symmetric monoidal (∞,1)-category of spectra
A binary operation on a set $S$ is a function $(-)\cdot (-) \colon S \times S \to S$ from the Cartesian product $S \times S$ to $S$. A magma (or binary algebraic structure, or, alternatively, a mono-binary algebra) $(S,\cdot)$ is a set equipped with a binary operation on it.
A magma is called
unital if it has a neutral element; that is, an element $1 \in S$ such that $1 \cdot x = x = x \cdot 1$. Some authors mean by ‘magma’ what we call a unital magma (cf. Borceux-Bourn Def. 1.2.1). One can consider one-sided unital elements separately: $1_L \cdot x = x$ and/or $x = x \cdot 1_R$. Note that units may be far from unique.
commutative if the binary operation takes the same value when its two arguments are interchanged: $x \cdot y = y \cdot x$.
associative if the binary operation satisfies the associativity condition $(x \cdot y) \cdot z = x \cdot (y \cdot z)$.
invertible if it has an inverse element.
an absorption magma if it has an element $0 \in S$ such that the binary operation satisfies the absorption condition: $0 \cdot x = x \cdot 0 = 0$.
a quasigroup if one-sided multiplication by any element is a bijection.
A magma has a square root function $\sqrt{-}:S \to S$ if for all $x \in S$, $\sqrt{x \cdot x} = x$ and $\sqrt{x} \cdot \sqrt{x} = x$.
The term ‘magma’ is from Bourbaki and is intended to suggest the fluidity of the concept; special cases include unital magmas, semigroups/monoids, quasigroups, groups, and so on. The term ‘groupoid’ is also used, but here that word means something else (see also related discussion at historical notes on quasigroups).
More generally, in any multicategory $M$, a magma object or magma in $M$ is an object $X$ of $M$ equipped with a multimorphism $m: X, X \to X$ in $M$. Here the multimorphism from $X$ and $X$ to $X$ is a binary operation in $M$. In particular, for $M$ a monoidal category, a magma structure on $X$ is a morphism $m\colon X \otimes X \to X$, and in a closed category, a magma structure on $X$ is a morphism $m\colon X \to [X, X]$.
Every magma $(M,\cdot)$ has a morphism $(-)^2:M \to M$ called the square and defined as $x^2 := x \cdot x$ for all $x \in M$.
An action of a set $A$ on another set $B$ is a function $act:A \times B \to B$. So this means that a magma is just an action of a set on itself.
There exists a function on the binary operation set $B:(M\times M\to M)\to (M\times M\to M)$ called the braiding that takes every binary operation on the set to its opposite binary operation, where for every magma operation $m:M\times M\to M$ and for all elements $a,b$ in $M$, $B(m)(a,b) = m(b,a)$. The set $M$ with the opposite binary operation is the opposite magma of $M$. $B$ is an involution; the opposite of an opposite magma is the original magma itself; this follows from the fact that Set is a symmetric monoidal category. Fixed points of $B$ are called commutative binary operations.
The free magma on one generator $(M, \cdot, 1)$ is a model of nesting parentheses around a magma, and is important in the study of higher category theory as composition of morphisms in (n,r)-categories and (infinity,n)-categories are not associative, but rather satisfy coherence laws such as the pentagon identity which relate the various ways to nest parentheses around the tensor product magma in an (infinity,n)-category with respect to homotopy equivalence. Many other higher categorical objects have coherence theorems which also deal with nested parentheses around a binary operation.
In foundations of mathematics such as intensional Martin-Loef dependent type theory, where functions do not preserve equality, one could distinguish between ordinary magmas as defined above, and extensional magmas, whose binary operation preserves equality.
For a magma $(A, \cdot)$ and for all elements $a, b, c:A$, a magma is left extensional if $a = c$ implies $a \cdot b = c \cdot b$, and a magma is right extensional if $a = c$ implies $b \cdot a = b \cdot c$. A magma is extensional if it is both left and right extensional.
The wikipedia entry magma is quite useful, having a list and a table of various subclasses of magmas, hence of binary algebraic structures.
Formalization in cubical homotopy type theory:
Last revised on February 2, 2023 at 19:53:26. See the history of this page for a list of all contributions to it.