This is about the mathematical concepts of a "magma", a "loop" and a "monoid", which are descriptions of certain properties that the combining of things may enjoy.

One of the most basic notions of mathematics are sets with subsets and powersets. From these, one can define a binary relation between sets $X$ and $Y$ to be a subset of $X\times Y$ and then a map from $X$ to $Y$ to be a binary relation between $X$ and $Y$ which is left-total (each element of $X$ is related to at least one element of $Y$) and right-unique (aka "functional", each element of $X$ is related to at most one element of $Y$).

Magmas

A binary operation by $X$ on $Y$ is a map $X \times Y \to Y$. A binary operation of $X$ on itself is called an inner binary operation on $X$. A magma $(X,\ast)$ is a set $X$ with an inner binary operation $\ast : X \times X \to X$.

Examples of magmas: the natural numbers with addition $(\mathbb{N},+)$ (but not the natural numbers with subtraction) and the set of words over an alphabet, with concatenation. The former is a special case of the latter if we encode numbers unary, i.e. $0$ is the empty word, $1$ is just $|$ ($2$ is $||$) and every other number $n$ is a word of length $n$ consisting only of $|$s. The alphabet is $\{|\}$ and $(\{|\}^{\mathbb{N}},concat)$ is essentialy the same as $(\mathbb{N},+)$.

Concatenation can be done in more than one dimension, i.e. we can consider not only words $w = w_1w_2w_3\cdots w_n$ over an alphabet $A$ (such that $w_i \in A$) with concatenation ("horizontal composition"), but 2-dimensional words composed of letters $w_{i,j}\in A$, such that each $w_{i,j}$ with fixed $i$ and varying $j$ is an ordinary word, and each $w_{i,j}$ with fixed $j$ and varying $i$ is a word written vertically. Such a 2-dimensional word has a "rectangular shape". The set of all 2-dimensional words over an alphabet $A$ has two magma structures (horizontal composition and vertical composition) and it can be interesting how they interact.

A classical example of a set with two magma structures is the set of loops in a pointed topological magma (continuous paths from the basepoint to the basepoint), with one operation given by the pointwise operation of the topological magma and the other given by concatenation of loops.

Other magmas: the natural numbers with multiplication, the natural numbers with exponentiation $(a,b) \mapsto a^b$, the complex numbers with multiplication, the octonions with multiplication, the sedenions with multiplication. If you haven't seen the octonions or sedenions before, look up the Cayley–Dickson construction in Wikipedia.

A magma homomorphism $f : (X,\ast) \to (Y,\ast)$ is a map $f : X \to Y$ such that $\forall x,y \in X : f(x \ast y) = f(x) \ast f(y)$. The unary encoding of natural numbers discussed above is a magma isomorphism $(\mathbb{N},+) \to (\{|\}^{\mathbb{N}},concat)$.

A magma operation $\circ$ of a magma $(M,\ast)$ on a set $X$ is a binary operation $\circ$ of $M$ on $X$ that is compatible with the magma operation $\ast$, i.e. $\forall m,n \in M\ \forall x \in X : (m\ast n) \circ x = m \circ (n \circ x)$.

There are some elementary properties one can study for any magma $(X,\ast)$ one encounters:

• Associativity: Is $\forall x,y,z \in X: (x\ast y)\ast z = x\ast(y\ast z)$?
• Neutral Elements: $\exists x \in X\ \forall y \in X : x\ast y = y$?
• Units: $\exists x \in X\ \exists y \in X\ \forall z \in X : (x\ast y) \ast z = z$?
• The dual magma $(X,\ast^\vee)$ with $\ast^\vee(x,y) := \ast(y,x)$.
• Commutativity: Is $\forall x,y \in X : x\ast y = y \ast x$?
• Solvability of Equations: Does there exist a unique inner binary operation $solv_\ast : X \times X \to X$ such that $\forall x,y,z \in X : x \ast y = z \Rightarrow x = solv_\ast(y,z)$?

A magma with the property that equations of the form $a\ast x = b$ and $x\ast a = b$ are uniquely solvable is called a quasigroup. A loop is a quasigroup that posesses a neutral element.

