With the introduced properties of binary operations we are now able to define the first simple algebraic structures (or *algebras*). The following table shows which algebras will be introduced soon and which properties they require.

Algebra $(X,\ast)$ | Closure | Associativity | Neutral Element | Existence of Inverse | Cancellation | Commutativity. |
---|---|---|---|---|---|---|

Magma | ✔ | (✔) | (✔) | (✔) | (✔) | (✔) |

Semigroup | ✔ | ✔ | (✔) | (✔) | (✔) | (✔) |

Monoid | ✔ | ✔ | ✔ | (✔) | (✔) | (✔) |

When in the table the entry "(✔)" is used, then it means that the defined binary operation "$\ast$" might not fulfill the property at all, which is required, if the sigh "✔" is used.

If an optional operation is fulfilled anyway, then the name of the algebraic structure is modified. For instance, if in a given semigroup the all elements $x,y\in X$ commute, i.e. $x\ast y=y\ast x,$ then we call such a semigroup a **commutative semigroup**. For some combinations, the name might even change. For instance, if $X$ is a monoid, and inverses exist for all elements of $X,$ then we have a *group* in place, but we will introduce groups in a later chapter. In this chapter, we want to concentrate on the tree simple-structure algebras: magma, semigroup, and monoid.

Examples: 1

- Definition: Magma
- Axiom: Axioms of Magma
- Definition: Semigroup
- Axiom: Axioms of Semigroup
- Definition: Monoid
- Axiom: Axioms of Monoid
- Definition: Exponentiation in a Monoid

Chapters: 1