- #1
calvino
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I have the following problems to do:
Problem 1: Let S be a monoid. Find a subgroup G of S with the property that any monoid homomorphism f: H-->S (H any group) has its image in G.
Problem 2: Find a monoid S such that there is no group G that contains S as a submonoid
Problem 3: Let X be a set, and let ~ be the least congruence relation on F_Mon(X) with xy ~ yx for all x, y in X. Prove that F_Mon(X)/~ is a free Abelian monoid over X.
any help?
For problem 1, I was thinking simply the group consisting of the identity of S. since homomorphisms will take e_H (id in H) to e_S, the id in G and S... but i have a feeling also that that is wrong since we are looking for a subgroup that will hold all of H's image (at least that's what i believe the question is asking)
For 2, I was trying to think of some sort of monoid with elements that do not have inverses. that way no GROUP could contain it. I was looking at functions with composition... not sure where to look.
NOTE: i added Problem 3, which I am currently working on. So i'll keep you updated. for now, any insight would be great. thanks
Problem 1: Let S be a monoid. Find a subgroup G of S with the property that any monoid homomorphism f: H-->S (H any group) has its image in G.
Problem 2: Find a monoid S such that there is no group G that contains S as a submonoid
Problem 3: Let X be a set, and let ~ be the least congruence relation on F_Mon(X) with xy ~ yx for all x, y in X. Prove that F_Mon(X)/~ is a free Abelian monoid over X.
any help?
For problem 1, I was thinking simply the group consisting of the identity of S. since homomorphisms will take e_H (id in H) to e_S, the id in G and S... but i have a feeling also that that is wrong since we are looking for a subgroup that will hold all of H's image (at least that's what i believe the question is asking)
For 2, I was trying to think of some sort of monoid with elements that do not have inverses. that way no GROUP could contain it. I was looking at functions with composition... not sure where to look.
NOTE: i added Problem 3, which I am currently working on. So i'll keep you updated. for now, any insight would be great. thanks
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