An attempt at writing maths in HTML

Some definitions:

A group is a non-empty set G with an operation ⋅ such that:

• the set is closed under the operation (closure)
  
• a⋅(b⋅c) = (a⋅b)⋅c (associativity)
  
• ∃e ∈ G such that ∀a ∈ G a⋅e = e⋅a = a (identity)
  
• ∀g ∈ G ∃g^-1 ∈ G such that g⋅g^-1 = g^-1⋅g = e (inverses)

A semi-group is a non-empty set with an operation such that

• the set is closed under the operation (closure)
  
• a⋅(b⋅c) = (a⋅b)⋅c (associativity)

Or, a semi-group is a non-empty set which is closed under an associative operation, or a semi-group is an associative groupoid.

A groupoid or magma is a non-empty set equipped with a single binary operation M × M → M. An operation is closed by definition (so I could have removed that axiom from each of the definitions above).

A monoid is a semi-group with identity.

A ring is a non-empty set S with two operations (called addition (+) and multiplication (×), however they might be defined) such that:

• (S, +) is an Abelian group with identity 0.
  
• (S, ×) is a monoid with identity 1.
  
• ∀a,b,c ∈ S, a(b+c) = ab + ac

A commutative ring is a ring where (S, ×) is commutative.