An associative magma is sometimes called a semigroup, although that word is also used for monoids, which are associative magmas that posess a neutral element, i.e. an element $e \in X$ such that $\ast e : X \to X$ and $e \ast : X \to X$ are both equal to the identity map.

Groups can be defined as monoids in which every element is a unit, i.e. every $x \in X$ has an inverse element $x^{-1} \in X$ which satisfies $x \ast x^{-1} = e$ and $x^{-1} \ast x = e$. Groups can also be defined as associative loops.

Loops

A Loop is a quasigroup with a neutral element, but it needn't be associative or commutative. In a loop, every element has a left-inverse and a right-inverse, since $x \ast y = e$ and $y \ast x = e$ are uniquely solvable, but unlike a group, these one-sided inverses don't coincide in general.

In a loop $L$, the left-associator $A(a,b,c) \in L$ of three elements $a,b,c \in L$, defined by $\left( a \ast (b \ast c) \right) \ast A(a,b,c) = \left( (a \ast b) \ast c \right)$, measures how non-associative the loop is (together with the right-associator).

A quasigroup $Q$ has the Moufang property (or satisfies the Moufang identity) if $\forall a,b,c \in Q : (a \ast b) \ast (c \ast a) = (a \ast (b \ast c)) \ast a$, which is a weak form of associativity. The Moufang property implies that there exists a neutral element (so the quasigroup is a loop, after all): since $a \ast e = a$ is uniquely solvable and $(x \ast a) \ast (e \ast x) = (x \ast (a \ast e)) \ast x$ is uniquely solvable, we have $(e \ast x) = x$, for all $x$, so $e$ is left-neutral. Now take $f$ defined by $f \ast e = e$, which can be shown by the same method to be right-neutral, so we conclude with $f = f \ast e = e$.

A loop that enjoys the Moufang property is called a Moufang loop. In a Moufang loop, left-inverses coincide with right-inverses. In particular, the left-associator and the right-associator coincide (but the associator is still non-trivial in general).

The nonzero quaternions with quaternion multiplication form a non-commutative associative loop. The nonzero octonions form a non-associative Moufang loop. In general, the set of invertible elements of an alternative algebra forms a Moufang loop. The integers with subtraction form a loop with neutral element $0$ that does not have the Moufang property.

Moufang's theorem states that the subloop generated by any two elements of a Moufang loop is an associative loop, i.e. a group. This theorem makes working with loops a lot more convenient.

Monoids

A monoid is an associative magma with a neutral element, but it needn't be commutative or have any inverses. A monoid homomorphism is a magma homomorphism that maps neutral elements to neutral elements.

An archetypical monoid is the set of endomorphisms of some thing, since composition of morphisms is required to be associative and there is the identity morphism. In other words, any monoid $M$ can be seen as a category with one object $1$ and $Hom(1,1) = M$ and vice versa. Observe that a group in this formalism is a category with one object such that all morphisms are invertible.

This points to an obvious generalization of monoids: categories. The corresponding generalization of groups is called groupoid.

To every commutative monoid $M$ one can associate an abelian group $G(M)$ such that the monoid homomorphisms into any other abelian group $H$ are in bijection with the group homomorphisms $G(M) \to H$. This can be done by the construction familiar from building integers $\mathbb{Z}$ out of natural numbers $\mathbb{N}$ and is called the Grothendieck group. Other prominent examples of this construction lie in the fundamentals of K-Theory.

The Eckmann-Hilton argument

The Eckmann-Hilton theorem states for a set $X$ with two inner binary operations $\ast, \circ$ that have a neutral element $1 \in X$ and distribute over each other, i.e. $(a \ast b) \circ (c \ast d) = (a \circ c) \ast (b \circ d)$, that the two operations actually coincide, their neutral elements coincide and that this operation is associative and commutative.

The classical application of the theorem is the case of the monoid of homotopy classes of loops of an H-space, which shows that, up to homotopy, composition of loops is the same as multiplication of loops. As a corollary, all higher homotopy groups $\pi_n(X)$ of a topological space $X$, for $n \geq 2$, are abelian.

It is very nice to work out the Eckmann-Hilton argument as an exercise given a few hints: visualize the two operations as horizontal and vertical composition and look at expressions of the form $(\alpha \ast 1) \circ (1 \ast \beta)$.