Notes on Abstract Algebra
A binary operation on a set S is a binary function : S × S → S.
A magma (M, *) is a set M together with a binary operation * on M.
A semigroup (S, *) is an associative magma, that is to say that it is a magma that satisfies this axiom:
- For all a, b and c in S, (a * b) * c = a * (b * c)
A monoid (M, *) is a unital semigroup, that is to say that it is a semigroup that satisfies this additional axiom:
- There is an element e in M such that for all a in M, a * e = e * a = a
The element e in the above axiom is the identity element.
A group (G, *) is a monoid whose elements are invertible, that is to say that it is a monoid that satisfies this additional axiom:
- For all a in G, there is an element b in G such that a * b = b * a = e, where e is the identity element from the monoid axiom.
If for all a and b in G, a * b = b * a then the group is commutative and is called an abelian group.
A ring (R, +, *) is a set R with two binary operations + and * (commonly referred to as addition and multiplication) such that:
- (R, +) is an abelian group. The identity element is called 0.
- (R, *) is a monoid. The identity element is called 1.
- Multiplication distributes over addition, which is to say:
- a * (b + c) = (a * b) + (a * c)
- (a + b) * c = (a * c) + (b * c)
If multiplication is commutative, then it is a commutative ring.
A field is a commutative ring such that 0 does not equal 1 and every element except 0 has a multiplicative inverse.
A vector space over a field F is a set V together with two operations:
- Vector addition: V × V → V denoted v + w where v, w ∈ V
- Scalar multiplication: F × V → V denoted a v where a ∈ F, v ∈ V
which satisfy these axioms:
- V under vector addition is an abelian group
- Scalar multiplication is associative: a(b v) = (ab) v
- 1 v = v, where 1 is the multiplicative identity in F
- Scalar multiplication distributes over vector addition: a(v + w) = a v + a w
- Scalar multiplication distributes over scalar addition: (a + b)v = a v + b v
The elements of V are called vectors and the elements of F are called scalars.