Magma (algebra)
Prejudice to the special areas
- Mathematics Abstract Algebra
- Group Theory
- Category theory
Includes as special cases
- Semigroup ( axioms EA) Monoid ( EAN ) Group ( Eani ) Abelian group ( EANIK )
- Commutative monoid ( EANK ) integers (N, )
In mathematics, a magma is an algebraic structure consisting of a set together with a two-digit internal linking. It is also groupoid, or sometimes called Binar operative.
A generalization of the magma is the pseudo - magma, in which the link does not have to be declared around the underlying quantity, that may be partially.
Definition
A magma is a pair consisting of a set ( the underlying set ) and a two-digit internal link
For the combination of two elements, one also writes short.
The empty set can be approved as a support amount; it is a trivial way a magma.
If the link is commutative, it does magma commutative or abelian; it is associative, this means magma or associative semigroup.
Examples
The following examples are magmas that are not semigroups:
- : The integers with the subtraction
- : The real numbers not equal with the Division
- The natural numbers with exponentiation, ie with the link
- The real numbers with the formation of the arithmetic mean as a link
- All floating point ( float) to any base, exponent and mantissa lengths with the multiplication are real, unitary, commutative magmas when ( the seclusion because of) the NaNs and ∞, it is. Thus, the floating-point multiplication is associative, not yet they generally has a unique inverse, although actually applies both for some cases.
The following examples are not magmas because the specified link is not defined for all possible values (that is, they are pseudo - magmas ):
- The natural numbers with the subtraction.
- The real numbers with the division.
- All Gleitkommamultiplikationen without NaNs or ∞.
Properties
The basic set is closed under an inner join by definition. Otherwise, a magma must have no special properties. By adding further conditions specific structures are defined, all of which are in turn magmas. Typical examples are:
- Semigroup: a magma whose operation is associative
- Monoid: a semigroup with a neutral element
- Quasigroup: a magma in which all equations of the form or clearly resolvable by
- Loop: a quasigroup with a neutral element
- Group: a monoid, in which each element has an inverse
- Abelian group: a group whose link is commutative
Free Magma
For each non-empty set, one can define the free magma over the set of all finite binary trees whose leaves are labeled with elements of. The product of two trees and the tree whose root has the left subtree and the right subtree. Can you write down the elements of the free magma through fully parenthesized expressions.
For example, let Then contains the free magma on, among other things, the pairwise distinct elements