A magma is a structure $(M,*)$ where $M$ is a set and $*$ is a binary operation under which $M$ is closed.
Usually we simply write $M$ to refer to the structure $(M,*)$ (abuse of notation).
If $*$ denotes multiplication or a general abstract operation, we typically write $ab$ instead of $a*b$ (juxtaposition).
Let $(M,*)$ be a magma. If $*$ has a certain property, e.g. associative, then we may also say $M$ has this property.
- In the context of algebra, abelian is a synonym for commutative.
Let $M$ be a magma and $e\in M$.
$e$ is left-neutral if for all $a\in M$,
\[ea = a.\]$e$ is right-neutral if for all $a\in M$,
\[ae = a.\]$e$ is neutral if it is both left-neutral and right-neutral.
$M$ is (left-/right-)unital if it has a (left-/right-)neutral element.
$M$ is uniquely (left-/right-)unital if it has exactly one (left-/right-)neutral element.
Let $M$ be a uniquely unital magma with neutral element $e$. Let $a$, $x\in M$.
$x$ is a left-inverse of $a$ if
\[xa = e.\]$a$ is left-invertible if it has a left-inverse.
$x$ is a right-inverse of $a$ if
\[ax = e.\]$a$ is right-invertible if it has a right-inverse.
$x$ is an inverse of $a$ if $x$ is both a left-inverse and a right-inverse of $a$.
$a$ is invertible if it has an inverse.
$M$ is (left-/right-)invertible if every element of $M$ is (left-/right-)invertible.
$M$ is uniquely (left-/right-)invertible if every element of $M$ has exactly one (left-/right-)inverse.
If $M$ is uniquely invertible, we generally write $a^{-1}$ for the unique inverse of $a$.
The invertible subset of $M$ is
\[\inv M = \{a\in M:\text{$a$ is invertible}\}.\]
DEF-MAG-HOM. Magma Homomorphism.
A magma homomorphism between two magmas $(M,*)$ and $(N,\diamond)$ is a function $f:M\to N$ such that for all $a$, $b\in M$,
\[f(a*b) = f(a)\diamond f(b).\]