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ITI

Institut für Theoretische Informatik

Dr. Jürgen Koslowski

Algebraic Automata Theory

Sheet 5, 2017-11-23

Exercise 1[16Points]

Anideal of a monoid hM,·, ei is a subset A⊆M that satisfies M·A⊆A⊇A·M. 1. Show that the set M-idl of all ideals of M is a submonoid of the power-set MP.

2. Analize the closure properties of Midl wrt. the Boolean operations union, intersection, and complement.

3. How do ideals behave wrt. residuation?

4. How do ideals behave wrt. direct and inverse images?

Exercise 2[14Points]

Monoids with an absorbing element 0 or ⊥ are also known asbinoids. Their homomorphisms have to preserve the absorbing element.

1. Show that a monoid can have at most one absorbing element.

2. Check, whether or not the category bin of binoids and their homomorphisms is a full subcategory ofmon.

3. Try to analyze the notion of ideal in bin.

due on Thursday, 2017-11-30, 13:15,

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