Magma (math)
touches the specialties 
includes as special cases 

In mathematics , a magma ( neuter , plural magma ) is an algebraic structure consisting of a set together with a twodigit internal link . It is also called a groupoid , sometimes binary or operational .
A generalization of the magma is the pseudomagma , 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 twodigit internal link
For , the connection of two elements , one also writes briefly .
The empty set can also be accepted as a carrier set ; the couple is a magma in a trivial way.
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 .
Then and only then is a submagma of if and is closed with respect to , i.e. i.e., it applies
 for everyone .
is then also called the upper magma of .
 The average of submagmas is a submagma.
 Each subset of a magma is contained in a smallest sub magma that contains. This submagma is called by created.
Examples
The following examples are magmas that are not semigroups:
 : the whole numbers with the subtraction
 : the real numbers not equal to the division
 The natural numbers with the exponentiation , i.e. with the link
 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 :
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 pseudomagmas):
 The natural numbers with subtraction.
 The real numbers with division.
 All floating point multiplications without NaNs or ∞.
Examples of submagmas are
 (the rational numbers not equal to the division ) is a submagma of (see above).
 The magma with the following table is a submagma of the abovementioned magma :
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
 Loop : a quasigroup 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
Morphisms
Are two magmas, it means a picture a morphism, if all the following applies: .
 Is , that's called endomorphism .
 If a morphism is bijective as a mapping, the reverse mapping is also a morphism. In this case it is called an isomorphism .
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 .
 In the above example there is only one morphism . If there is a morphism, it follows:
 . It is therefore only an option.
 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 .
 The embedding of a magma in an upper magma is always a morphism.
Free magma
For each nonempty 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.
Let, for example, then the free magma contains, among other things, the elements that are different in pairs
Remarks
 ↑ The term groupoid is also used for a mathematical structure in category theory, see groupoid (category theory) .
 ↑ Nicolas Bourbaki: in "Elements of Mathematics Algebra I" in Chapter I "Algebraic Structures"
literature
 Nicolas Bourbaki : Elements of Mathematics: Algebra I. SpringerVerlag, Berlin Heidelberg New York London Paris Tokyo, 1989, ISBN 9783540642435 .
 Nicolas Bourbaki : Elements of Mathematics: Algebra I. Hermann, Paris / AddisonWesley, Reading, Massachusetts, 1974.
 Lothar Gerritzen: Basic terms in algebra. Friedr. Vieweg & Sohn, Braunschweig / Wiesbaden 1994. ISBN 3528065192 .
 Th. Ihringer: General Algebra . Heldermann, Lemgo 2003; ISBN 3885381109 .
 Georges Papy: Simple connection structures: groupoids. Vandenhoeck & Ruprecht, Göttingen 1969.