I've been taking baby steps in abstract algebra, to try to learn more about the mathematics that underlie some of my favorite hashing algorithms.

Definitions:

The elements of P are known as the positive elements of S; all other nonzero elements of S are known as the negative elements of S.

If a and b are elements of S, and if a + -b is positive, then we write a > b and b < a.

The commutative law for addition: a + b = b + a

The commutative law for multiplication: a × b = b × a

The associative law for addition: (a + b) + c = a + (b + c)

The associative law for multiplication: (a × b) × c = a × (b × c)

The (left) distributive law for multiplication over addition: a × (b + c) = (a × b) + (a × c)

Here's a table of algabraic structures and their properties. (It currently looks quite ugly because of the default bootstrap styling; maybe I'll fix it later.)

ordered field | field | division ring (sfield) | integral domain | ring with unity | commutative ring | ring | Abelian or commutative group | group | Abelian or commutative semigroup | semigroup | |
---|---|---|---|---|---|---|---|---|---|---|---|

If a and b are in S, then a + b = b + a |
X | X | X | X | X | X | X | ||||

If a and b are in S, then a × b = b × a |
X | X | X | X | X | ||||||

If a, b, and c are in S, then (a + b) + c = a + (b + c) |
X | X | X | X | X | X | X | ||||

If a, b, and c are in S, then (a × b) × c = a × (b × c) |
X | X | X | X | X | X | X | X | X | X | X |

If a, b, and c are in S, then a × (b + c) = (a × b) + (a × c) and (b + c) × a = (b × a) + (c × a) |
X | X | X | X | X | X | X | ||||

S contains an element 0 (zero) such that for any element a of S, a + 0 = a |
X | X | X | X | X | X | X | ||||

S contains an element 1 (unity) different from 0 (zero), such that for any element a of S, a × 1 = a |
X | X | X | X | X | X | |||||

For each element a in S, there exist an element -a in S such that a + -a = 0 |
X | X | X | X | X | X | X | ||||

If a, b, and c are in S, c ≠ 0, and c × a = c × b or a × c = b × c then a = b |
X | X | X | X | |||||||

For each element a ≠ 0 in S, there exists an element a ^{-1} in Ssuch that a × a ^{-1} = 1 |
X | X | X | X | X | ||||||

There exists a subset P, not containing 0, of the set S such that if a ≠ 0, then one and only one of a and -a is in P |
X | ||||||||||

There exists a subset P, not containing 0, of the set S such that if a and b are in P, then a + b and a × b are in P |
X |