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Algebraic Automata Theory Sheet 0, 2017-10-22

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ITI

Institut für Theoretische Informatik

Dr. Jürgen Koslowski

Algebraic Automata Theory

Sheet 0, 2017-10-22

Exercise 1 [10 POINTS]

As we have seen, monoids precisely the are 1-object categories, while monoid-homomorphisms are precisely the functors between those. Characterize the natural transformations in this setting.

Exercise 2 [10 POINTS]

Check that the family of functions

( P (X )) δ

X

P (X ) , X ∈ set

that map a string of n subsets A i ⊆ X - to their concatenation, i.e., a subset of X n ⊆ X , constitutes a distributive law between the free monoid monad and the power-set monad. What about a distributive law in the opposite direction?

Exercise 3 [15 POINTS]

Consider Rat as a functor on suitable category. Try to find a monad-structure on Rat . How does this relate to other monads induced by F and P ?

due on Thursday, 2017-10-27, 13:15,

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