  Effect systems are lightweight extensions to type systems that can verify a
wide range of important properties with modest developer burden. But our
general understanding of effect systems is limited primarily to systems where
the order of effects is irrelevant. Understanding such systems in terms of a
semilattice of effects grounds understanding of the essential issues, and
provides guidance when designing new effect systems. By contrast, sequential
effect systems -- where the order of effects is important -- lack an
established algebraic structure on effects.
  We present an abstract polymorphic effect system parameterized by an effect
quantale -- an algebraic structure with well-defined properties that can model
the effects of a range of existing sequential effect systems. We define effect
quantales, derive useful properties, and show how they cleanly model a variety
of known sequential effect systems.
  We show that for most effect quantales, there is an induced notion of
iterating a sequential effect; that for systems we consider the derived
iteration agrees with the manually designed iteration operators in prior work;
and that this induced notion of iteration is as precise as possible when
defined. We also position effect quantales with respect to work on categorical
semantics for sequential effect systems, clarifying the distinctions between
these systems and our own in the course of giving a thorough survey of these
frameworks. Our derived iteration construct should generalize to these semantic
structures, addressing limitations of that work. Finally, we consider the
relationship between sequential effects and Kleene Algebras, where the latter
may be used as instances of the former.
