transformations) for all \mathbf {C}-objects X: \mu _X : \mathcal {M}(\mathcal {M}(X)) \rightarrow \mathcal {M}(X) and \eta _X : X \rightarrow \mathcal {M}(X), such that the following diagrams commute:

1. \mu _X \circ \mathcal {M}(\mu _X) = \mu _X \circ \mu _{\mathcal {M}(X)}
2. \mu _X \circ \mathcal {M}(\eta _X) = \mu _X \circ \eta _{\mathcal {M}(X)} = \mathsf {id}_{\mathcal {M}(X)}

The flat domain lifting operation (\cdot )_\bot forms a monad, where the endofunctor \mathcal {M}(X) = X_\bot , the operation \mu _X : (X_\bot )_\bot \rightarrow X_\bot collapses the two \bot values, and \eta _X : X \rightarrow X_\bot is simply \eta _X(x) = x.

For a monad (\mathcal {M}, \mu , \eta ) over the category \mathbf {C}, the Kleisli category has the same objects as \mathbf {C}, but the morphisms A \xrightarrow {h} B are those morphisms h of \mathbf {C} that have the form A \xrightarrow {h} \mathcal {M}(B). Composition is defined as follows, for morphisms A \xrightarrow {f} \mathcal {M}(B) and B \xrightarrow {g} \mathcal {M}(C): g \circ _K f = \mu _C \circ \mathcal {M}(g) \circ f Identity morphisms for this category are just \eta .

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}

Instantiating our monadic semantics where the monad \mathcal {M} is flat domain lifting, we recover our original semantics of \mathcal {C}.

Let us instantiate our monadic semantics with the monad \mathcal {M}(X) = X_\bot \oplus \mathbf {1}, where the added value (from \mathbf {1}) is called \mathsf {fail}. The operation \mu _X : (X_\bot \oplus \mathbf {1})_\bot \oplus \mathbf {1} \rightarrow X_\bot \oplus \mathbf {1} collapses the two possible \bot values to \bot and the two possible \mathsf {fail} values to \mathsf {fail}, and the operation \eta _X : X \rightarrow X_\bot \oplus \mathbf {1} is the straightforward injection.

Then we may extend our language with lightweight exception constructs: \mathcal {C} ::= \dots \mid \mathsf {throw} \mid \mathsf {try}\ \mathcal {C}_1\ \mathsf {catch}\ \mathcal {C}_2 and define their semantics, leaving our existing semantics unchanged: \begin {array}{lcl} \llbracket \mathsf {throw} \rrbracket \ \sigma &=& \mathsf {fail} \\ \llbracket \mathsf {try}\ c_1\ \mathsf {catch}\ c_2 \rrbracket \ \sigma &=& \begin {cases} \llbracket c_2 \rrbracket \ \sigma & \text {if}\ \llbracket c_1 \rrbracket \ \sigma = \mathsf {fail} \\ \llbracket c_1 \rrbracket \ \sigma & \text {otherwise}\end {cases} \end {array}