Magma (algebra) . Algebraic structure of the form (A, *) where A is a non-empty set, in which an internal binary operation * has been defined, or an internal composition or combination law [1]
Summary
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- 1 Introduction
- 2 Definitions
- 3 More Definitions
- 4 Non-associativity
- 1 More examples
- 5 See also
- 6 Sources
- 7 References
Introduction
In abstract algebra , a magma is an especially elementary type of algebraic structure .
Specifically, a magma consists of a set X with a single binary operation on it. It is usually (but not always) interpreted as a form of multiplication . No axiom is imposed on the operation when defining a magma. Magmas are not usually studied as such; instead, different types of magmas are considered, depending on which axioms are required for the operation.
Definitions
Commonly studied types of magmas include:
- quasigroups – non-empty magmas where divisionis always possible;
- loop (algebra) | loops – quasigroups with identity elements;
- semigroups – magmas where the operation is associativity;
- monoids – semigroups with identity element;
- groups- monoids with inverse elements , or equivalently, associative quasigroups (which are always loops);
- abelian group – groups where the operation is commutativity | commutative]].
The term “magma” was introduced by Bourbaki . Previously, the term “group” was common, and is still sometimes used. In this encyclopedia, however, we reserve “group” for a different algebraic concept, described in group.
There is what we can call a free magma on any set X . It can be described, in familiar terms in computer science, as the magma of the Trees (programming) with operation given by the juxtaposition (ordered) of the trees by the root. It therefore has a foundational role in syntax .
More Definitions
A magma is called
- medial (algebra) if it satisfies the identity xy.uz = xu.yz (ie (x * y) * (u * z) = (x * u) * (y * z)),
- semimedial if it satisfies the identity xx.yz = xy.xz,
- semimedial if it satisfies the identity yz.xx = yx.zx,
- semi-medial if it is both left and right semi-medial,
- autodistributive if it satisfies the identity x.yz = xy.xz,
- autodistributive if it satisfies the identity yz.x = yx.zx,
- self-distributive if it is both left and right distributive,
- commutative if it satisfies xy = yx,
- idempotent if it satisfies xx = x,
- unipotent if it satisfies xx = yy,
- powerful if it satisfies xx.y = yy.x = xx,
- alternative (algebra) if it satisfies xx.y = x.xy & x.yy = xy.y,
- a semigroup if it satisfies x.yz = xy.z (associativity),
- a semigroup with left zerosif it satisfies x = xy,
- a semigroup with right zerosif it satisfies x = yx,
- a semigroup with null multiplicationif it satisfies xy = uv,
- entropic (algebra) if it is universal algebra of a cancellativemagma .
No Associativity
A binary operation * on a set S that does not satisfy the associative law is called non-associative. Symbolically, for such an operation the order of evaluation matters. Subtraction and division of real numbers are well-known examples of non-associative operations:
In general, parentheses should be used to indicate the order of evaluation if a non-associative operation appears more than once in an expression. However, mathematicians agree on a particular order of evaluation for several common non-associative operations. This has the status of a convention, not a mathematical truth. A left-associable operation is conventionally evaluated from left to right, that is, while a right-associable operation is conventionally evaluated from right to left:
Left-associable and right-associable operations occur; the examples are given below.
More examples
Left-associable operations include the following.
- Subtraction and division of real numbers:
Right-associable operations include the following.
- Exponentiation of real numbers:
The reason exponentiation is right-associable is that a repeated left-associable operation of the exponent would be less useful. Multiple occurrences could be rewritten with multiplication: (xy) z = x (yz).
- The assignment operator in many programming languages is right-associable.
For example, in the C language
x = y = z; means x = (y = z); y no (x = y) = z;
In other words, the declaration would assign the value of za to both x and y.
Non-associative operations for which no conventional order of evaluation is defined include the following.
- Take the average of real numbers:
- Take the relative complement of sets:
