$\calA$ is a reduction system for $\tau \in \Mod({\Sigma})$ and ${\Sigma}_{\calA}$ is the compactification of ${\Sigma}-\calA$, then $\tau$ induces a mapping class ${\operatorname{\rho}}_{\!\calA}(\tau) \in \Mod({\Sigma}_{\calA})$; this element ${\operatorname{\rho}}_{\!\calA}(\tau)$ is called the *reduction of $\tau$ along $\calA$*. Although the reduction is meant to be applied to reducible elements along non-empty reduction system, we allow a reduction system $\calA$ of an element $\tau$ to be empty for the logical convenience; if $\calA=\nil$, the reduction ${\operatorname{\rho}}_{\!\calA}(\tau)$ coincides with $\tau$.

Given an element $\tau \in \Mod({\Sigma})$, we can always take some positive power $\tau^N$ so that $\tau^N$ takes each connected component of ${\Sigma}$ to itself and each component of $\partial {\Sigma}$ to itself. If the restrictions of $\tau^N$ to some components are periodic, we can take yet higher power $\tau^{N'}$ so that the restriction of $\tau^{N'}$ to each component is either trivial or aperiodic.

-   An element $\tau \in \Mod({\Sigma})$ is *adequately reduced* if there is some power $\tau^N$, taking each component of ${\Sigma}$ to itself and each component of $\partial {\Sigma}$ to itself, such that the restriction of $\tau^N$ to each component of ${\Sigma}$ is either (i) trivial or (ii) aperiodic and irreducible.

-   Given an element $\tau \in \Mod({\Sigma})$, a (possibly empty) 