1

Section 10.1 What Is an Algebra?To most people the word algebra means high school algebra. But there are many kinds of algebra. An algebra consists of one or more sets, called carriers, and one or more operations on the sets. The collection of operations is often called the signature of the algebra. Denote an algebra by listing the carriers, then a semicolon, then the operators in the signature.

Examples. N; +N; 0, + Q; +, ·, 0, 1S, lists(S); , cons, head, tailA*; , cat.

To solve problems with an algebra means to be able to apply properties of the operations. For example, in the algebra Q; +, ·, 0, 1, the following properties hold:

x + 0 = x, x·1 = x, x + y = y + x, x·y=y·x, x· (y + z) = x·y + x·z, and so on.

Abstract versus Concrete. Some algebras are abstract in the sense that the carriers are not concrete known sets. So the operations must be described by axioms.

Example. Consider the algebra S; a, ƒ, where a S and let ƒ3(x) = ƒ6(x) for all x S. In this case we can say that S contains the elements a, ƒ(a), ƒ2(a), ƒ3(a), ƒ4(a), ƒ5(a).

Properties of Binary Operations. Let ° be a binary operation on S.

° is commutative if x ° y = y ° x for all x, y S.

° is associative if (x ° y) ° z = x ° (y ° z) for all x, y S.

An element e S is an identity for ° if x ° e = e ° x = x for all x S.

An element z S is a zero for ° if x ° z = z ° x = z for all x S.

If x, y S, and e is an identity, then y is an inverse of x if x ° y = y ° x = e .

2

Binary Operation Tables

If ° is a binary operation on a finite set S, then we can represent ° in

table form, where the entry in row x and column y is the value x ° y.

Example. Let S = N4 = {0, 1, 2, 3} and let x ° y = (x + y) mod 4.

Then the operation table for ° is pictured.

€

o 0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 2

Quiz (1 minute). For the table pictured in the previous example, state whether ° satisfies any of the following properties: commutative, associative, identity, zero, inverses.Answer. Is commutative; is associative; has identity 0; no zero;

has inverses 1 ° 3 = 0, 2 ° 2 = 0, and 0 ° 0 = 0.

Quiz (6 minutes). Let S = {a, b, c} and let ° be a binary operation on S. In each case, find an

operation table for ° that satisfies the given conditions.

1. b is a zero, a is an identity, every element except b has an inverse, and ° is commutative.

2. ° is commutative but not associative.

3. ° is associative but not commutative.

In (2) we have b ° (b ° c) = b ° a = b and (b ° b) ° c = b ° c = a. So ° is not associative.

In (3) we have x ° y = y for x, y S. So (x ° y) ° z = z and x ° (y ° z) = x ° z = z for

x, y, z S. So ° is associative.

Answers.

€

o a b ca a b cb b b ac c a b

(2)

€

o a b ca a b cb a b cc a b c

(3)

€

o a b ca a b cb b b bc c b a

(1)

3

Using Properties of OperationsAlgebras are useful when we apply properties of the operations to solve problems.

Example. Suppose we want to prove the following statement about integers:

x + x = x implies x = 0.

Here is an “equational proof” with reasons in parentheses.

Proof: x = x + 0 (0 is identity for +)= x + (x + –x) (–x is the inverse of x with respect to +)= (x + x) + –x (+ is associative)= x + –x (hypothesis)= 0 (–x is the inverse of x with respect to +). QED.

Notice in the example that we used several properties of the algebra Z; 0, +. Namely, that + has an identity, + is associative, and an inverse exists for every integer.

Quiz (3 minutes). Given the algebra Z; 0, +. Prove the cancellation law:

x + z = y + z implies x = y.

Answer. x = x + 0 (0 is identity for +)= x + (z + –z) (–z is the inverse of z with respect to +)= (x + z) + –z (+ is associative)= (y + z) + –z (hypothesis)= y + (z + –z) (+ is associative)= y + 0 (–z is the inverse of z with respect to +)= y (0 is identity for +). QED.