1. History and terminology

2. Definition

3. Morphism of magmas

4. Notation and combinatorics

5. Free magma

6. Types of magma

7. Classification by properties

8. Generalizations

9. See also

10. References

11. Further reading

In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation. The binary operation must be closed by definition but no other properties are imposed.

History and terminology

The term groupoid was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid (translated from the German Gruppoid). The term was then appropriated by B. A. Hausmann and Øystein Ore (1937)^[1] 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.^[2]

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]."^[3] It also appears in Bourbaki's Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970.^[4]

Definition

A magma is a set M matched with an operation, •, that sends any two elements {{nowrap|a, b ∈ M}} to another element, {{nowrap|a • b}}. The symbol, •, is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation {{nowrap|(M, •)}} must satisfy the following requirement (known as the magma or closure axiom):

For all a, b in M, the result of the operation {{nowrap|a • b}} is also in M.

And in mathematical notation:

.

If • is instead a partial operation, then {{math|S}} is called a partial magma^[5] or more often a partial groupoid.^[5][6]

Morphism of magmas

A morphism of magmas is a function, {{nowrap|f : M → N}}, mapping magma M to magma N, that preserves the binary operation:

f (x •_M y) = f(x) •_N f(y)

where •_M and •_N denote the binary operation on M and N respectively.

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:

{{math|1= (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, {{math|1=xy • z = (x • y) • z}}. For example, the above is abbreviated to the following expression, still containing parentheses:

{{math|1= (a • bc)d}}.

A way to avoid completely the use of parentheses is prefix notation, in which the same expression would be written {{math|1= ••a•bcd}}. Another way, familiar to programmers, is postfix notation (Reverse Polish notation), in which the same expression would be written {{math|1= 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 {{math|n}} applications of the magma operator is given by the Catalan number, {{math|C_n}}. Thus, for example, {{math|1=C₂ = 2}}, which is just the statement that {{math|(ab)c}} and {{math|a(bc)}} are the only two ways of pairing three elements of a magma with two operations. Less trivially, {{math|1=C₃ = 5}}: {{math|((ab)c)d}}, {{math|(a(bc))d}}, {{math|(ab)(cd)}}, {{math|a((bc)d)}}, and {{math|a(b(cd))}}.

There are magmas with elements so 1, 1, 16, 19683, 4294967296, ... {{OEIS|A002489}} magmas with 0, 1, 2, 3, 4, ... elements. The corresponding numbers of non-isomorphic magmas are 1, 1, 10, 3330, 178981952, ... {{OEIS|A001329}} and of simultaneously non-isomorphic and non-antiisomorphic magmas are 1, 1, 7, 1734, 89521056, ... {{OEIS|A001424}}.^[7]

Free magma

A free magma, M_X, on a set, X, is the "most general possible" magma generated by X (i.e., there are no relations or axioms imposed on the generators; see free object). It can be described as the set of non-associative words on X with parentheses retained.^[8]

It can also be viewed, in terms familiar in computer science, as the magma of binary trees with leaves labelled by elements of X. The operation is that of joining trees at the root. It therefore has a foundational role in syntax.

A free magma has the universal property such that, if {{nowrap|f : X → N}} is a function from X to any magma, N, then there is a unique extension of f to a morphism of magmas, f ′

f ′ : M_X → N.

Types of magma

Magmas are not often studied as such; instead there are several different kinds of magma, depending on what axioms the operation is required to satisfy. Commonly studied types of magma include:

//Quasigroup">Quasigroup: A magma where division is always possible

//Loop (algebra)">Loop: A quasigroup with an identity element

//Semigroup">Semigroup: A magma where the operation is associative

//Semilattice">Semilattice: A semigroup where the operation is commutative and idempotent

//Monoid">Monoid: A semigroup with an identity element

//Group (mathematics)">Group: A monoid with inverse elements, or equivalently, an associative loop, or a non-empty associative quasigroup

//Abelian group">Abelian group: A group where the operation is commutative

Note that each of divisibility and invertibility imply the cancellation property.

