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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:

  1. A set of elements, A
  2. 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

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:

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 +.

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:

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 +.

Mnemonic: Monoid, the i is for identity

Group

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

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 +.

(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:

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.

Example

The set of integers with addition as the operation, 0 as the identity, negation as the inverse.

References

Wikipedia is a fantastic resource and goes into much further depth, and has an illustrative table comparing the aforementioned structures.

Magma

Semigroup

Monoid

Group

Abelian group