# Magma (math)

 touches the specialties mathematics includes as special cases Semigroup (axioms EA) Monoid (EAN) Group (EANI) Abelian Group (EANIK) commutative semigroup (EAK) commutative monoid (EANK) natural numbers ( N , +) Quasi-group (equations solvable)

In mathematics , a magma ( neuter , plural magma ) is an algebraic structure consisting of a set together with a two-digit internal link . It is also called a groupoid , sometimes binary or operational .

A generalization of the magma is the pseudo-magma , in which the link no longer has to be explained on the entire underlying set, i.e. it can be partial .

## Definitions

### magma

A magma is a pair consisting of a quantity (the carrier quantity ) and a two-digit internal link${\ displaystyle (M, *),}$ ${\ displaystyle M}$ ${\ displaystyle * \, \ colon \, M \ times M \ rightarrow M.}$ For , the connection of two elements , one also writes briefly . ${\ displaystyle a * b}$ ${\ displaystyle a, b \ in M}$ ${\ displaystyle from}$ The empty set can also be accepted as a carrier set ; the couple is a magma in a trivial way. ${\ displaystyle \ emptyset}$ ${\ displaystyle M}$ ${\ displaystyle (\ emptyset, \ emptyset)}$ If the connection is commutative , the magma is called commutative or Abelian; if it is associative , the magma is called associative or semigroup .

### Sub magma

• Be a magma. A magma is called a sub- magma of , if and , d. that is, the shortcut is the restriction of on .${\ displaystyle {\ boldsymbol {M}} = (M, *)}$ ${\ displaystyle {\ boldsymbol {U}} = (U, \ circ)}$ ${\ displaystyle {\ boldsymbol {M}}}$ ${\ displaystyle U \ subseteq M}$ ${\ displaystyle \ circ = * | _ {U \ times U}}$ ${\ displaystyle \ circ}$ ${\ displaystyle *}$ ${\ displaystyle U \ times U}$ Then and only then is a sub-magma of if and is closed with respect to , i.e. i.e., it applies ${\ displaystyle {\ boldsymbol {U}}}$ ${\ displaystyle {\ boldsymbol {M}}}$ ${\ displaystyle U \ subseteq M}$ ${\ displaystyle U}$ ${\ displaystyle *}$ ${\ displaystyle a \ circ b = a * b \ in U}$ for everyone .${\ displaystyle a, b \ in U}$ ${\ displaystyle {\ boldsymbol {M}}}$ is then also called the upper magma of . ${\ displaystyle {\ boldsymbol {U}}}$ • The average of sub-magmas is a sub-magma.
• Each subset of a magma is contained in a smallest sub- magma that contains. This sub-magma is called by created.${\ displaystyle U \ subset M}$ ${\ displaystyle U}$ ${\ displaystyle U}$ ## Examples

The following examples are magmas that are not semigroups:

• ${\ displaystyle (\ mathbb {Z}, -)}$ : the whole numbers with the subtraction
• ${\ displaystyle (\ mathbb {R} \ setminus \ {0 \}, /)}$ : the real numbers not equal to the division${\ displaystyle 0}$ • The natural numbers with the exponentiation , i.e. with the link${\ displaystyle a * b = a ^ {b}}$ • The real numbers with the formation of the arithmetic mean as a link
• All floating point representations ( floating point numbers ) for arbitrary bases, exponent and mantissa lengths with the multiplication are real, unitary, commutative magmas if you add the NaNs and ∞ (for the sake of seclusion) . Floating point multiplication is neither associative nor does it generally have a clear inverse, even if both are actually true in some cases.
• Finite magmas are often represented with link tables, e.g. B. for the magma :${\ displaystyle A = (\ {a, b, c, d \}, *)}$ ${\ displaystyle *}$ a b c d
a a b c a
b c d b c
c c a a c
d a d d b

The following examples are not magmas because the specified link is not defined for all possible values ​​(they are therefore pseudo-magmas):

• The natural numbers with subtraction.
• The real numbers with division.
• All floating point multiplications without NaNs or ∞.

