In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed.
History and terminology The term groupoid was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid. The term was then appropriated by B. A. Hausmann and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a groupoid in the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term groupoid is "perhaps most often used in modern mathematics" in the sense given to it in category theory. According to Bergman and Hausknecht (1996): "There is no generally accepted word for a set with a not necessarily associative binary operation. The word groupoid is used by many universal algebraists, but workers in category theory and related areas object strongly to this usage because they use the same word to mean 'category in which all morphisms are invertible'. The term magma was used by Serre [Lie Algebras and Lie Groups, 1965]." It also appears in Bourbaki's Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970.
Definition A magma is a set M with an operation • that sends any two elements a, b ∈ M to another element, a • b ∈ M. The symbol • is a general placeholder for a properly defined operation. This requirement that for all a, b in M, the result of the operation a • b also be in M, is known as the magma or closure property. In mathematical notation:
∀ a , b : a , b ∈ M ⟹ a ⋅ b ∈ M . {\displaystyle \forall a,b\colon a,b\in M\implies a\cdot b\in M.}
If • is instead a partial operation, then (M, •) is called a partial magma or, more often, a partial groupoid.
Morphism of magmas A morphism of magmas is a function f : M → N that maps a magma (M, •) to a magma (N, ∗) and preserves the binary operation:
f (x • y) = f(x) ∗ f(y). For example, with M equal to the positive real numbers and • as the geometric mean, N equal to the real number line, and ∗ as the arithmetic mean, a logarithm f is a morphism of the magma (M, •) to (N, ∗).
proof: log x y = log x + log y 2 . {\displaystyle \log {\sqrt {xy}}\ =\ {\frac {\log x+\log y}{2}}.}
Note that these commutative magmas are not associative; nor do they have an identity element. This morphism of magmas has been used in economics since 1863 when William Stanley Jevons calculated the rate of inflation in 39 commodities in England in his A Serious Fall in the Value of Gold Ascertained.
Notation and combinatorics The magma operation may be applied repeatedly, and in the general, non-associative case, the order matters, which is notated with parentheses. Also, the operation • is often omitted and notated by juxtaposition:
(a • (b • c)) • d ≡ (a(bc))d. A shorthand is often used to reduce the number of parentheses, in which the innermost operations and pairs of parentheses are omitted, being replaced just with juxtaposition: xy • z ≡ (x • y) • z. For example, the above is abbreviated to the following expression, still containing parentheses:
(a • bc)d. A way to avoid completely the use of parentheses is prefix notation, in which the same expression would be written ••a•bcd. Another way, familiar to programmers, is postfix notation (reverse Polish notation), in which the same expression would be written abc••d•, in which the order of execution is simply left-to-right (no currying). The set of all possible strings consisting of symbols denoting elements of the magma, and sets of balanced parentheses is called the Dyck language. The total number of different ways of writing n applications of the magma operator is given by the Catalan number Cn. Thus, for example, C2 = 2, which is just the statement that (ab)c and a(bc) are the only two ways of pairing three elements of a magma with two operations. Less trivially, C3 = 5: ((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), and a(b(cd)). For an n-element set, there are nn2 magmas associated with that set, so for n = 0, 1, 2, 3, 4, ..., there are 1, 1, 16, 19683, 4294967296, ... (sequence A002489 in the OEIS) magmas. The corresponding numbers of magmas up to isomorphism are 1, 1, 10, 3330, 178981952, ... (sequence A001329 in the OEIS) and the numbers of magmas up to isomorphism and antiisomorphism are 1, 1, 7, 1734, 89521056, ... (sequence A001424 in the OEIS).
Free magma
A free magma M X {\displaystyle M_{X}} on a set X {\displaystyle X} is the "most general possible" magma generated by X {\displaystyle X} (i.e., there are no relations or axioms imposed on the generators; see free object). The binary operation on M X {\displaystyle M_{X}} is formed by wrapping each of the two operands in parentheses and juxtaposing them in the same order. For example
a ⋅ b = ( a ) ( b ) {\displaystyle a\cdot b=(a)(b)} ,
a ⋅ ( a ⋅ b ) = ( a ) ( ( a ) ( b ) ) {\displaystyle a\cdot (a\cdot b)=(a)((a)(b))} ,
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