Classification by properties

A magma {{math|(S, •)}}, with {{math|x, y, u, z}} ∈ {{math|S}}, is called

//Medial magma">Medial: If it satisfies the identity, {{math|xy • uz ≡ xu • yz}}

Left semimedial

If it satisfies the identity, {{math|xx • yz ≡ xy • xz}}

Right semimedial

If it satisfies the identity, {{math|yz • xx ≡ yx • zx}}

Semimedial

If it is both left and right semimedial

Left distributive

If it satisfies the identity, {{math|x • yz ≡ xy • xz}}

Right distributive

If it satisfies the identity, {{math|yz • x ≡ yx • zx}}

Autodistributive

If it is both left and right distributive

//Commutative magma">Commutative: If it satisfies the identity, {{math|xy ≡ yx}}

//Idempotent">Idempotent: If it satisfies the identity, {{math|xx ≡ x}}

//Unipotent">Unipotent: If it satisfies the identity, {{math|xx ≡ yy}}

Zeropotent

If it satisfies the identities, {{math|xx • y ≡ xx ≡ y • xx}}^[9]

//Alternativity">Alternative: If it satisfies the identities {{math|xx • y ≡ x • xy}} and {{math|x • yy ≡ xy • y}}

//Power-associative">Power-associative: If the submagma generated by any element is associative

A

//semigroup">semigroup, or associative: If it satisfies the identity, {{math|x • yz ≡ xy • z}}

A left unar

If it satisfies the identity, {{math|xy ≡ xz}}

A right unar

If it satisfies the identity, {{math|yx ≡ zx}}

Semigroup with zero multiplication, or

//null semigroup">null semigroup: If it satisfies the identity, {{math|xy ≡ uv}}

{{anchor|unital}}Unital

If it has an identity element

Left-

//cancellative">cancellative: If, for all {{math|x, y}}, and, {{math|z}}, {{math|xy {{=}} xz}} implies {{math|y {{=}} z}}

Right-cancellative

If, for all {{math|x, y}}, and, {{math|z}}, {{math|yx {{=}} zx}} implies {{math|y {{=}} z}}

Cancellative

If it is both right-cancellative and left-cancellative

A

//Null semigroup#Left zero semigroup">semigroup with left zeros: If it is a semigroup and, for all {{math|x}}, the identity, {{math|x ≡ xy}}, holds

A

//Null semigroup#Right zero semigroup">semigroup with right zeros: If it is a semigroup and, for all {{math|x}}, the identity, {{math|x ≡ yx}}, holds

Trimedial

If any triple of (not necessarily distinct) elements generates a medial submagma

Entropic

If it is a homomorphic image of a medial cancellation magma.^[10]

Generalizations

See n-ary group.

See also

• Magma category
• Auto magma object
• Universal algebra
• Magma computer algebra system, named after the object of this article.
• Commutative non-associative magmas
• Algebraic structures whose axioms are all identities
• Groupoid algebra

References

• {{springer|id=m/m110040|author=M. Hazewinkel|title=Magma}}
• {{springer|id=Groupoid|author=M. Hazewinkel|title=Groupoid}}
• {{springer|id=f/f110190|author=M. Hazewinkel|title=Free magma}}
• {{mathworld|urlname=Groupoid|title=Groupoid}}

Further reading

• {{citation |first=Richard Hubert |last=Bruck |authorlink=Richard Hubert Bruck |title=A survey of binary systems |year=1971 |publisher=Springer |isbn=978-0-387-03497-3 |edition=3rd}}

• Bosna, KS
• Bosna (ridge)
• Bosnasaray
• Bosna (song)
• Bosna village
• Bosna (village)
• Bosngun
• Bosnia and Herzegovina at the 2016 European Athletics Championships
• Bosnia and Herzegovina at the 2016 Summer Paralympics
• Bosnia and Herzegovina at the 2016 Winter Youth Olympics
• Bosnia and Herzegovina at the 2017 Summer Universiade
• Bosnia and Herzegovina at the 2017 World Aquatics Championships
• Bosnia and Herzegovina at the 2017 World Championships in Athletics
• Bosnia and Herzegovina at the 2017 World Games
• Bosnia and Herzegovina at the 2018 European Athletics Championships
• Bosnia and Herzegovina at the 2018 European Championships
• Bosnia and Herzegovina at the 2018 Mediterranean Games
• Bosnia and Herzegovina at the 2018 Summer Youth Olympics
• Bosnia and Herzegovina at the 2018 Winter Olympics
• Bosnia and Herzegovina at the 2018 Winter Paralympics
• Bosnia and Herzegovina at the European Games
• Bosnia and Herzegovina at the European Youth Olympic Festival
• Bosnia and Herzegovina at the IAAF World Championships in Athletics
• Bosnia and Herzegovina at the Summer Olympics
• Bosnia and Herzegovina at the Winter Olympics