reduction system $\calA$ of $\tau$ is said to be an *adequate reduction system* if the reduction ${\operatorname{\rho}}_{\!\calA}(\tau)$ is adequately reduced.

By definition, an empty reduction system for an adequately reduced element is indeed an adequate reduction system. Thurston observed that every element $\tau \in \Mod(S)$ either is adequately reduced or has a non-empty adequate reduction system [@Thurston:Surfaces].

Generally, adequate reduction systems of $\tau$ are not unique. The crucial fact shown in [@Birman-Lubotzky-McCarthy:AbSol] is that there is a canonical choice of an adequate reduction system for each element $\tau \in \Mod({\Sigma})$; it is indeed the unique minimal adequate reduction system for the element. We call such system the *canonical reduction system* for $\tau$, and denote it by $\calC$ (the reference to $\tau$ should always be clear from the context).

For a connected surface ${\Sigma}$, the canonical reduction system for $\tau \in \Mod({\Sigma})$ is empty if and only if $\tau$ is periodic or irreducible. More generally, for any surface ${\Sigma}$, which may be connected or disconnected, the canonical reduction system for $\tau \in \Mod({\Sigma})$ is empty if and only if $\tau$ is adequately reduced. Adequately reduced elements are more general than irreducible ones, but their important properties can be derived from those 