algebra

$(A,\circ )$ … algebraic structure or “grupoid” or “algebra” or “magma” or …

An algebraic structure is a set $A\ne \varnothing$ equipped with one or more binary operations $\circ :A\times A\to A$.
Example: $(\mathbb{N},+),(\mathbb{N},\cdot ),(\mathbb{Z},\cdot ),\dots$

closure

Closure

$(A^A,\circ )$ … set of all functions from $A$ to $A$ with composition as operation
$B^A:=\{f:A\to B\}$ … set of all functions from $A$ to $B$
$B\ne \varnothing ,B\subseteq A$

$$\text{closure}:=\forall x,y\in B:x\circ y\in B$$

$B$ is closed under function composition $\circ$ (or simply “closed”), which means any operation on elements of $B$ results in an element of $B$.
$B\subseteq A$ is called a substructure of $(A,\circ )$.

In simple words: A set is closed under an operation if the result of the operation on any points in the set is within the same set.

EXAMPLE

$(\mathbb{N},+)\le (\mathbb{Z},+)$
$\mathbb{N}\subseteq \mathbb{Z}$ is closed under addition.
$\mathbb{Q}\setminus \{0\},\cdot \le (\mathbb{Q},\cdot )$

The set of all bijective functions $S_A$ is a substructure of $(A^A,\circ )$.

$S_A:=\{f|f:A\mapsto A,f\text{ bijective}\}$, $(S_A,\circ )\le (A^A,\circ )$
This means that the composition of two bijective functions is also bijective.
I.e. the set of all bijective functions is closed under composition.

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semigroup

Semigroup: $(A,\circ )$ is a semigroup if $\circ$ is associative.

Example: $(\mathbb{N}\setminus \{0\},+)$

https://de.wikipedia.org/wiki/Halbgruppe

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Monoid

$(A,\circ )$ is a monoid if $\circ$ is associative and $A$ has an neutral element.
Example: $(\mathbb{N},+)$, $(A^A,\circ )$ (set of all functions from $A$ to $A$ with composition)

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Group

$(A,\circ )$ is a group iff
$\circ$ is associative,
$A$ has an neutral element,
every element has an inverse element.

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Abelian group

A group $(A,\circ )$ is abelian iff:
$\circ$ is associative
$A$ has an neutral element
every element has an inverse element
and $\circ$ is commutative.

The group operation in abelian groups is usually denoted with $+$.

Example: $(\mathbb{Z},+)$

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Subgroup

A subgroup is a substructure which also has group properties:
$(G,\circ )$ group, $U\subseteq G$ subgroup $⟺(U,\circ )$ group.
We denote it as $U\le G$.

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