Math 360, Fall 2018, Assignment 4

I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.

- Jonathan Swift, Gulliver's Travels

Read:

1. Section 2.
2. Section 3.

Carefully define the following terms, then give one example and one non-example of each:

1. Binary operation (on a set $S$).
2. Binary structure.
3. Associative (binary structure).
4. Commutative (binary structure).
5. Identity element (in a binary structure).
6. Inverse (of an element of a binary structure with identity).
7. Semigroup.
8. Monoid.
9. Group.
10. Abelian group.
11. Isomorphism (from one binary structure to another).
12. Isomorphic (binary structures).
13. Structural property (of a binary structure).

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning the number of distinct binary operations on a set with $n$ elements.
2. Theorem concerning the uniqueness of identity elements.
3. Theorem concerning the uniqueness of inverses.

Solve the following problems:

1. Section 2, problems 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 17, 18, 19, and 20.
2. Section 3, problems 2, 3, 4, 5, 6, 10, and 17.
3. Give examples of all of the following: (i) a binary structure which is not a semigroup; (ii) a semigroup which is not a monoid; (iii) a monoid which is not a group; (iv) a group which is not abelian; and (v) an abelian group.

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