Magma

Definition

A set equipped with a binary operation is termed a magma.

Magmas are also sometimes termed groupoids, but the term groupoid has another more common use.

Relation with other structures

Additional structures and conditions

The following are the common choices of additional conditions usually imposed on the binary operation of a magma:

• The existence of a neutral element, i.e., an identity element for the binary operation. For a binary operation $*$, a left neutral element is an element $e$ such that $e * x = x$ for all $x$ and a right neutral element is an element $e$ such that $x * e = e$ for all $x$. A (two-sided) neutral element is an element that is both left and right neutral.
• The associativity condition, which states that $a * (b * c) = (a * b) * c$ for all $a,b,c$ in the underlying set of the magma. There are many weaker versions of associativity, which require only certain kinds of expressions to associate.
• The commutativity condition, which states that $a * b = b * a$ for all $a,b$ in the underlying set of the magma. There are weaker versions of commutativity, which requre only certain kinds of expressions to commute.
• The existence of a nil element, i.e., a zero element for the binary operation. $n$ is a left nil for $*$ if $n * x = n$ for all $x$, and a right nil for $*$ if $x * n = n$ for all $x$.
• The existence of inverse elements with respect to a two-sided neutral element. A left inverse for $a$ with respect to $e$ is a $b$ such that $b * a = e$, and a right inverse is a $b$ such that $a * b = e$.
• Cancellation. An element $a$ is left-cancellative if $a * b = a * c \implies b = c$ and right-cancellative if $b * a = c * a \implies b = c$.
• Existence of solutions to left/right quotient equations: Whether an equation of the form $a * x = b$ can be solved, for given $a,b$. Similarly, whether $y * a = b$ can be solved for given $a,b$.

Terminology for magmas with these additional conditions

A Yes means that the structure/condition is always true for that type of magma while a No means that it is not necessarily true. The collection of Yeses in any row completely defines the type of magma being considered.

Name	Two-sided neutral element	Associativity	Commutativity	Inverses	Cancellation	Existence of quotients
Magma	No	No	No	No	No	No
Unital magma	Yes	No	No	No	No	No
Cancellative magma	No	No	No	No	Yes	No
Semigroup	No	Yes	No	No	No	No
Cancellative semigroup	No	Yes	No	No	Yes	No
Monoid	Yes	Yes	No	No	No	No
Quasigroup	No	No	No	No	Yes	Yes
Algebra loop	Yes	No	No	Yes	Yes	Yes
Group	Yes	Yes	No	Yes	Yes	Yes
Abelian group	Yes	Yes	Yes	Yes	Yes	Yes