  Loop agreement is a family of wait-free tasks that includes set agreement and
simplex agreement, and was used to prove the undecidability of wait-free
solvability of distributed tasks by read/write memory. Herlihy and Rajsbaum
defined the algebraic signature of a loop agreement task, which consists of a
group and a distinguished element. They used the algebraic signature to
characterize the relative power of loop agreement tasks. In particular, they
showed that one task implements another exactly when there is a homomorphism
between their respective signatures sending one distinguished element to the
other. In this paper, we extend the previous result by defining the composition
of multiple loop agreement tasks to create a new one with the same combined
power. We generalize the original algebraic characterization of relative power
to compositions of tasks. In this way, we can think of loop agreement tasks in
terms of their basic building blocks. We also investigate a category-theoretic
perspective of loop agreement by defining a category of loops, showing that the
algebraic signature is a functor, and proving that our definition of task
composition is the "correct" one, in a categorical sense.
