MAGMA (ALGEBRA)

In mathematics, particularly in abstract algebra, a 'magma' (or 'groupoid') is a basic kind of algebraic structure. Specifically, a magma consists of a set ''M'' equipped with a single binary operation ''M'' × ''M'' → ''M''. A binary operation is closed by definition, but no other axioms are imposed on the operation.
The term ''magma'' for this kind of structure was introduced by Bourbaki. The term ''groupoid'' is an older, but still commonly used alternative which was introduced by Øystein Ore. However, ''groupoid'' also refers to an entirely different algebraic structure described at groupoid.

Contents

Types of magmas

Morphism of magmas

Combinatorics and parentheses

Free magma

More definitions

See also

References

Types of magmas

Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what axioms one might require of the operation.
Commonly studied types of magmas include

★ quasigroups—nonempty magmas where division is always possible;

★ loops—quasigroups with identity elements;

★ semigroups—magmas where the operation is associative;

★ monoids—semigroups with identity elements;

★ groups—monoids with inverse elements, or equivalently, associative loops (which are always quasigroups);

★ abelian groups—groups where the operation is commutative.
:

::''From magma to group, via two alternative paths. Key:''
::''M = magma, d = divisibility, a = associativity,''
::''Q = quasigroup, S = semigroup, e = identity.''
::''L = loop, i = inversibility, N = monoid, G = group''
::''Note that both divisibility and inversibility imply''
::''the existence of the cancellation property.''

Morphism of magmas

A morphism of magmas is a function $f:M\; o\; N$ mapping magma $M$ to magma $N$, that preserves the binary operation:
:$f(x\; ;$
★ _M ;y) = f(x) ;
★ _N; f(y)
where $$
★ _M and $$
★ _N denote the binary operation on $M$, respectively on $N$.

Combinatorics and parentheses

For the general, non-associative case, the magma operation may be repeatedly iterated. To denote pairings, parentheses are used. The resulting string consists of symbols denoting elements of the magma, and balanced sets of parenthesis. The set of all possible strings of balanced parenthesis is called the Dyck language. The total number of different ways of writing ''n'' applications of the magma operator is given by the Catalan number $C\_n$. Thus, for example, $C\_2=2$, which is just the statement that $(ab)c$ and $a(bc)$ are the only two ways of pairing three elements of a magma with two operations.
A shorthand is often used in order to avoid as much parenthesis as possible. This is accomplished by using juxtaposition in place of the operation. For example, if the magma operation is
★ , then ''xy''
★ ''z'' abbreviates (''x''
★ ''y'')
★ ''z''. Further abbreviations are possible by inserting spaces, for example by writing ''xy''
★ ''z''
★ ''wv'' in place of ((''x''
★ ''y'')
★ ''z'')
★ (''w''
★ ''v''). Of course, for more complex expressions the use of parenthesis turns out to be unavoidable. A way to avoid completely the use of parentheses is prefix notation, which is, however, counterintuitive.

Free magma

A 'free magma' $M\_X$ on a set ''X'' is the "most general possible" magma generated by the set ''X'' (that is there are no relations or axioms imposed on the generators; see free object). It can be described, in terms familiar in computer science, as the magma of binary trees with leaves labeled by elements of ''X''. The operation is that of joining trees at the root. It therefore has a foundational role in syntax.
A free magma has the freeness property such that, if $f:X\; o\; N$ is a function from the set ''X'' to any magma ''N'', then there is a unique extension of $f$ to a morphism of magmas $f^prime$
:$f^prime:M\_X\; o\; N$
''See also'': free semigroup, free group, Hall set

More definitions

A magma (''S'',
★ ) is called

★ ''unital'' if it has an identity element,

★ ''medial'' if it satisfies the identity ''xy''
★ ''uz'' = ''xu''
★ ''yz'' (i.e. (''x''
★ ''y'')
★ (''u''
★ ''z'') = (''x''
★ ''u'')
★ (''y''
★ ''z'') for all ''x'', ''y'', ''u'', ''z'' in ''S''),

★ ''left semimedial'' if it satisfies the identity ''xx''
★ ''yz'' = ''xy''
★ ''xz'',

★ ''right semimedial'' if it satisfies the identity ''yz''
★ ''xx'' = ''yx''
★ ''zx'',

★ ''semimedial'' if it is both left and right semimedial,

★ ''left distributive'' if it satisfies the identity ''x''
★ ''yz'' = ''xy''
★ ''xz'',

★ ''right distributive'' if it satisfies the identity ''yz''
★ ''x'' = ''yx''
★ ''zx'',

★ ''autodistributive'' if it is both left and right distributive,

★ ''commutative'' if it satisfies the identity ''xy'' = ''yx'',

★ ''idempotent'' if it satisfies the identity ''xx'' = ''x'',

★ ''unipotent'' if it satisfies the identity ''xx'' = ''yy'',

★ ''zeropotent'' if it satisfies the identity ''xx''
★ ''y'' = ''yy''
★ ''x'' = ''xx'',

★ ''alternative'' if it satisfies the identities ''xx''
★ ''y'' = ''x''
★ ''xy'' and ''x''
★ ''yy'' = ''xy''
★ ''y'',

★ ''power-associative'' if the submagma generated by any element is associative,

★ ''left-cancellative'' if for all ''x'', ''y'', and ''z'', ''xy'' = ''xz'' implies ''y'' = ''z''

★ ''right-cancellative'' if for all ''x'', ''y'', and ''z'', ''yx'' = ''zx'' implies ''y'' = ''z''

★ ''cancellative'' if it is both right-cancellative and left-cancellative

★ a ''semigroup'' if it satisfies the identity ''x''
★ ''yz'' = ''xy''
★ ''z'' (associativity),

★ a ''semigroup with left zeros'' if it satisfies the identity ''x'' = ''xy'',

★ a ''semigroup with right zeros'' if it satisfies the identity ''x'' = ''yx'',

★ a ''semigroup with zero multiplication'' if it satisfies the identity ''xy'' = ''uv'',

★ a ''left unar'' if it satisfies the identity ''xy'' = ''xz'',

★ a ''right unar'' if it satisfies the identity ''yx'' = ''zx'',

★ ''trimedial'' if any triple of its (not necessarily distinct) elements generates a medial submagma,

★ ''entropic'' if it is a homomorphic image of a medial cancellation magma.

See also

★ Magma category

★ Auto magma object

★ Universal algebra

★ Magma computer algebra system, named after the object of this article.

★ An example of a commutative non-associative magma

References

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