../HW9/Solutions are available.

1. Maximization

1. Let A be the semigroup (N, max), where max(a,b) = a if a > b or b otherwise. Prove that every subset of N yields a subsemigroup of A.

2. Now let M be the monoid (N, max, 0) obtained by extending A with the identity 0. Determine which subsets of N yield submonoids of M.

2. Egalitarian semigroups

Define an egalitarian semigroup to be a semigroup in which wx=yz for any elements w, x, y, and z.

1. Prove or disprove: If A and B are egalitarian semigroups, then any function f:A->B is a homomorphism.

2. Give a formal definition of a free egalitarian semigroup F(S) over a set S, by defining both its set of elements and the effect of its semigroup operation. Prove that the structure you defined satisfies the definition of a free algebra given in AlgebraicStructures: specifically, that F(S) is an egalitarian semigroup that contains all the elements of S, and for any function f from S to the carrier of some egalitarian semigroup G, there is a unique homomorphism f^*:F(S)->G such that f^*(x)=f(x) for all x in S.

3. Quotients

Let A be the free monoid over {a,b} and B be the free monoid over {b}. Define a function f:A->B where for each sequence x, f(x) is the sequence obtained by removing all occurrences of a from x: for example, f(ababab) = f(bbab) = bbb and f(aaa) = <>.

1. Show that f is a homomorphism. (Hint: Use the fact that A is a free algebra.)

2. Let ~ be the kernel of f. Show that A/~ is isomorphic to (N,+,0).

4. Back to the center

Let G be a group, and let C = { a∈G | ax = xa for all x in G }. Show that C is an abelian subgroup of G.