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I have encountered these algebraic structures in my readings often, but can’t seem to remember them. I hope that jotting them down in this manner will help me to recall and recognize the patterns.

The list of algebraic structures here are in order from most general (less requirements) to least general (more requirements).

These structures are also called group-like, because they similar to a group (which is a structure we will get to), but with either more or less requirements.

Group-like algebraic structures

A group-like algebraic structure is made up of 2 components:

A set of elements, A
An operation, •

The operation combines any two elements in the set to form a third element: given the elements a and b, you can combine them to get a • b.

Below I will list out the different kinds of structures classified as group-like, and enumerate requirements the structure must fulfill.

Magma

A magma is the simplest structure: the operation only has 1 requirement, which is

Closure, for any a and b, a • b must be in the set of elements

In other words, you can never get out of this structure by applying operations.

Semigroup

A semigroup is a magma with 1 more requirement on the operation:

Associativity, which is that for any a, b, and c, (a • b) • c = a • (b • c)

The order in which the operation is applied does not matter, as long as the order of the operands stay the same.

Example

The set of positive integers with addition as the operation.

Let the set A be the set of positive integers: 1, 2, …, and the operation be +.

Closure is satisfied: a + b is still a positive integer in A
Associativity is satisfied: (a + b) + c = a + (b + c), since + is associative

Mnemonic: semi is half, half is 1/2, so a semigroup has 2 requirements.

Monoid

A monoid is a semigroup with 1 more requirement on the set of elements:

Identity, there must be an identity element, e, such that for all a, e • a = a = a • e

In other words, an operation applied to the identity effectively does nothing.

Example

The set of natural numbers with addition as the operation, 0 as the identity.

Let the set A be the set of natural numbers: 0, 1, 2, …, and the operation be +.

Closure and associativity are satisfied, see semigroup
Identity is satisfied with 0 as the identity element: 0 + a = a = a + 0

Mnemonic: Monoid, the i is for identity

Group

A group is a monoid with 1 more requirement on the set of elements:

Inverse, for every a in the set, there is an inverse -a, such that a • -a = -a • a = e

Applying the operation to an element an its inverse gives you the identity, they cancel out.

Example

The set of integers with addition as the operation, 0 as the identity, the inverse of each integer is its negation.

Let the set A be the set of integers: …, -2, -1, 0, 1, 2, …, and the operation be +.

Closure, associativity, and identity are satisfied, see monoid
Inverse is satisfied with -a being the inverse of a: a + -a = -a + a = 0

(Observe that the set of integers can form a group with another operation: multiplication, and 1 as the identity)

Abelian group

An Abelian group is a group with 1 more requirement on the operation:

Commutativity, for all a and b, a • b = b • a

Note that we had some restricted form of this in monoids (for the identity) and in groups (for inverses). This restriction now extends to all elements in the set.