../HW9/Solutions are available.

# 1. Maximization

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.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.

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

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) = <>.

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