Examples of sub-magmas are

• ${\ displaystyle (\ mathbb {Q} \ setminus \ {0 \}, /)}$ (the rational numbers not equal to the division ) is a sub-magma of (see above).${\ displaystyle 0}$ ${\ displaystyle (\ mathbb {R} \ setminus \ {0 \}, /)}$ • The magma with the following table is a sub-magma of the above-mentioned magma :${\ displaystyle B = (\ {a, c \}, \ circ)}$ ${\ displaystyle A = (\ {a, b, c, d \}, *)}$ ${\ displaystyle \ circ}$ a c
a a c
c c a

## properties

The basic set is closed by definition under an internal link . Otherwise a magma does not have to have any special properties. By adding further conditions, more specific structures are defined, which in turn are all magmas. Typical examples are:

• Semigroup : a magma whose link associatively is
• Monoid : a semigroup with a neutral element
• Quasigroup : a magma in which all equations of the form or clear after resolvable${\ displaystyle ax = b}$ ${\ displaystyle xa = b}$ ${\ displaystyle x}$ • Loop : a quasi-group with a neutral element
• Group : a monoid in which each member is an inverse has
• Abelian group : a group whose link commutative is
• Medial magma: a magma in which the equation applies to all elements${\ displaystyle (a \ star b) \ star (c \ star d) = (a \ star c) \ star (b \ star d)}$ ## Morphisms

Are two magmas, it means a picture a morphism, if all the following applies: . ${\ displaystyle (A, \ star), (B, \ circ)}$ ${\ displaystyle f \ colon A \ rightarrow B}$ ${\ displaystyle a, a '\ in A}$ ${\ displaystyle f (a \ star a ') = f (a) \ circ f (a')}$ • Is , that's called endomorphism .${\ displaystyle A = B}$ ${\ displaystyle f \ colon A \ rightarrow A}$ • If a morphism is bijective as a mapping, the reverse mapping is also a morphism. In this case it is called an isomorphism .${\ displaystyle f \ colon A \ rightarrow B}$ ${\ displaystyle f}$ ### Examples of morphisms

• The identity on a magma is always a morphism.
• The concatenation of morphisms is a morphism. The class of magmas together with the class of morphisms form a category .
• If a magma has only one element, there is exactly one morphism for each magma .${\ displaystyle E}$ ${\ displaystyle A}$ ${\ displaystyle f \ colon A \ rightarrow E}$ • In the above example there is only one morphism . If there is a morphism, it follows: ${\ displaystyle A \ rightarrow B}$ ${\ displaystyle f \ colon A \ rightarrow B}$ • ${\ displaystyle f (a) = f (a \ star a) = f (a) \ circ f (a)}$ . It is therefore only an option.${\ displaystyle f (a) = a}$ • Since there is a commutative magma, it follows . Suppose it is . In this case follows on the one hand . On the other hand follows . That is a contradiction. So is . There now follows .${\ displaystyle B}$ ${\ displaystyle f (b) = f (a \ star b) = f (b \ star a) = f (c)}$ ${\ displaystyle f (b) = c}$ ${\ displaystyle f (d) = f (b \ star b) = c \ circ c = a}$ ${\ displaystyle f (d) = f (d \ star b) = a \ circ c = c}$ ${\ displaystyle f (b) = f (c) = a}$ ${\ displaystyle f (d) = a}$ • The embedding of a magma in an upper magma is always a morphism.

## Free magma

For each non-empty set can be the free magma on defined as the set of all finite binary trees whose leaves with elements of are labeled. The product of two trees and is the tree whose root has the left subtree and the right subtree . The elements of the free magma can be written down using fully bracketed expressions. ${\ displaystyle X}$ ${\ displaystyle X}$ ${\ displaystyle X}$ ${\ displaystyle AB}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle A}$ ${\ displaystyle B}$ Let, for example, then the free magma contains, among other things, the elements that are different in pairs${\ displaystyle X = \ {a, b, c \}.}$ ${\ displaystyle X}$ ${\ displaystyle a, \, b, \, c, \, ab, \, ba, \, (ab) c, \, a (bc), \, (aa) (bb), \, (a (ab )) b, \, (ab) (ab).}$ ## Remarks

1. The term groupoid is also used for a mathematical structure in category theory, see groupoid (category theory) .
2. Nicolas Bourbaki: in "Elements of Mathematics Algebra I" in Chapter I "Algebraic Structures"

## literature

• Nicolas Bourbaki : Elements of Mathematics: Algebra I. Springer-Verlag, Berlin Heidelberg New York London Paris Tokyo, 1989, ISBN 978-3-540-64243-5 .
• Nicolas Bourbaki : Elements of Mathematics: Algebra I. Hermann, Paris / Addison-Wesley, Reading, Massachusetts, 1974.
• Lothar Gerritzen: Basic terms in algebra. Friedr. Vieweg & Sohn, Braunschweig / Wiesbaden 1994. ISBN 3-528-06519-2 .
• Th. Ihringer: General Algebra . Heldermann, Lemgo 2003; ISBN 3-88538-110-9 .
• Georges Papy: Simple connection structures: groupoids. Vandenhoeck & Ruprecht, Göttingen 1969.