Magma (algebra)

A Magma is an algebraic structure of the form ${displaystyle (A,circledcirc)}$ with A is a set where an internal binary operation has been defined: ${displaystyle circledcirc}$.

${displaystyle {begin{array}{rccl}circledcirc: strangerAtimes A hypolongrightarrow &Alongrightarrow (a,b) strangerlongmapsto &c=acircledcirc bend{array}}}}}}$

This law of composition being an internal operation:

The term magma is due to the association of French mathematicians calling themselves Nicolas Bourbaki. For some time it competed, to reflect the same concept, with the word grupoid, which has other meanings in mathematics (see groupoid article), so its use as a synonym for magma is not advisable.

Definitions

From magma to group.

Commonly studied magma types include:

• quasigroups — non-empty magmas where division is always possible.
• loops — quasigroups with neutral elements.
• semigroups — magma where the operation is associative.
• monoids — semi-groups with neutral element.
• groups — monoids with symmetrical elements, or equivalently, associative groups (which are always loops).
• Abelian groups — groups where the operation is commutative.

The term "magma" It was introduced by Bourbaki. The term "groupoid" was previously used, and is still sometimes used. In this encyclopedia, however, we reserve the term groupoid for a different algebraic concept.

There exists what we can call a free magma over any set X and that can be described in terms familiar to computer science as the magma of binary trees with operation given by the (ordered) juxtaposition of the trees by the root. It therefore has a foundational role in syntax.

More Definitions

A magma is called:

• medial if it satisfies xy.uz=xu.yz (i.e. (x*y)*(u*z)=(x*u)*(y*z)
• semimedial left if it satisfies xx.yz=xy.xz,
• semimedial right if it meets the identity yz.xx=yx.zx,
• semimedial if it is, at the same time, left and right halfway,
• left distribution if it satisfies x.yz=xy.xz,
• right distribution if it meets yz.x=yx.zx identity,
• self-distributive if it is, at the same time, left and right distribution,
• commutative if satisfying xy=yx,
• idempotent if you satisfy xx=x,
• unipotent if you satisfy xx=yyy,
• zeropotent if it satisfies xx.y=yy.x=xx,
• alternative if xx.y=x.xy > x.yy=xy.y,
• a semi-group if satisfying x.yz=xy.z (associative),
• One semi-group with left zeros or left cancelling elements if it satisfies x=xy,
• One semi-group with zero rights or cancellation of rights if it satisfies x=yx,
• One semi-group with null multiplication if it satisfies xy=uv,
• entropic if it is homomorphic image of an unpaid magma.

Non-associativity

A binary operation * on a set S that does not satisfy the associative law is called non-associative. symbolically,

${displaystyle (x*y)*zneq x*(y*z)qquad {mbox{for some}x,y,zin S}$

for such an operation the order of evaluation matters. Subtraction and division of real numbers are well-known examples of non-associative operations:

${displaystyle left.{begin{matrix}(x-y)-zneq x-(y-z)quad \x/y)/zneq x/(y/z)qquad qquad end{matrix}}{right}{mbox{for some}x,y,zin mathbb {$

In general, parentheses should be used to indicate the order of evaluation if a non-associative operation occurs 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,

${displaystyle left.{begin{matrix}x*y*z=(x*y)*zqquad qquad quad quad ,w*x*y*z=(w*x)}{quad \{mbox{etc}{qquad qquad qquad xquad xquad qquad qquad$

while a right-associable operation is conventionally evaluated from right to left:

${displaystyle left.{begin{matrix}x*y*z=x*(y*z)qquad quad quad quad ,w*x*y*z=w*(y*z)}{quad \{mbox{etc}{qquad qquad qquad xquad xquad qquad qquad$

Left-binding and right-binding operations occur; examples are given below.

More examples

Left-associable operations include the following.

• Subtraction and division of real numbers:

${displaystyle x-y-z=(x-y)-zqquad {mbox{for all}x,y,zin mathbb {R};}$

${displaystyle x/y/z=(x/y)/zqquad qquad quad {mbox{for all }x,y,zin mathbb {R} {mbox{ with }yneq 0,zneq 0. !$

Right-binding operations include the following.

• Exponentiation of real numbers:

${displaystyle x^{y^{z}}=x^{(y^{z}}}}}.$

The reason exponentiation is right-associable is that a repeated left-associative operation on the exponent would be less useful. Multiple occurrences could be rewritten with multiplication:

${displaystyle (x^{y})^{z}=x^{(yz)}}.$

• The assignment operator in many programming languages is right-associable.

For example, in the C language

x = y = z;meansx = (y = z);and no(x = y) = z;

That is, the declaration would assign the value of z to both x and y.

Non-associative operations for which no conventional evaluation order is defined include the following.

• Take the average real numbers:

${displaystyle {(x+y)/2+z over 2}neq {x+(y+z)/2 over 2}neq {x+y+z over 3}qquad {mbox{for some}}x,y,zin mathbb {R}. !$

• Take the relative complement of sets:

${displaystyle (Abackslash B)backslash Cneq Abackslash (Bbackslash C)qquad {mbox{for some sets}A,B,C.}$